# A novel distribution with upside down bathtub shape hazard rate: properties,
estimation and applications

Tuhin Subhra Mahato, Subhankar Dutta, and Suchandan Kayal Email address :
<EMAIL_ADDRESS>author :
<EMAIL_ADDRESS>address<EMAIL_ADDRESS><EMAIL_ADDRESS>
Department of Mathematics, National Institute of Technology Rourkela,
Rourkela-769008, Odisha, India

Abstract

In this communication, we introduce a new statistical model and study its
various mathematical properties. The expressions for hazard rate, reversed
hazard rate, and odd functions are provided. We explore the asymptotic
behaviours of the density and hazard functions of the newly proposed model.
Further, moments, median, quantile, and mode are obtained. The cumulative
distribution and density functions of the general $k$th order statistic are
provided. Sufficient conditions, under which the likelihood ratio order
between two inverse generalized linear failure rate (IGLFR) distributed random
variables holds, are derived. In addition to these results, we introduce
several estimates for the parameters of IGLFR distribution. The maximum
likelihood and maximum product spacings estimates are proposed. Bayes
estimates are calculated with respect to the squared error loss function.
Further, asymptotic confidence and Bayesian credible intervals are obtained.
To observe performance of the proposed estimates, we carry out a Monte Carlo
simulation using $R$ software. Finally, two real life data sets are considered
for the purpose of illustration.

Keywords: IGLFR distribution; Order statistics; Stochastic orderings; Maximum
likelihood estimate; Bayes estimate; Mean squared error.

##  1 Introduction

The statistical literature on lifetime models has been enriched with several
continuous distributions and it is still developing. Many distributions, which
have been useful in various real-life situations are introduced in past two
decades. For example, Abouammoh and Alshingiti (2009) proposed a new
distribution named as generalized inverted exponential (GIE) distribution and
estimated its reliability parameter using various estimation techniques.
Further, De Gusmao et al. (2011) introduced and studied a three-parameter
generalized inverse Weibull distribution. The authors observed that the
failure rate of this distribution is decreasing and unimodal. Tahir et al.
(2018) proposed an inverted model called the inverted Nadarajah-Haghighi
distribution. Some properties of this distribution and estimates of model
parameters have been explored by the authors. Basheer (2019) introduced a new
generalized alpha power inverse Weibull distribution. The author has used
alpha power transformation method in order to yield this distribution. Various
characterization results and statistical properties of the alpha power inverse
Weibull distribution have been proposed. Finally, the author employed several
estimation techniques for estimation of the unknown parameters of alpha power
inverse Weibull distribution. Hassan and Abd-Allah (2019) proposed a new
three-parameter lifetime distribution named as the inverse power Lomax
distribution. This distribution can be obtained using inverse transformation
from power Lomax distribution. Rao and Mbwambo (2019) introduced a
generalization of the inverse Rayleigh distribution known as exponentiated
inverse Rayleigh distribution. Some statistical properties of the newly
proposed distribution have been investigated. In particular, the mode,
quantiles, moments, reliability, and hazard functions have been obtained.
Various estimates for the model parameters have been proposed.

In statistical literature, inverted distributions have been generated by using
the inverse transformation to display different properties of the probability
density function and hazard rate functions and also allowed to explore more
applicability in real-life phenomenon. The inverted distributions are applied
in several areas related to econometrics, biological sciences, medical
research, and survival analysis. To know more details about inverted
distributions, one may be referred to Sheikh et al. (1987) and Lehmann and
Shaffer (1998). There are some well-known inverted distributions and these are
listed below:

  * •

Inverted Weibull distribution with cumulative distribution function (CDF):
$F_{IW}(x)=\exp(-\alpha x^{-\theta}),~{}x,\alpha,\theta>0$. (For instance, see
Keller and ARR (1982))

  * •

Inverted Rayleigh distribution with CDF: $F_{IR}(x)=\exp(-\alpha
x^{-2}),~{}x,\alpha>0$. (For instance, see Voda (1972))

  * •

Inverted Gamma distribution with CDF: $F_{IG}(x)=\Gamma(\alpha,\theta
x^{-1})\Gamma(\alpha),~{}x,\alpha,\theta>0$. (For instance, see Lin et al.
(1989))

  * •

Inverted power Lindley distribution with CDF: $F_{IPL}(x)=(1+\frac{\lambda
x^{-\alpha}}{1+\lambda})exp(-\lambda x^{-\alpha}),~{}x,\alpha,\lambda>0$. (For
instance, see Barco et al. (2017))

The purpose of this paper is two folds. First, we introduce a new distribution
and study its various statistical properties. Secondly, estimators for the
unknown parameters of IGLFR distribution have been obtained from both
frequentist and Bayesian approaches and establish a recommendation for
choosing the most appropriate estimator, which we believe would be of
significant interest to applied statisticians. The maximum likelihood
estimates (MLEs) and maximum product spacing estimates (MPSEs) have been
considered as frequentist estimates. Furthermore, the Bayes estimates of the
unknown parameters have been computed under the assumptions of independent
gamma priors based on squared error loss function (SELF). A Monte Carlo
simulation study has been carried out to compare the performance of the
estimates. Finally, two real life data sets have been analyzed for
illustrative purpose.

The article is organized as follows. In Section $2$, we first provide
construction of the newly proposed IGLFR distribution. Some special cases are
presented. Then, several statistical properties are explored. In Section $3$,
we present estimation techniques for the computation of the unknown
parameters. In Section $4$, simulation study has been carried out to see
comparative performance of the proposed estimates. Section $5$ presents real
life data analysis. Finally, some concluding remarks are added in Section $6$.

##  2 The proposed IGLFR distribution

In this section, we introduce a new continuous distribution referred to as the
IGLFR distribution and study its properties. The new three-parameter
distribution is derived using an inverse transformation to the generalized
linear failure rate (GLFR) distribution proposed by Sarhan and Kundu (2009).
The cumulative distribution function (CDF) of a GLFR distributed random
variable $Y$ is given by

$\displaystyle G(y;\alpha,\beta,\theta)=\left[1-\exp\left\\{-\left(\alpha
y+\frac{\beta
y^{2}}{2}\right)\right\\}\right]^{\theta},~{}y>0,~{}\alpha,\beta,\theta>0.$
(2.1)

Note that in (2.1), $\theta$ is known as the shape parameter. The probability
density function (PDF) of GLFR distribution is

$\displaystyle g(y;\alpha,\beta,\theta)=\theta(\alpha+\beta
y)\left[1-\exp\left\\{-\left(\alpha y+\frac{\beta
y^{2}}{2}\right)\right\\}\right]^{\theta-1}\exp\left\\{-\left(\alpha
y+\frac{\beta y^{2}}{2}\right)\right\\},~{}y>0.$ (2.2)

###  2.1 Definition

Let $X=\frac{1}{Y}$ be a transformation proposed by Tahir et al. (2018) and
here $Y$ follows GLFR distribution with CDF (2.1). Then, the CDF of $X$ is

$\displaystyle F(x)$ $\displaystyle=$ $\displaystyle P(X\leq
x)=P\left(\frac{1}{Y}\leq
x\right)=P\left(Y\geq\frac{1}{x}\right)=1-P\left(Y<\frac{1}{x}\right)$ (2.3)
$\displaystyle=$ $\displaystyle
1-\left[1-\exp\left\\{-\left(\frac{\alpha}{x}+\frac{\beta}{2x^{2}}\right)\right\\}\right]^{\theta},~{}~{}x>0.$

Thus, the PDF of $X$ can be easily obtained from (2.3), and is given by

$\displaystyle
f(x;\alpha,\beta,\theta)=\theta\left(\frac{\alpha}{x^{2}}+\frac{\beta}{x^{3}}\right)\exp\left\\{-\left(\frac{\alpha}{x}+\frac{\beta}{2x^{2}}\right)\right\\}\left[1-\exp\left\\{-\left(\frac{\alpha}{x}+\frac{\beta}{2x^{2}}\right)\right\\}\right]^{\theta-1},$
(2.4)

where $x>0$ and $\alpha,~{}\beta,~{}\theta>0.$ We call the model associated
with CDF and PDF given by (2.3) and (2.4), respectively as the IGLFR
distribution, and denote $X\sim IGLFR(\alpha,\beta,\theta)$. The PDF and CDF
of $IGLFR(\alpha,\beta,\theta)$ distribution are plotted in Figure $1$ and
Figure $2$, respectively for various parameter values.

Figure 1: PDFs of IGLFR($\alpha$,$\beta$,$\theta$) distribution, for different
values of parameters.

Figure 2: CDFs of IGLFR($\alpha$,$\beta$,$\theta$) distribution, for different
values of parameters.

Some special cases:

  * •

Let $\alpha=0$, $\beta=\frac{2}{\lambda^{2}},$ and $\theta=\alpha>0.$ Then,
the IGLFR distribution with PDF given by (2.4) reduces to the PDF of
generalized inverted Rayleigh distribution (see Bakoban and Abubaker (2015)) .

  * •

Consider $\alpha=\lambda>0$, $\beta=0$, and $\theta=\alpha>0.$ Using these
particular values of the parameters, the PDF of IGLFR distribution becomes the
PDF of generalized inverted exponential distribution (see Krishna and Kumar
(2013)).

  * •

Let $\alpha=0,$ $\beta=2\sigma^{2}$, and $\theta=\alpha>0$. Then, the IGLFR
distribution reduces to the exponentiated inverse Rayleigh distribution (see
Rao and Mbwambo (2019)).

  * •

For $\alpha=0,$ $\beta=\lambda$ and $\theta=1$, the IGLFR distribution reduces
to inverse Rayleigh distribution (see Malik and Ahmad (2018)).

  * •

Let $\alpha=\alpha>0,$ $\beta=\theta>0$, and $\theta=1$. Then, the IGLFR
distribution becomes modified inverse Rayleigh distribution (see Khan (2014)).

  * •

Consider $\alpha=\frac{1}{\beta}$, $\beta=0$, and $\theta=1.$ Then, the IGLFR
distribution reduces to inverted exponential distribution (see Dey (2007)).

######  Remark 2.1.

For a positive integer $\theta$, the CDF of $IGLFR(\alpha,\beta,\theta)$
distribution represents the CDF of the minimum of a simple random sample of
size $\theta$ from the modified inverse Rayleigh distribution (MIRD) with
survival function given by (see Khan (2014))

$\displaystyle\bar{K}(x)=1-\exp\left\\{-\left(\frac{\alpha}{x}+\frac{\beta}{2x^{2}}\right)\right\\},~{}~{}x>0,~{}\alpha,\beta>0.$

Thus, we can say that the $IGLFR(\alpha,\beta,\theta)$ distribution provides
the CDF of a series system when each component of the system has the MIRD.

######  Remark 2.2.

Let $T=X^{\frac{1}{\lambda}}$, where $\lambda>0$ and $X\sim
IGLFR(\alpha,\beta,\theta)$. Then, the CDF of $T$ is obtained as

$K_{T}(x)=1-\left[1-\exp\left\\{-\left(\frac{\alpha}{x^{\lambda}}+\frac{\beta}{2x^{2\lambda}}\right)\right\\}\right]^{\theta},~{}~{}x>0.$

To the best of our knowledge, the CDF of $T$ does not belong to a known family
of distributions. However, if $\beta=0,$ then the CDF of $T$ reduces to CDF of
the well-known inverse exponentiated Weibull distribution.

###  2.2 Hazard rate, reversed hazard rate and odd functions

In survival analysis, the hazard function plays a central role. Suppose at
time $t$, an item is working. Under this prior information, the hazard
function represents the failure probability of the item in next $dt$ units of
time. The hazard rate function uniquely determines a CDF. The hazard function
of IGLFR distribution is obtained as

$\displaystyle
h(x;\alpha,\beta,\theta)=\frac{\theta~{}{(\frac{\alpha}{x^{2}}+\frac{\beta}{x^{3}})}~{}{e^{-(\frac{\alpha}{x}+\frac{\beta}{2x^{2}})}}}{1-e^{-(\frac{\alpha}{x}+\frac{\beta}{2x^{2}})}}=\theta~{}h(x;\alpha,\beta,1).$
(2.5)

In the context of reliability or maintenance applications, the hazard rate
function describes how likely is the occurence of a failure in the immediate
future. From (2.5), it is clear that the IGLFR distribution is a proportional
hazard family. The plots of $h(x;\alpha,\beta,\theta)$ for different choices
of the parameters are depicted in Figure $3$.

Figure 3: Hazard rate functions of IGLFR($\alpha$,$\beta$,$\theta$), for
different values of the parameters.

As a dual of hazard function, the reversed hazard rate function characterizes
the probability of an immediate past failure, under the information that a
failure has already occured. The reversed hazard rate function of a random
variable $X$ following IGLFR($\alpha$,$\beta$,$\theta$) is given by

$\displaystyle\text{r}(x;\alpha,\beta,\theta)=\frac{\theta~{}{(\frac{\alpha}{x^{2}}+\frac{\beta}{x^{3}})}~{}{e^{-(\frac{\alpha}{x}+\frac{\beta}{2x^{2}})}}~{}\Big{[}1-e^{-(\frac{\alpha}{x}+\frac{\beta}{2x^{2}})}\Big{]}^{\theta-1}}{1-\Big{(}1-e^{-(\frac{\alpha}{x}+\frac{\beta}{2x^{2}})}\Big{)}^{\theta}},~{}x>0.$
(2.6)

The reversed hazard rate function of the IGLFR($\alpha$,$\beta$,$\theta$)
distribution is plotted in Figure $4$, for different choices of the
parameters.

Figure 4: Reversed hazard rate functions of IGLFR($\alpha$,$\beta$,$\theta$).

The odd function is

$\displaystyle\text{O}(x;\alpha,\beta,\theta)=\frac{1}{\Big{[}1-e^{-(\frac{\alpha}{x}+\frac{\beta}{2x^{2}})}\Big{]}^{\theta}}-1,~{}x>0.$
(2.7)

###  2.3 Asymptotic behaviour of the PDF and hazard function

Here, we will study behaviour of the PDF and hazard function of the newly
proposed model when $x$ tends to $0$ and $\infty.$ After some calculations, we
obtain

$\displaystyle\lim_{x\rightarrow 0}f(x)$ $\displaystyle=$
$\displaystyle\lim_{x\rightarrow
0}\left[\theta\left(\frac{\alpha}{x^{2}}+\frac{\beta}{x^{3}}\right)\exp\left\\{-\left(\frac{\alpha}{x}+\frac{\beta}{2x^{2}}\right)\right\\}\left[1-\exp\left\\{-\left(\frac{\alpha}{x}+\frac{\beta}{2x^{2}}\right)\right\\}\right]^{\theta-1}\right]$
(2.8) $\displaystyle=$ $\displaystyle 0$

and

$\displaystyle\lim_{x\rightarrow\infty}f(x)$ $\displaystyle=$
$\displaystyle\lim_{x\rightarrow
0}\left[\theta\left(\frac{\alpha}{x^{2}}+\frac{\beta}{x^{3}}\right)\exp\left\\{-\left(\frac{\alpha}{x}+\frac{\beta}{2x^{2}}\right)\right\\}\left[1-\exp\left\\{-\left(\frac{\alpha}{x}+\frac{\beta}{2x^{2}}\right)\right\\}\right]^{\theta-1}\right]$
(2.9) $\displaystyle=$ $\displaystyle 0.$

These results show that the proposed model is non-monotonic and has atleast
one mode. To verify this, we use graphical approach (see Figure $1$), which
indicates that the proposed model has only one mode, that is, the IGLFR
distribution is unimodal. Further,

$\displaystyle\lim_{x\rightarrow 0}\text{h}(x;\alpha,\beta,\theta)$
$\displaystyle=$ $\displaystyle\lim_{x\rightarrow
0}\left[\frac{\theta~{}{(\frac{\alpha}{x^{2}}+\frac{\beta}{x^{3}})}~{}{e^{-(\frac{\alpha}{x}+\frac{\beta}{2x^{2}})}}}{1-e^{-(\frac{\alpha}{x}+\frac{\beta}{2x^{2}})}}\right]$
(2.10) $\displaystyle=$ $\displaystyle 0$

and

$\displaystyle\lim_{x\rightarrow\infty}\text{h}(x;\alpha,\beta,\theta)$
$\displaystyle=$
$\displaystyle\lim_{x\rightarrow\infty}\left[\frac{\theta~{}{(\frac{\alpha}{x^{2}}+\frac{\beta}{x^{3}})}~{}{e^{-(\frac{\alpha}{x}+\frac{\beta}{2x^{2}})}}}{1-e^{-(\frac{\alpha}{x}+\frac{\beta}{2x^{2}})}}\right]$
(2.11) $\displaystyle=$ $\displaystyle 0.$

From the above limits, similar observation for the hazard function can be
noticed, that is, it has upside down bathtub shape. It is worth mentioning
that the GLFR model has increasing, decreasing and bathtub shaped hazard
functions (see Sarhan and Kundu (2009)). However, the newly proposed model has
upside down bathtub shape hazard function. There are various real life
examples, where distributions with upside down bathtub shape hazard function
is useful for analyse. For example, in biology, the upside down bathtub-shaped
hazard rate model can be used in the course of a disease whose mortality
reaches a peak after some time, and then decreases gradually.

###  2.4 Moments, median, quantile and mode

In this subsection, we will discuss about moments, median, quantile and mode
of the newly proposed IGLFR distribution. As expected, the mean and other
moments are difficult to derive in closed form. The $r$th order moment of the
IGLFR distribution is given by

$\displaystyle\mu_{r}^{\prime}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}x^{r}f(x)dx$ (2.12) $\displaystyle=$
$\displaystyle\int_{0}^{\infty}x^{r}\theta\left(\frac{\alpha}{x^{2}}+\frac{\beta}{x^{3}}\right)\exp\left\\{-\left(\frac{\alpha}{x}+\frac{\beta}{2x^{2}}\right)\right\\}\left[1-\exp\left\\{-\left(\frac{\alpha}{x}+\frac{\beta}{2x^{2}}\right)\right\\}\right]^{\theta-1}dx$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}x^{r-3}\theta\left(\alpha
x+\beta\right)\exp\left\\{-\left(\frac{\alpha}{x}+\frac{\beta}{2x^{2}}\right)\right\\}\left[1-\exp\left\\{-\left(\frac{\alpha}{x}+\frac{\beta}{2x^{2}}\right)\right\\}\right]^{\theta-1}dx.$

Further, we know that

$\left[1-\exp\left\\{-\left(\frac{\alpha}{x}+\frac{\beta}{2x^{2}}\right)\right\\}\right]^{\theta-1}=\sum_{i=0}^{\infty}(-1)^{i}{\theta-1\choose
i}\exp\left\\{-i\left(\frac{\alpha}{x}+\frac{\beta}{2x^{2}}\right)\right\\}.$

Substituting this in (2.12), we obtain

$\displaystyle\mu_{r}^{\prime}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}x^{r-3}\theta\left(\alpha
x+\beta\right)\sum_{i=0}^{\infty}(-1)^{i}{\theta-1\choose
i}\exp\left\\{-(i+1)\left(\frac{\alpha}{x}+\frac{\beta}{2x^{2}}\right)\right\\}dx$
(2.13) $\displaystyle=$
$\displaystyle\theta\sum_{i=0}^{\infty}(-1)^{i}{\theta-1\choose
i}\int_{0}^{\infty}x^{r-3}\left(\alpha
x+\beta\right)\exp\left\\{-(i+1)\left(\frac{\alpha}{x}+\frac{\beta}{2x^{2}}\right)\right\\}dx.$

Let the median of $X$ be M. Then, M must satisfy the following equation:

$\int_{0}^{\text{M}}f(x;\alpha,\beta,\theta)~{}dx=\int_{\text{M}}^{\infty}f(x;\alpha,\beta,\theta)~{}dx.$
(2.14)

Substituting the PDF given by (2.4) in the last equation, and then after some
calculations, we obtain

$\text{Median
}=\frac{-\alpha-\sqrt{\alpha^{2}-2\beta\log\bigg{[}1-\Big{(}\frac{1}{2}\Big{)}^{\frac{1}{\theta}}\bigg{]}}}{2\log\bigg{[}1-\Big{(}\frac{1}{2}\Big{)}^{\frac{1}{\theta}}\bigg{]}}.$
(2.15)

Suppose U $\thicksim$ Uniform (0,1), then to generate quantile from IGLFR
distribution, we invert the CDF of IGLFR(x;$\alpha,\beta,\theta$). Let
$x=F^{-1}(q)$ be quantile of the IGLFR(x;$\alpha,\beta,\theta$) then from the
equation $F(x;\alpha,\beta,\theta)=q$, the $q$th quantile is turned out as

$\ \
x_{q}=\frac{-\alpha-\sqrt{\alpha^{2}-2\beta\log\bigg{[}1-\big{(}1-q\big{)}^{\frac{1}{\theta}}\bigg{]}}}{2\log\bigg{[}1-\big{(}1-q\big{)}^{\frac{1}{\theta}}\bigg{]}},\
\text{since}\ 0<x<1.$ (2.16)

There are different methods which are used to find skewness and kurtosis in a
certain distribution. Quantile based measures, such as Bowley skewness [Kenney
(1939)] and Moors kurtosis [Moors (1988)], can quantify asymmetry and the
peakedness of a given distribution. Based on the quantiles, the coefficients
of skewness and kurtosis are provided by

$\displaystyle\text{Skewness}=$
$\displaystyle\frac{(Q_{3}-Q_{2})-(Q_{2}-Q_{1})}{(Q_{3}-Q_{2})+(Q_{2}-Q_{1})}$
$\displaystyle\text{Kurtosis}=$
$\displaystyle\frac{(E_{7}-E_{5})-(E_{3}-E_{1})}{(E_{6}-E_{2})}.$

where $Q_{i}$ is the $i^{th}$ quartile, $Q_{i}=F^{-1}(\frac{i}{4})$ and
$E_{i}$ is the $i^{th}$ octile, $E_{i}=F^{-1}(\frac{i}{8})$.

The mode of the distribution can be obtained using the following approach:

$\displaystyle f^{\prime}(x)$ $\displaystyle=0,$ $\displaystyle
f^{\prime\prime}(x)$ $\displaystyle<0.$

The first order derivative of the PDF $f(x)$ given by (2.4) with respect to
$x$ is given by

$\displaystyle\theta~{}{e^{-(\frac{\alpha}{x}+\frac{\beta}{2x^{2}})}}~{}\Big{[}1-e^{-(\frac{\alpha}{x}+\frac{\beta}{2x^{2}})}\Big{]}^{\theta-2}~{}\Bigg{[}\bigg{(}\frac{-2\alpha
x^{3}-3\beta x^{2}+\alpha^{2}x^{2}+\beta^{2}+2\alpha\beta x}{x^{6}}\bigg{)}$
(2.17)
$\displaystyle+e^{-(\frac{\alpha}{x}+\frac{\beta}{2x^{2}})}\bigg{(}\frac{2\alpha
x^{3}+3\beta x^{2}-\theta(\beta^{2}+2\alpha\beta
x+\alpha^{2}x^{2})}{x^{6}}\bigg{)}\Bigg{]}$ $\displaystyle=0.$

The root of (2.17), which satisfies $f^{\prime\prime}(x)<0$ is known as the
mode of the IGLFR distribution. We note that it is difficult to get the roots
of (2.17) in explicit form. Thus, for some particular choices of the
parameters, the mode can be obtained by adopting some numerical techniques.

###  2.5 Order statistics

In this subsection, we derive explicit expression for the PDF of the $k$th
order statistic $X_{k:n}$ in a random sample of size $n$ from the IGLFR
distribution. Note that the $k$th order statistic has an important
interpretation in reliability theory. The lifetime of a $(n-k+1)$-out-of-$n$
system is denoted by $X_{k:n}$. Using PDF and CDF of the IGLFR distribution,
the CDF and PDF of the $k$th order statistic $X_{k:n}$ are respectively given
by

$\displaystyle F_{\text{k}:n}(x)$ $\displaystyle=$
$\displaystyle\sum_{i=k}^{n}{n\choose
i}\Bigg{\\{}1-\Big{[}\gamma(\alpha,\beta,x)\Big{]}^{\theta}\Bigg{\\}}^{i}\Big{[}\gamma(\alpha,\beta,x)\Big{]}^{(n-i)\theta},$
(2.18) $\displaystyle f_{\text{k}:n}(x)$ $\displaystyle=$
$\displaystyle\frac{n!}{(\text{k}-1)!(\text{n}-\text{k})!}~{}\theta~{}{(\frac{\alpha}{x^{2}}+\frac{\beta}{x^{3}})}~{}{[1-\gamma(\alpha,\beta,x)]}~{}\Big{[}\gamma(\alpha,\beta,x)\Big{]}^{\theta(n-\text{k}+1)-1}$
(2.19)
$\displaystyle\times\Bigg{\\{}1-\Big{[}\gamma(\alpha,\beta,x)\Big{]}^{\theta}\Bigg{\\}}^{(\text{k}-1)}.$

Denote
$\gamma(\alpha,\beta,x)=1-\exp\\{-(\frac{\alpha}{x}+\frac{\beta}{2x^{2}})\\}.$
By substituting $k=1$ and $k=n$ in the last two equations, we respectively
obtain the CDFs and PDFs of the minimum and maximum order statistics, which
are given by

$\displaystyle F_{1:n}(x)$
$\displaystyle=1-\Big{[}\gamma(\alpha,\beta,x)\Big{]}^{n\theta},$
$\displaystyle f_{1:n}(x)$
$\displaystyle=n\theta~{}{\left(\frac{\alpha}{x^{2}}+\frac{\beta}{x^{3}}\right)}~{}{[1-\gamma(\alpha,\beta,x)]}~{}\Big{[}\gamma(\alpha,\beta,x)\Big{]}^{n\theta-1},$

and

$\displaystyle F_{n:n}(x)$
$\displaystyle=\Bigg{\\{}1-\Big{[}\gamma(\alpha,\beta,x)\Big{]}^{\theta}\Bigg{\\}}^{n},$
$\displaystyle f_{n:n}(x)$
$\displaystyle=n\theta~{}{\left(\frac{\alpha}{x^{2}}+\frac{\beta}{x^{3}}\right)}~{}{[1-\gamma(\alpha,\beta,x)]}~{}\Big{[}\gamma(\alpha,\beta,x)\Big{]}^{(\theta-1)}~{}\Bigg{\\{}1-\Big{[}\gamma(\alpha,\beta,x)\Big{]}^{\theta}\Bigg{\\}}^{(n-1)},$

respectively. Note that the minimum and maximum order statistics respectively
represent the lifetimes of a series and parallel systems.

###  2.6 Stochastic ordering

To study a comparative behaviour, the concept of stochastic ordering is a
well-recognized tool in reliability theory. In this subsection, we discuss
stochastic ordering results between two IGLFR distributed random variables.
Let $X$ and $Y$ be two absolutely continuous random variables with PDFs
$f_{X}(.)$, $f_{Y}(.),$ and CDFs $F_{X}(.)$, $F_{Y}(.),$ respectively. Then,
$X$ is said to be smaller than $Y$ in the sense of

  * •

the likelihood ratio ordering, abbreviated by $X\leq_{lr}Y$, if
$\frac{f_{Y}(x)}{f_{X}(x)}$ is non-decreasing in $x$;

  * •

the usual stochastic order, abbreviated by $X\leq_{st}Y$, if $F_{Y}(x)\leq
F_{X}(x)$, for all $x;$

  * •

the hazard rate order, abbreviated by $X\leq_{hr}Y$, if $h_{X}(x)\leq
h_{Y}(x),$ for all $x$, where $h_{X}$ and $h_{Y}$ are the hazard rate
functions of $X$ and $Y,$ respectively;

  * •

the reverse hazard rate order, abbreviated by $X\leq_{rh}Y$, if $r_{X}(x)\leq
r_{Y}(x),$ for all $x$, where $r_{X}$ and $r_{Y}$ are the reversed hazard rate
functions of $X$ and $Y,$ respectively.

For various other stochastic orders and their applications, readers are
referred to Shaked and Shanthikumar (2007). The following result provides
sufficient conditions such that the likelihood ratio order between two IGLFR
distributed random variables exists.

######  Theorem 2.1.

Let X $\thicksim$ IGLFR($\alpha_{1}$,$\beta_{1}$,$\theta_{1}$) and Y
$\thicksim$ IGLFR($\alpha_{2}$,$\beta_{2}$,$\theta_{2}$). If
$\alpha_{1}=\alpha_{2}=\alpha$, $\beta_{1}=\beta_{2}=\beta,$ and
$\theta_{1}\geq\theta_{2},$ then $X\leq_{lr}Y$.

###### Proof.

We have

$\displaystyle\frac{f_{Y}(x)}{f_{X}(x)}=\frac{\theta_{2}~{}{(\frac{\alpha_{2}}{x^{2}}+\frac{\beta_{2}}{x^{3}})}~{}{e^{-(\frac{\alpha_{2}}{x}+\frac{\beta_{2}}{2x^{2}})}}~{}\Big{[}1-e^{-(\frac{\alpha_{2}}{x}+\frac{\beta_{2}}{2x^{2}})}\Big{]}^{\theta_{2}-1}}{\theta_{1}~{}{(\frac{\alpha_{1}}{x^{2}}+\frac{\beta_{1}}{x^{3}})}~{}{e^{-(\frac{\alpha_{1}}{x}+\frac{\beta_{1}}{2x^{2}})}}~{}\Big{[}1-e^{-(\frac{\alpha_{1}}{x}+\frac{\beta_{1}}{2x^{2}})}\Big{]}^{\theta_{1}-1}}.$
(2.20)

Further,

$\displaystyle\frac{d}{dx}\text{log}\Bigg{(}\frac{f_{Y}(x)}{f_{X}(x)}\Bigg{)}$
$\displaystyle=$
$\displaystyle\frac{\alpha_{2}}{\alpha_{2}x+\beta_{2}}-\frac{\alpha_{1}}{\alpha_{1}x+\beta_{1}}+\frac{\alpha_{2}-\alpha_{1}}{x^{2}}-\frac{\beta_{2}-\beta_{1}}{x^{3}}$
$\displaystyle-\frac{(\theta_{2}-1)~{}e^{-(\frac{\alpha_{2}}{x}+\frac{\beta_{2}}{2x^{2}})}~{}(\frac{\alpha_{2}}{x^{2}}+\frac{\beta_{2}}{x^{3}})}{1-e^{-(\frac{\alpha_{2}}{x}+\frac{\beta_{2}}{2x^{2}})}}+\frac{(\theta_{1}-1)~{}e^{-(\frac{\alpha_{1}}{x}+\frac{\beta_{1}}{2x^{2}})}~{}(\frac{\alpha_{1}}{x^{2}}+\frac{\beta_{1}}{x^{3}})}{1-e^{-(\frac{\alpha_{1}}{x}+\frac{\beta_{1}}{2x^{2}})}}.$

From (2.6), clearly, if $\alpha_{1}=\alpha_{2}=\alpha$,
$\beta_{1}=\beta_{2}=\beta$, and $\theta_{1}\leq\theta_{2}$, then
$\frac{d}{dx}\text{log}\Big{(}\frac{f_{Y}(x)}{f_{X}(x)}\Big{)}\geq 0$, which
proves the desired ordering. ∎

In order to illustrate Theorem 2.1, the following numerical example is
considered.

######  Example 2.1.

Take two IGLFR distributed random variables $X$ and $Y$, such that $X\sim
IGLFR(0.5,0.8,2)$ and $Y\sim IGLFR(0.5,0.8,1.2)$. Here,
$\theta_{1}=2>\theta_{2}=1.2$. Now, we plot the ratio of the PDFs of $X$ and
$Y$ in Figure $5,$ which ensures that $X\leq_{lr}Y$ holds.

Figure 5: Plot of the ratio of the PDFs of $X$ and $Y$ as in Example 2.1.

######  Remark 2.3.

This implication is well-known: $X\leq_{st}Y\Leftarrow X\leq_{rh}Y\Leftarrow
X\leq_{lr}Y\Rightarrow X\leq_{hr}Y\Rightarrow X\leq_{st}Y.$ Thus, under
similar conditions as in Theorem 2.1, the usual stochastic, reversed hazard
rate and hazard rate orders also hold.

##  3 Estimation

###  3.1 Frequentist approach

Here, we propose various estimates of parameters of the proposed model.
Mainly, we emply two approaches: $(i)$ maximum likelihood estimation and
$(ii)$ maximum product spacings estimation. Irshad et al. (2021) derived MLEs
of the parameters of a newly proposed exponentiated unit Lindley distribution.
Maya et al. (2022) obtained MLEs for unit Muth distribution. First, we derive
MLEs of the unknown model parameters for IGLFR distribution.

####  3.1.1 Maximum likelihood estimation

Here, we propose MLEs of the parameters $\alpha$, $\beta$, and $\theta$ for
IGLFR distribution. Consider a random sample $(X_{1},\ldots,X_{n})$ drawn from
the IGLFR distribution with PDF given by (2.4). The likelihood function is

$\displaystyle
L(\alpha,\beta,\theta|data)=\theta^{n}\prod_{i=1}^{n}\left(\frac{\alpha}{x_{i}^{2}}+\frac{\beta}{x_{i}^{3}}\right)\exp\left\\{-\left(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}}\right)\right\\}\left[1-\exp\left\\{-\left(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}}\right)\right\\}\right]^{\theta-1},$
(3.1)

where data=$(x_{1},\ldots,x_{n})$. The log-likelihood function is obtained as

$\displaystyle l(\alpha,\beta,\theta)$ $\displaystyle=$ $\displaystyle
n\log\theta+\sum_{i=1}^{n}\log\left(\frac{\alpha}{x_{i}^{2}}+\frac{\beta}{x_{i}^{3}}\right)-\sum_{i=1}^{n}\left(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}}\right)+(\theta-1)$
(3.2)
$\displaystyle\times\sum_{i=1}^{n}\log\left[1-\exp\left\\{-\left(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}}\right)\right\\}\right].$

The normal equations can be obtained after differentiating log-likelihood
function with respect to the parameters $\alpha$, $\beta$, and $\theta$, and
equating them to zero, which are given by

$\displaystyle\frac{\partial l}{\partial\alpha}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\frac{\frac{1}{x_{i}^{2}}}{\frac{\alpha}{x_{i}^{2}}+\frac{\beta}{x_{i}^{3}}}-\sum_{i=1}^{n}\frac{1}{x_{i}}+(\theta-1)~{}\sum_{i=1}^{n}\frac{\frac{1}{x_{i}}\exp\\{-(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}})\\}}{1-\exp\\{-(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}})\\}}=0,$
(3.3) $\displaystyle\frac{\partial l}{\partial\beta}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\frac{\frac{1}{x_{i}^{3}}}{\frac{\alpha}{x_{i}^{2}}+\frac{\beta}{x_{i}^{3}}}-\sum_{i=1}^{n}\frac{1}{2x_{i}^{2}}+(\theta-1)~{}\sum_{i=1}^{n}\frac{\frac{1}{2x_{i}^{2}}\exp\\{-(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}})\\}}{1-\exp\\{-(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}})\\}}=0,$
(3.4) $\displaystyle\frac{\partial l}{\partial\theta}$ $\displaystyle=$
$\displaystyle\frac{n}{\theta}+\sum_{i=1}^{n}\log\left[1-\exp\left\\{-\left(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}}\right)\right\\}\right]=0.$
(3.5)

The above normal equations are difficult to solve simultaneously. Thus, we
need numerical technique. The values of $\alpha$ and $\beta$ can be obtained
from (3.3) and (3.4), using Newton-Raphson method. Then, substituting the
values of $\alpha$ and $\beta$ in (3.5), the value of $\theta$ can be
obtained. The MLEs of $\alpha$, $\beta$, and $\theta$ are respectively denoted
by $\hat{\alpha}$, $\hat{\beta}$, and $\hat{\theta}.$

####  3.1.2 Maximum product spacings estimation

We derive estimates of the unknown model parameters using maximum product
spacings (MPS) estimation method. We recall that the MPS estimation technique
is an alternative to MLE for the estimation of parameters of a continuous
univariate distribution. The MPS method was introduced by Cheng and Amin
(1979) and Cheng and Amin (1983). Later, this method was independently
developed as an approximation to the Kullback-Leibler measure by Ranneby
(1984). Consider an ordered random sample $X_{1:1}\leq\ldots\leq X_{n:n}$ from
the newly proposed distribution with CDF given by (2.3). For convenience, in
this subsection, we denote $X_{i}=X_{i:n},~{}i=1,\ldots,n.$ Further, the
spacings are denoted by

$\text{D}_{i}=\begin{cases}\text{F}(x_{1}),&\text{when}\ i=1,\\\
\text{F}(x_{i})-\text{F}(x_{i-1}),&\text{when}\ i=2,\ldots,n,\\\
1-\text{F}(x_{n}),&\text{when}\ i=n+1\end{cases}$

such that $\sum\text{D}_{i}=1$, see Pyke (1965). The geometric mean of the
spacings is given by

$\displaystyle\text{G}^{*}=\Bigg{[}\prod_{i=1}^{n+1}\text{D}_{i}\Bigg{]}^{\frac{1}{n+1}}=\Bigg{[}\text{F}(x_{1})\Big{\\{}\prod_{i=2}^{n}[\text{F}(x_{i})-\text{F}(x_{i-1})]\Big{\\}}[1-\text{F}(x_{n})]\Bigg{]}^{\frac{1}{n+1}}.$
(3.6)

Taking logarithm both sides of (3.6), we obtain

$\displaystyle g^{*}=\log G^{*}$ $\displaystyle=$
$\displaystyle\frac{1}{n+1}\Bigg{[}\log\Big{\\{}1-\big{[}\gamma(\alpha,\beta,x_{1})\big{]}^{\theta}\Big{\\}}+\log\big{[}\gamma(\alpha,\beta,x_{n})\big{]}^{\theta}$
(3.7)
$\displaystyle+\sum_{i=2}^{n}\log\Big{\\{}\big{[}\gamma(\alpha,\beta,x_{i})\big{]}^{\theta}-\big{[}\gamma(\alpha,\beta,x_{i-1})\big{]}^{\theta}\Big{\\}}\Bigg{]}.$

Now, differentiating (3.7) with respect to $\alpha,$ $\beta,$ and $\theta$
partially, we get

$\displaystyle\frac{\partial g^{*}}{\partial\alpha}$ $\displaystyle=$
$\displaystyle\frac{1}{n+1}\Bigg{[}-\frac{\frac{\theta}{x_{1}}\big{[}1-\gamma(\alpha,\beta,x_{1})\big{]}\big{[}\gamma(\alpha,\beta,x_{1})\big{]}^{\theta-1}}{1-\big{[}\gamma(\alpha,\beta,x_{1})\big{]}^{\theta}}+\frac{\frac{\theta}{x_{n}}[1-\gamma(\alpha,\beta,x_{n})]\big{[}\gamma(\alpha,\beta,x_{n})\big{]}^{\theta-1}}{\big{[}\gamma(\alpha,\beta,x_{n})\big{]}^{\theta}}$
$\displaystyle+\sum_{i=2}^{n}\frac{\frac{\theta}{x_{i}}[1-\gamma(\alpha,\beta,x_{i})]\big{[}\gamma(\alpha,\beta,x_{i})\big{]}^{\theta-1}-\frac{\theta}{x_{i-1}}[1-\gamma(\alpha,\beta,x_{i-1})]\big{[}\gamma(\alpha,\beta,x_{i-1})\big{]}^{\theta-1}}{\big{[}\gamma(\alpha,\beta,x_{i-1})\big{]}^{\theta}-\big{[}\gamma(\alpha,\beta,x_{i})\big{]}^{\theta}}\Bigg{]},$

$\displaystyle\frac{\partial g^{*}}{\partial\beta}$ $\displaystyle=$
$\displaystyle\frac{1}{n+1}\Bigg{[}-\frac{\frac{\theta}{2x_{1}^{2}}[1-\gamma(\alpha,\beta,x_{1})]\big{[}\gamma(\alpha,\beta,x_{1})\big{]}^{\theta-1}}{1-\big{[}\gamma(\alpha,\beta,x_{1})\big{]}^{\theta}}+\frac{\frac{\theta}{2x_{n}^{2}}[1-\gamma(\alpha,\beta,x_{n})]\big{[}\gamma(\alpha,\beta,x_{n})\big{]}^{\theta-1}}{\big{[}\gamma(\alpha,\beta,x_{n})\big{]}^{\theta}}$
$\displaystyle+\sum_{i=2}^{n}\frac{\frac{\theta}{2x_{i}^{2}}[1-\gamma(\alpha,\beta,x_{i})]\big{[}\gamma(\alpha,\beta,x_{i})\big{]}^{\theta-1}-\frac{\theta}{2x_{i-1}^{2}}[1-\gamma(\alpha,\beta,x_{i-1})]\big{[}\gamma(\alpha,\beta,x_{i-1})\big{]}^{\theta-1}}{\big{[}\gamma(\alpha,\beta,x_{i-1})\big{]}^{\theta}-\big{[}\gamma(\alpha,\beta,x_{i})\big{]}^{\theta}}\Bigg{]},$

and

$\displaystyle\frac{\partial g^{*}}{\partial\theta}$ $\displaystyle=$
$\displaystyle\frac{1}{n+1}\Bigg{[}\frac{-\big{[}\gamma(\alpha,\beta,x_{1})\big{]}^{\theta}\log\big{[}\gamma(\alpha,\beta,x_{1})\big{]}}{1-\big{[}\gamma(\alpha,\beta,x_{1})\big{]}^{\theta}}+\frac{\big{[}\gamma(\alpha,\beta,x_{n})\big{]}^{\theta}\log\big{[}\gamma(\alpha,\beta,x_{n})\big{]}}{\big{[}\gamma(\alpha,\beta,x_{n})\big{]}^{\theta}}$
(3.10)
$\displaystyle+\sum_{i=2}^{n}\frac{\big{[}\gamma(\alpha,\beta,x_{i-1})\big{]}^{\theta}\log\big{[}\gamma(\alpha,\beta,x_{i-1})\big{]}-\big{[}\gamma(\alpha,\beta,x_{i})\big{]}^{\theta}\log\big{[}\gamma(\alpha,\beta,x_{i})\big{]}}{\big{[}\gamma(\alpha,\beta,x_{i-1})\big{]}^{\theta}-\big{[}\gamma(\alpha,\beta,x_{i})\big{]}^{\theta}}\Bigg{]}.$

Clearly, no explicit solutions can be obtained from the nonlinear equations
$\frac{\partial g^{*}}{\partial\alpha}=0$, $\frac{\partial
g^{*}}{\partial\beta}=0$, and $\frac{\partial g^{*}}{\partial\theta}=0$. Thus,
it is required to adopt a suitable numerical technique, say Newton-Raphson
method. The solution for $\alpha,$ $\beta,$ and $\theta$ is the MPS estimates.

####  3.1.3 Asymptotic confidence intervals

Here, we derive asymptotic confidence intervals of $\alpha,$ $\beta,$ and
$\theta$ of the IGLFR distribution. Here, the MLEs of the parameters can not
be obtained explicitely. Thus, it is impossible to obtain their exact
distributions. Under some regularity conditions, the MLEs
$(\hat{\alpha},\hat{\beta},\hat{\theta})$ asymptotically follow tri-variate
normal distribution with mean vector $(\alpha,\beta,\theta)$ and variance-
covariance matrix $\text{I}_{0}^{-1}$, where

$\displaystyle\text{I}_{0}^{-1}=$
$\displaystyle\begin{pmatrix}-\frac{\partial^{2}\log\text{L}}{\partial\alpha^{2}}&-\frac{\partial^{2}\log\text{L}}{\partial\alpha\partial\beta}&-\frac{\partial^{2}\log\text{L}}{\partial\alpha\partial\theta}\\\
\\\
-\frac{\partial^{2}\log\text{L}}{\partial\alpha\partial\beta}&-\frac{\partial^{2}\log\text{L}}{\partial\beta^{2}}&-\frac{\partial^{2}\log\text{L}}{\partial\beta\partial\theta}\\\
\\\
-\frac{\partial^{2}\log\text{L}}{\partial\theta\partial\alpha}&-\frac{\partial^{2}\log\text{L}}{\partial\theta\partial\beta}&-\frac{\partial^{2}\log\text{L}}{\partial\theta^{2}}\end{pmatrix}^{-1}=$
$\displaystyle\begin{pmatrix}\text{Var}(\hat{\alpha})&\text{Cov}(\hat{\alpha},\hat{\beta})&\text{Cov}(\hat{\alpha},\hat{\theta})\\\
\\\
\text{Cov}(\hat{\alpha},\hat{\beta})&\text{Var}(\hat{\beta})&\text{Cov}(\hat{\beta},\hat{\theta})\\\
\\\
\text{Cov}(\hat{\alpha},\hat{\theta})&\text{Cov}(\hat{\beta},\hat{\theta})&\text{Var}(\hat{\theta})\end{pmatrix}$

and

$\displaystyle\frac{\partial^{2}\log\text{L}}{\partial\alpha^{2}}=$
$\displaystyle-\sum_{i=1}^{n}\frac{\frac{1}{x_{i}^{4}}}{\big{(}\frac{\alpha}{x_{i}^{2}}+\frac{\beta}{x_{i}^{3}}\big{)}^{2}}-(\theta-1)~{}\sum_{i=1}^{n}\frac{e^{-(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}})}\frac{1}{x_{i}^{2}}}{\big{[}1-e^{-(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}})}\big{]}^{2}},$
$\displaystyle\frac{\partial^{2}\log\text{L}}{\partial\alpha\partial\beta}=$
$\displaystyle-\sum_{i=1}^{n}\frac{\frac{1}{x_{i}^{5}}}{\big{(}\frac{\alpha}{x_{i}^{2}}+\frac{\beta}{x_{i}^{3}}\big{)}^{2}}-(\theta-1)~{}\sum_{i=1}^{n}\frac{e^{-(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}})}\frac{1}{2x_{i}^{3}}}{\big{[}1-e^{-(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}})}\big{]}^{2}},$
$\displaystyle\frac{\partial^{2}\log\text{L}}{\partial\alpha\partial\theta}=$
$\displaystyle\sum_{i=1}^{n}\frac{e^{-(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}})}\frac{1}{x_{i}}}{\big{[}1-e^{-(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}})}\big{]}},~{}~{}~{}~{}\frac{\partial^{2}\log\text{L}}{\partial\beta\partial\theta}=\sum_{i=1}^{n}\frac{e^{-(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}})}\frac{1}{2x_{i}^{2}}}{\big{[}1-e^{-(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}})}\big{]}},$
$\displaystyle\frac{\partial^{2}\log\text{L}}{\partial\beta^{2}}=$
$\displaystyle-\sum_{i=1}^{n}\frac{\frac{1}{x_{i}^{6}}}{\big{(}\frac{\alpha}{x_{i}^{2}}+\frac{\beta}{x_{i}^{3}}\big{)}^{2}}-(\theta-1)~{}\sum_{i=1}^{n}\frac{e^{-(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}})}\frac{1}{4x_{i}^{4}}}{\big{[}1-e^{-(\frac{\alpha}{x_{i}}+\frac{\beta}{2x_{i}^{2}})}\big{]}^{2}},~{}~{}\frac{\partial^{2}\log\text{L}}{\partial\theta^{2}}=-\frac{n}{\theta^{2}}.$

Thus, the $100(1-\nu)$% confidence intervals for the parameters $\alpha$,
$\beta$, and $\theta$ are respectively obtained as

$\hat{\alpha}\pm\text{Z}_{\frac{\nu}{2}}\sqrt{\text{Var}(\hat{\alpha})},~{}~{}\hat{\beta}\pm\text{Z}_{\frac{\nu}{2}}\sqrt{\text{Var}(\hat{\beta})},~{}~{}\hat{\theta}\pm\text{Z}_{\frac{\nu}{2}}\sqrt{\text{Var}(\hat{\theta})},$

where $Z_{\frac{\nu}{2}}$ is the upper $\frac{\nu}{2}$th percentile of the
standard normal distribution.

###  3.2 Bayesian approach

In many real life applications, the parameters of a model can not be fixed. It
varies and as a result, the model parameters have some randomness. Bayesian
analysis include the randomness of the parameter into study. Bayesian approach
allows us to consider the prior beliefs (in the form of prior distribution)
about the parameter with the available data information (likelihood function).
We note that if proper information about the unknown parameters are available,
then proper prior distributions are taken into account, otherwise, improper
prior distributions are considered in the study. In this subsection, we will
derive Bayes estimates of $\alpha$, $\beta$, and $\theta$ with respect to
SELF. Let $\delta$ be an estimator of the parameter $\lambda.$ Then, the SELF
is defined as

$\displaystyle L(\lambda,\delta)=(\delta-\lambda)^{2}.$ (3.11)

We recall that the SELF is symmetric. Further, under SELF, the mean of the
posterior distribution is a Bayes estimator. Note that here the conjugate
priors do not exist. Thus, according to Kundu and Pradhan (2009), we assume
independent gamma priors for the unknown parameters. A random variable $T$ is
said to follow gamma distribution with parameters $m$ and $n$ if then PDF is

$\displaystyle
f^{*}(t|m,n)=\frac{1}{\Gamma(m)n^{m}}t^{m-1}\exp\\{-\frac{t}{n}\\},~{}t>0,~{}m,n>0.$
(3.12)

Henceforth, we denote $T\sim Gamma(m,n)$ if $T$ has the PDF given by (3.12).
Consider $\alpha\sim Gamma(a,b)$, $\beta\sim Gamma(c,d),$ and $\theta\sim
Gamma(p,q)$. After some calculations, the joint posterior distribution of
$\alpha,~{}\beta,$ and $\theta$ is obtained as

$\pi(\alpha,\beta,\theta|data)=\frac{k\prod_{i=1}^{n}\Big{[}{(\frac{\alpha}{x_{i}^{2}}+\frac{\beta}{x_{i}^{3}})}~{}{[1-\gamma(\alpha,\beta,x_{i})]}~{}\big{[}\gamma(\alpha,\beta,x_{i})\big{]}^{\theta-1}\Big{]}}{\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}k\prod_{i=1}^{n}\Big{[}{(\frac{\alpha}{x_{i}^{2}}+\frac{\beta}{x_{i}^{3}})}~{}{[1-\gamma(\alpha,\beta,x_{i})]}~{}\big{[}\gamma(\alpha,\beta,x_{i})\big{]}^{\theta-1}\Big{]}\
d\alpha\ d\beta\ d\theta},$ (3.13)

where
$k=k(\alpha,\beta,\theta)=\frac{1}{\Gamma(a)b^{a}}\frac{1}{\Gamma(c)d^{c}}\frac{1}{\Gamma(p)q^{p}}\alpha^{a-1}\beta^{c-1}\theta^{p-1+n}\exp\\{-\big{(}\frac{\alpha}{b}+\frac{\beta}{d}+\frac{\theta}{q}\big{)}\\}.$
Thus, the Bayes estimate of $\alpha$ with respect to SELF, denoted by
$\hat{\alpha}_{\text{BE}}$, is obtained as

$\displaystyle\hat{\alpha}_{\text{BE}}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\alpha~{}\pi(\alpha,\beta,\theta|data)\
d\alpha\ d\beta\ d\theta,$ (3.14) $\displaystyle=$
$\displaystyle\frac{\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\alpha
k\prod_{i=1}^{n}\Big{[}{(\frac{\alpha}{x_{i}^{2}}+\frac{\beta}{x_{i}^{3}})}~{}{[1-\gamma(\alpha,\beta,x_{i})]}\big{[}\gamma(\alpha,\beta,x_{i})\big{]}^{\theta-1}\Big{]}\
d\alpha\ d\beta\
d\theta}{\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\text{k}~{}\prod_{i=1}^{n}\Big{[}{(\frac{\alpha}{x_{i}^{2}}+\frac{\beta}{x_{i}^{3}})}~{}{[1-\gamma(\alpha,\beta,x_{i})]}~{}\Big{[}\gamma(\alpha,\beta,x_{i})\Big{]}^{\theta-1}\Big{]}\
d\alpha\ d\beta\ d\theta},$ $\displaystyle=$ $\displaystyle\eta_{1}(data).$

The Bayes estimates of $\beta$ and $\theta$, respectively denoted by
$\hat{\beta}_{\text{BE}}$ and $\hat{\theta}_{\text{BE}}$ can be obtained
similarly. Here, we omit the expressions for $\hat{\beta}_{\text{BE}}$ and
$\hat{\theta}_{\text{BE}}$ to avoid repetitions. We observe that it is
impossible to obtain the Bayes estimates of $\alpha$, $\beta,$ and $\theta$ in
closed form. Thus, we will employ MCMC technique to compute the approximate
Bayes estimates, which is discussed in the next subsection.

####  3.2.1 MCMC Method

Here, MCMC approach will be employed to compute the approximate Bayes
estimates of the parameters under SELF. From the posterior density function
given by (3.13), we obtain the following conditional posterior density
functions:

$\pi_{1}(\alpha|\beta,\theta,data)\propto~{}\alpha^{a-1}\exp\\{-\frac{\alpha}{b}\\}\prod_{i=1}^{n}\Bigg{[}{\Big{(}\frac{\alpha}{x_{i}^{2}}+\frac{\beta}{x_{i}^{3}}\Big{)}}~{}{\exp\\{-\frac{\alpha}{x_{i}}\\}}~{}\Big{[}\gamma(\alpha,\beta,x_{i})\Big{]}^{\theta-1}\Bigg{]},$
(3.15)
$\pi_{2}(\beta|\alpha,\theta,data)\propto~{}\beta^{c-1}\exp\\{-\frac{\beta}{d}\\}\prod_{i=1}^{n}\Bigg{[}{\Big{(}\frac{\alpha}{x_{i}^{2}}+\frac{\beta}{x_{i}^{3}}\Big{)}}~{}{\exp\\{-\frac{\beta}{2x_{i}^{2}}\\}}~{}\Big{[}\gamma(\alpha,\beta,x_{i})\Big{]}^{\theta-1}\Bigg{]}$
(3.16)

and

$\pi_{3}(\theta|\alpha,\beta,data)\propto~{}\theta^{p-1+n}\exp\\{-\frac{\theta}{q}\\}\prod_{i=1}^{n}\Big{[}\gamma(\alpha,\beta,x_{i})\Big{]}^{\theta-1}.$
(3.17)

The above density functions
$\pi_{1}(\alpha|\beta,\theta,data),\pi_{2}(\beta|\alpha,\theta,data),$ and
$\pi_{3}(\theta|\alpha,\beta,data)$ can not be written in the form of any well
known distributions. Thus, the MCMC samples can not be generated from these
densities. So, the Metropolis-Hastings algorithm is utilized to obtain MCMC
samples from the conditional density functions. We obtain the Bayes estimates
by using the following steps:

Step 1: Choose initial values as $\alpha^{(1)}=\hat{\alpha}$,
$\beta^{(1)}=\hat{\beta},$ and $\theta^{(1)}=\hat{\theta}$. Set $i=1$.
Step 2: Generate $\alpha^{(i)},~{}\beta^{(i)},$ and $\theta^{(i)}$ with normal
distribution as $\alpha^{(i)}$ $\thicksim$ N $(\alpha^{(i-1)}$,
$Var(\hat{\alpha}))$, $\beta^{(i)}$ $\thicksim$
N$(\beta^{(i-1)}$,Var($\hat{\beta}))$, and $\theta^{(i)}$ $\thicksim$
N$(\theta^{(i-1)}$,Var($\hat{\theta}))$.
Step 3: Compute
$\Omega_{\alpha}=min\Big{(}1,\frac{\pi_{1}(\alpha^{(i)}|\beta^{(i-1)},\theta^{(i-1)},data)}{\pi_{1}(\alpha^{(i-1)}|\beta^{(i-1)},\theta^{(i-1)},data)}\Big{)}$,
$\Omega_{\beta}=min\Big{(}1,\frac{\pi_{2}(\beta^{(i)}|\alpha^{(i-1)},\theta^{(i-1)},data)}{\pi_{2}(\beta^{(i-1)}|\alpha^{(i-1)},\theta^{(i-1)},data)}\Big{)},$
and
$\Omega_{\theta}=min\Big{(}1,\frac{\pi_{3}(\theta^{(i)}|\alpha^{(i-1)},\beta^{(i-1)},data)}{\pi_{3}(\theta^{(i-1)}|\alpha^{(i-1)},\beta^{(i-1)},data)}\Big{)}$.

Step 4: Generate samples for $\mathcal{T}_{1}$ $\thicksim$ Uniform(0,1),
$\mathcal{T}_{2}$ $\thicksim$ Uniform(0,1), and $\mathcal{T}_{3}$ $\thicksim$
Uniform(0,1).

Step 5: Set

$\alpha=\begin{cases}\alpha^{(i)},&\text{if}\
\mathcal{T}_{1}\leq\Omega_{\alpha},\\\ \alpha^{(i-1)},&\text{otherwise}.\\\
\end{cases}$ $\beta=\begin{cases}\beta^{(i)},&\text{if}\
\mathcal{T}_{2}\leq\Omega_{\beta},\\\ \beta^{(i-1)},&\text{otherwise}.\\\
\end{cases}$ $\theta=\begin{cases}\theta^{(i)},&\text{if}\
\mathcal{T}_{3}\leq\Omega_{\theta},\\\ \theta^{(i-1)},&\text{otherwise}.\\\
\end{cases}$

Step 6: Set i=i+1.

Step 7: Repeat steps 1 to 6, $K$ times to get
$\alpha^{(1)},\cdots,\alpha^{(K)}$ ; $\beta^{(1)},\cdots,\beta^{(K)}$ and
$\theta^{(1)}\cdots,\theta^{(K)}$.

Thus, under SELF, the Bayes estimates of $\alpha,\beta,$ and $\theta$ are
respectively given as

$\hat{\alpha}_{SEL}=\frac{1}{K}~{}\sum_{i=1}^{K}\alpha^{(i)},~{}\hat{\beta}_{SEL}=\frac{1}{K}~{}\sum_{i=1}^{K}\beta^{(i)},~{}\mbox{and}~{}\hat{\theta}_{SEL}=\frac{1}{K}~{}\sum_{i=1}^{K}\theta^{(i)}.$

The MCMC method is also used to derive $100(1-\frac{\nu}{2})$% credible
intervals of the parameters $\alpha$, $\beta$, and $\theta$ as

$\Big{(}{\alpha}_{(t)},~{}{\alpha}_{\big{(}t+(1-\nu)K\big{)}}\Big{)},~{}\Big{(}{\beta}_{(t)},~{}{\beta}_{\big{(}t+(1-\nu)K\big{)}}\Big{)},~{}\mbox{and}~{}\Big{(}{\theta}_{(t)},~{}{\theta}_{\big{(}t+(1-\nu)K\big{)}}\Big{)}.$

##  4 Simulation study

Here, a Monte Carlo simulation is executed to see the comparative performance
of the proposed estimators. On the basis of average bias and mean squared
error (MSE), we will examine the estimators’ performance. Sample sizes are
considered as $n=20,~{}30,~{}40,~{}50,~{}60,~{}70,~{}80,$ and $90.$ $10,000$
independent samples of size $n$ are generated from IGLFR distribution with two
sets of parameters $\alpha=0.5$, $\beta=0.5$, $\theta=1$ and $\alpha=0.3$,
$\beta=0.5$, $\theta=0.2$. We report average bias, obtained by MLE, MPS and
Bayes estimates along with their MSEs presented in the parentheses in Table
$1$, Table $2$, and Table $3$, for different values of $\alpha,~{}\beta,$ and
$\theta$. All simulations have done by using $R$ software. To compute MLE and
MPSE, ’nleqslv’ package is used. To obtain Bayes estimates, MCMC samples have
been generated by using ’coda’ package. The $95\%$ asymptotic confidence and
Bayesian credible intervals along with interval lengths (ALs) and coverage
probabilities (CPs) are presented in Tables $4$, $5$, and $6$. From the
tables, following observations are noticed.

  * •

From Table $1$, Table $2$, and Table $3$, we clearly observe that the mean
squared errors and average biases decrease as the sample size increases.

  * •

From Table $4$, Table $5$, and Table $6$, we notice that when the sample size
increases the average confidence/credible lengths decrease. There is no
specific pattern of increasing or decreasing values of CPs when the sample
size increase.

  * •

For small sample sizes, the Bayesian credible intervals or the asymptotic
confidence intervals are slightly skewed, and they become symmetric for large
sample sizes.

  * •

Based on average bias and MSEs, in general, the Bayes estimates give better
results than the MLEs and MPS estimates.

  * •

Based on average bias and MSEs, in general, the MLEs give better performance
than the MPS estimates.

  * •

The Bayesian credible intervals provide superior result than the asymptotic
confidence intervals based on the average interval lengths.

  * •

Based on CP, BCIs perform better than ACIs.

From above observations, we can summarize that Bayes estimators have better
performance than the classical estimators. In classical estimators, one can
prefer MPSE over MLE. Bayesian estimation contains more information than the
maximum likelihood estimation. In terms of AL and CP of interval estimates, an
experimenter can choose BCIs over ACIs.

Table 1: Average bias and MSEs (shown in parentheses) of the estimates for the
IGLFR distribution with $\alpha=0.5$, $\beta=0.5$, and $\theta=1$ with
different values of $n$.

| MLE | MPSE | Bayes
---|---|---|---
$n$ | $\alpha$ | $\beta$ | $\theta$ | $\alpha$ | $\beta$ | $\theta$ | $\alpha$ | $\beta$ | $\theta$
20 | 0.1529 | -0.0121 | 0.2827 | 0.4100 | -0.2377 | 0.6820 | 0.0019 | 0.0310 | 0.0268
| (0.2746) | (0.1470) | (0.6573) | (0.3628) | (0.1707) | (1.1797) | (0.0290) | (0.0408) | (0.0545)
30 | 01090 | 0.0034 | 0.1822 | 0.3788 | -0.2327 | 0.5338 | -0.0037 | 0.0297 | 0.0240
| (0.1898) | (0.0959) | (0.3146) | (0.3227) | (0.1298) | (0.6972) | (0.0286) | (0.0360) | (0.0478)
40 | 0.0896 | -0.0181 | 0.1273 | 0.3277 | -0.2177 | 0.4110 | 0.0011 | 0.0132 | 0.0143
| (0.1673) | (0.0792) | (0.2163) | (0.2601) | (0.1147) | (0.4390) | (0.0290) | (0.0283) | (0.0376)
50 | 0.0437 | 0.0032 | 0.0764 | 0.2763 | -0.1833 | 0.3396 | -0.0064 | 0.0180 | 0.0089
| (0.1378) | (0.0673) | (0.1594) | (0.2173) | (0.0953) | (0.3387) | (0.0289) | (0.261) | (0.0373)
60 | 0.0661 | -0.0207 | 0.0894 | 0.2761 | -0.1898 | 0.3150 | 0.0022 | 0.0055 | 0.0162
| (0.1244) | (0.0565) | (0.1393) | (0.1971) | (0.0895) | (0.2638) | (0.0288) | (0.0219) | (0.0341)
70 | 0.0502 | -0.0084 | 0.0733 | 0.2513 | -0.1681 | 0.2862 | -0.0049 | 0.0157 | 0.0109
| (0.1145) | (0.0511) | (0.1190) | (0.1843) | (0.0813) | (0.2356) | (0.0268) | (0.0224) | (0.0305)
80 | 0.0399 | -0.0066 | 0.0580 | 0.2230 | -0.1509 | 0.2484 | -0.0031 | 0.0101 | 0.0103
| (0.1084) | (0.0510) | (0.0995) | (0.1649) | (0.0775) | (0.1936) | (0.0288) | (0.0216) | (0.0297)
90 | 0.0259 | -0.0053 | 0.0389 | 0.2105 | -0.1471 | 0.2233 | -0.0066 | 0.0083 | 0.0024
| (0.0931) | (0.0430) | (0.0866) | (0.1392) | (0.0656) | (0.1553) | (0.0255) | (0.0181) | (0.0276)
100 | 0.0300 | -0.0083 | 0.0386 | 0.2034 | -0.1418 | 0.2091 | -0.0066 | 0.0096 | 0.0011
| (0.0859) | (0.0384) | (0.0770) | (0.1299) | (0.0594) | (0.1381) | (0.0248) | (0.0174) | (0.0247)
150 | 0.0004 | 0.0063 | 0.0180 | 0.1426 | -0.0990 | 0.1446 | -0.0185 | 0.0142 | -0.0082
| (0.0631) | (0.0284) | (0.0498) | (0.0819) | (0.0386) | (0.0745) | (0.0238) | (0.0150) | (0.0196)

Table 2: Average bias and MSEs (shown in parentheses) of the estimates for the
IGLFR distribution with $\alpha=0.3$, $\beta=0.5$, and $\theta=0.2$ with
different values of $n$.

| MLE | MPSE | Bayes
---|---|---|---
$n$ | $\alpha$ | $\beta$ | $\theta$ | $\alpha$ | $\beta$ | $\theta$ | $\alpha$ | $\beta$ | $\theta$
20 | 0.1389 | 0.2113 | 0.0179 | 0.2014 | -0.1460 | 0.0521 | -0.0568 | -0.0448 | -1.314
| (0.2283) | (0.6509) | (0.0056) | (0.2170) | (0.2477) | (0.0970) | (0.0535) | (0.0579) | (3.1959)
30 | 0.0885 | 0.1094 | 0.0169 | 0.1615 | -0.1386 | 0.0363 | -0.0491 | -0.0463 | -1.3032
| (0.1313) | (0.3323) | (0.0255) | (0.1364) | (0.1986) | (0.0053) | (0.0497) | (0.0550) | (3.0672)
40 | 0.0937 | 0.0771 | 0.0083 | 0.1679 | -0.1355 | 0.0270 | -0.0603 | -0.0370 | -1.3147
| (0.1089) | (0.2033) | (0.0039) | (0.1256) | (0.1491) | (0.0032) | (0.0517) | (0.0535) | (2.8042)
50 | 0.0387 | 0.0659 | 0.0068 | 0.1138 | -0.1168 | 0.0196 | -0.0470 | -0.0305 | -1.1898
| (0.0726) | (0.0256) | (0.0130) | (0.0861) | (0.0975) | (0.0023) | (0.0484) | (0.0498) | (2.373)
60 | 0.0398 | 0.0396 | 0.0099 | 0.1163 | -0.1210 | 0.0182 | -0.0446 | -0.0246 | -1.1871
| (0.0632) | (0.9200) | (0.0288) | (0.0813) | (0.0854) | (0.0019) | (0.0437) | (0.0476) | (2.3030)
70 | 0.0501 | 0.0209 | 0.0091 | 0.1241 | -0.1173 | 0.0192 | -0.0349 | -0.0340 | -1.0715
| (0.0686) | (0.0973) | (0.0615) | (0.0867) | (0.0878) | (0.0019) | (0.0446) | (0.0426) | (1.9222)
80 | 0.0274 | 0.0366 | 0.0063 | 0.0898 | -0.0919 | 0.0154 | -0.0345 | -0.0291 | -1.006
| (0.0533) | (0.0817) | (0.0078) | (0.0655) | (0.0713) | (0.0014) | (0.0426) | (0.0425) | (1.6935)
90 | 0.0290 | 0.0193 | 0.0060 | 0.0914 | -0.0989 | 0.0127 | -0.0331 | -0.0252 | -1.0627
| (0.0471) | (0.0796) | (0.0332) | (0.0573) | (0.0547) | (0.0011) | (0.0404) | (0.0391) | (1.7514)
100 | 0.0286 | 0.0089 | 0.0058 | 0.0961 | -0.1045 | 0.0129 | -0.0443 | -0.0173 | -0.9954
| (0.0455) | (0.0789 ) | (0.0186) | (0.0574) | (0.0544) | (0.0011) | (0.0361) | (0.0362) | (1.5309)
150 | 0.0043 | 0.0141 | -0.0009 | 0.0616 | -0.0627 | 0.0101 | -0.0055 | -0.0015 | -0.0808
| (0.0321) | (0.0467) | (0.0073) | (0.0386) | (0.0363) | (0.0024) | (0.0016) | (0.0012) | (0.0995)

Table 3: Average bias and MSEs (shown in parentheses) of the estimates for the
IGLFR distribution with $\alpha=1$, $\beta=0.7$, and $\theta=0.4$ with
different values of $n$.

| MLE | MPSE | Bayes
---|---|---|---
$n$ | $\alpha$ | $\beta$ | $\theta$ | $\alpha$ | $\beta$ | $\theta$ | $\alpha$ | $\beta$ | $\theta$
20 | 0.1299 | 0.3016 | 0.1220 | 0.1079 | 0.1345 | 0.0974 | -0.0978 | -0.0852 | -0.0869
| (0.6000) | (2.5044) | (0.0224) | (0.4964) | (0.9482) | (0.0360) | (0.1088) | (0.0864) | (0.3562)
30 | 0.1224 | 0.2930 | 0.1122 | 0.0966 | 0.1248 | 0.0735 | -0.0957 | -0.0740 | -0.0270
| (0.3851) | (1.2016) | (0.0144) | (0.3429) | (0.5968) | (0.0240) | (0.0922) | (0.0781) | (0.1109)
40 | 0.1108 | 0.2688 | 0.1046 | 0.0952 | 0.1195 | 0.0505 | -0.0891 | -0.0687 | -0.0221
| (0.3243) | (0.7481) | (0.0103) | (0.2905) | (0.4606) | (0.0138) | (0.0820) | (0.0691) | (0.0731)
50 | 0.1044 | 0.2421 | 0.1040 | 0.0937 | 0.1074 | 0.0449 | -0.0784 | -0.0500 | -0.0165
| (0.2685) | (0.6839) | (0.0086) | (0.2346) | (0.4379) | (0.0113) | (0.0730) | (0.0709) | (0.0670)
60 | 0.0978 | 0.2399 | 0.1004 | 0.0927 | 0.1038 | 0.0356 | -0.0705 | -0.0397 | -0.0093
| (0.2301) | (0.5042) | (0.0061) | (0.2147) | (0.3523) | (0.0078) | (0.0717) | (0.0731) | (0.0380)
70 | 0.0908 | 0.2664 | 0.0987 | 0.0894 | 0.0983 | 0.0350 | -0.0600 | -0.0509 | -0.0069
| (0.2125) | (0.4120) | (0.0060) | (0.1951) | (0.3147) | (0.0073) | (0.0601) | (0.0630) | (0.0270)
80 | 0.0897 | 0.2364 | 0.0941 | 0.0875 | 0.0958 | 0.0289 | -0.0402 | -0.0385 | -0.0064
| (0.1820) | (0.3720) | (0.0052 ) | (0.1797) | (0.2942) | (0.0065) | (0.0623) | (0.0709) | (0.0287)
90 | 0.0855 | 0.2226 | 0.0925 | 0.0841 | 0.0918 | 0.0289 | -0.0402 | -0.0385 | -0.0064
| (0.1728) | (0.3379) | (0.0045) | (0.1752) | (0.2802) | (0.0055) | (0.0539) | (0.0628) | (0.0239)
100 | 0.0839 | 0.2016 | 0.0888 | 0.0813 | 0.0909 | 0.0269 | -0.0398 | -0.0341 | -0.0016
| (0.1542) | (0.2763) | (0.0039) | (0.1617) | (0.2512) | (0.0048) | (0.0545) | (0.0580) | (0.0106)
150 | 0.0828 | 0.1937 | 0.0832 | 0.0805 | 0.0891 | 0.0207 | -0.0292 | -0.0099 | 0.0048
| (0.1016) | (0.1903) | (0.0023) | (0.1103) | (0.1786) | (0.0029) | (0.0416) | (0.0507) | (0.0016)

Table 4:  ALs of $95\%$ asymptotic confidence intervals (ACI) and Bayesian
credible intervals (BCI) and their corresponding CPs (in parentheses) with
$\alpha=0.5$, $\beta=0.5$, and $\theta=1$ with different values of $n$.

| ACI | BCI
---|---|---
$n$ | $\alpha$ | $\beta$ | $\theta$ | $\alpha$ | $\beta$ | $\theta$
20 | 1.9150 | 1.4573 | 2.7331 | 0.6034 | 0.7079 | 0.8788
| (0.9342) | (0.9298) | (0.9391) | (0.9485) | (0.9507) | (0.9527)
30 | 1.6524 | 1.267 | 2.1736 | 0.6033 | 0.6611 | 0.7868
| (0.9358) | (0.9314) | (0.9402) | (0.9473) | (0.9518) | (0.9511)
40 | 0.14596 | 1.1098 | 1.7382 | 0.6099 | 0.6210 | 0.7239
| (0.9414) | (0.9337) | (0.9430) | (0.9510) | (0.9522) | (0.9517)
50 | 1.3370 | 1.0681 | 1.5426 | 0.6226 | 0.604 | 0.7239
| (0.9421) | (0.9385) | (0.9457) | (0.9523) | (0.9531) | (0.9549)
60 | 1.3029 | 0.9975 | 1.3441 | 0.6292 | 0.5536 | 0.6723
| (0.9441) | (0.9383) | (0.9457) | (0.9529) | (0.9537) | (0.9525)
70 | 1.2344 | 0.9652 | 1.2958 | 0.5849 | 0.5586 | 0.6421
| (0.9427) | (0.9405) | (0.9453) | (0.9525) | (0.9537) | (0.9544)
80 | 1.1892 | 0.9032 | 1.216 | 0.6277 | 0.5614 | 0.6515
| (0.9398) | (0.9439) | (0.9458) | (0.9538) | (0.9521) | (0.9513)
90 | 1.1532 | 0.8651 | 1.1555 | 0.5766 | 0.5229 | 0.6069
| (0.9454) | (0.9429) | (0.9451) | (0.9509) | (0.9517) | (0.9526)
100 | 1.1215 | 0.8227 | 1.0825 | 0.5687 | 0.5021 | 0.6022
| (0.9433) | (0.9416) | (0.9447) | (0.9526) | (0.9532) | (0.9541)
150 | 0.9949 | 0.6847 | 0.8899 | 0.5625 | 0.4576 | 0.5388
| (0.9433) | (0.9415) | (0.9443) | (0.9524) | (0.9529) | (0.9539)

Table 5: ALs of $95\%$ asymptotic confidence intervals (ACI) and Bayesian
credible intervals (BCI) and their corresponding CPs (in parentheses) with
$\alpha=0.3$, $\beta=0.5$, and $\theta=0.2$ with different values of $n$.

| ACI | BCI
---|---|---
$n$ | $\alpha$ | $\beta$ | $\theta$ | $\alpha$ | $\beta$ | $\theta$
20 | 1.2813 | 2.0610 | 0.6583 | 0.6726 | 0.8700 | 0.4641
| (0.9415) | (0.9389) | (0.9422) | (0.9514) | (0.9532) | (0.9508)
30 | 1.0218 | 1.4866 | 0.6019 | 0.6737 | 0.8299 | 0.4058
| (0.9421) | (0.9406) | (0.9427) | (0.9521) | (0.9519) | (0.9525)
40 | 0.9584 | 1.3192 | 0.5763 | 0.6762 | 0.8352 | 0.3747
| (0.9433) | (0.9396) | (0.9418) | (0.9529) | (0.9517) | (0.9524)
50 | 0.7901 | 1.1588 | 0.5533 | 0.6486 | 0.8080 | 0.3716
| (0.9428) | (0.9419) | (0.9436) | (0.9537) | (0.9541) | (0.9538)
60 | 0.7628 | 0.3015 | 0.5425 | 0.6259 | 0.8386 | 0.03270
| (0.9435) | (0.9450) | (0.9441) | (0.9522) | (0.9537) | (0.9515)
70 | 0.7443 | 0.9798 | 0.5283 | 0.6411 | 0.7595 | 0.3166
| (0.9424) | (0.9443) | (0.9449) | (0.9553) | (0.9540) | (0.9521)
80 | 0.6848 | 0.8818 | 0.4665 | 0.6312 | 0.7588 | 0.3139
| (0.9445) | (0.9437) | (0.9456) | (0.9539) | (0.9512) | (0.9496)
90 | 0.6621 | 0.8065 | 0.4317 | 0.6257 | 0.7613 | 0.2987
| (0.9428) | (0.9440) | (0.9425) | (0.9519) | (0.9507) | (0.9515)
100 | 0.6368 | 0.7683 | 0.4019 | 0.5967 | 0.7438 | 0.2788
| (0.9433) | (0.9417) | (0.9445) | (0.9529) | (0.9513) | (0.9536)
150 | 0.4625 | 0.5832 | 0.3793 | 0.1038 | 0.0899 | 0.2367
| (0.9443) | (0.9429) | (0.9408) | (0.9531) | (0.9547) | (0.9512)

Table 6: ALs of $95\%$ asymptotic confidence intervals (ACI) and Bayesian
credible intervals (BCI) and their corresponding CPs (in parentheses) with
$\alpha=1$, $\beta=0.7$, and $\theta=0.4$ with different values of $n$.

| ACI | BCI
---|---|---
$n$ | $\alpha$ | $\beta$ | $\theta$ | $\alpha$ | $\beta$ | $\theta$
20 | 2.6780 | 4.1893 | 0.5645 | 1.1012 | 1.0216 | 0.8413
| (0.9432) | (0.9417) | (0.9455) | (0.9526) | (0.9517) | (0.9529)
30 | 2.3077 | 3.1644 | 0.4515 | 1.0359 | 0.9473 | 0.3928
| (0.9420) | (0.9423) | (0.9437) | (0.9536) | (0.9524) | (0.9515)
40 | 2.1733 | 2.6883 | 0.3796 | 1.0886 | 0.9107 | 0.3059
| (0.9427) | (0.9429) | (0.9433) | (0.9518) | (0.9525) | (0.9547)
50 | 2.0280 | 2.4362 | 0.3431 | 0.9761 | 0.9139 | 0.2753
| (0.9450) | (0.9439) | (0.9427) | (0.9538) | (0.9516) | (0.9534)
60 | 1.9420 | 2.2549 | 0.3087 | 0.9644 | 0.9401 | 0.2284
| (0.9416) | (0.9428) | (0.9436) | (0.9528) | (0.9529) | (0.9545)
70 | 1.7909 | 2.0937 | 0.2894 | 0.9151 | 0.8737 | 0.2290
| (0.9409) | (0.9434) | (0.9443) | (0.9547) | (0.9538) | (0.9512)
80 | 1.7050 | 2.0370 | 0.2686 | 0.8812 | 0.9474 | 0.2188
| (0.9426) | (0.9432) | (0.9445) | (0.9533) | (0.9504) | (0.9526)
90 | 1.6319 | 1.9473 | 0.2511 | 0.8696 | 0.8996 | 0.2017
| (0.9428) | (0.9439) | (0.9419) | (0.9533) | (0.9528) | (0.9538)
100 | 1.5480 | 1.8356 | 0.2378 | 0.8772 | 0.8790 | 0.1814
| (0.9416) | (0.9445) | (0.9461) | (0.9539) | (0.9528) | (0.9495)
150 | 1.2775 | 1.6382 | 0.1941 | 0.7568 | 0.8428 | 0.1370
| (0.9425) | (0.9437) | (0.9448) | (0.9516) | (0.9525) | (0.9524)

##  5 Real data analysis

Here, two real life data sets are analyzed to illustrate the applicability of
the proposed distribution.

###  5.1 Flood level data set

A real life data set related to flood level has been considered to illustrate
the applicability of IGLFR distribution. This data set collected by United
States Water Resources Council Hydrology Committee (1977) contains $39$
observations of the annual flood discharge rates ($ft^{3}/s$) of the Floyed
river. Flood rates of rivers have a significant socio-economic, political
impact and also has an important role in engineering phenomenon. The data set
is given below:
————————————————————————————————————————–
1460, 4050, 3570, 2060, 1300, 1390, 1720, 6280, 1360, 7440, 5320, 1400, 3240,
2710, 4520, 4840, 8320, 13900, 71500, 6250, 2260, 318, 1330, 970, 1920, 15100,
2870, 20600, 3810, 726, 7500, 7170, 2000, 829, 17300, 4740, 13400, 1940, 5660.
————————————————————————————————————————–

Table 7: Goodness-of-fit test for the proposed IGLFR with some other
distributions (Dist) models for flood level data.

| Estimates (SE) |
---|---|---
Dist | $\alpha$ | $\beta$ | $\theta$ | K-S |  $p$-value
IGLFR | 2377.2233 | 2.2279 | 1.1717 | 0.0851 | 0.9173
GIE | 1.1716 | 2377.6415 |  | 0.0876 | 0.8999
GIW | 9.6911 | 1.0181 | 242.0750 | 0.0892 | 0.8884
IW | 2441.9167 | 1.0180 |  | 0.0893 | 0.8876
IPL | 1.0181 | 2444.1216 |  | 0.0894 | 0.8871
IG | 210.5542 | 10.1148 |  | 0.0905 | 0.8718

To fit this data set with the proposed IGLFR distribution, we compare it with
some well-known inverse distributions such as inverse Weibull (IW),
generalized inverse exponential (GIE), inverse power Lindley (IPL), inverse
Gompertz (IG) and generalized inverse Weibull (GIW) distributions. The
goodness-of-fit test has been compared of the above mentioned distribution
models by using Kolmogorov-Smirnov (K-S) distance and corresponding $p$-value.
Table 7 represents the MLEs of the parameters and the corresponding K-S
distance and associated $p$-values of the competing model distributions. From
Table 7, it has been observed that the smallest K-S distance and the largest
$p$-value has been computed for IGLFR distribution. Figure $6$ contains the
empirical CDF (ECDF) plot, probability-probability (P-P) plot, quantile-
quantile (Q-Q) plot, boxplot, histogram with density plot and TTT plot for
IGLFR distribution under the given real life data. This shows that our
proposed IGLFR distribution is a better fit to the given flood level data than
the other above mentioned inverse distributions. The point and interval
estimates of the model parameters of IGLFR distribution for the flood level
data have been tabulated in Table 8. From Table 8, it has been observed that
BCIs perform better than ACIs in terms of interval length.

Table 8: Point and interval estimates of the parameters of IGLFR distribution
for flood level data.

Parameter | MLE | ACI | Bayes | BCI
---|---|---|---|---
$\alpha$ | 2377.2233 | (1421.6925, 3332.7552) | 2418.4050 | (2373.3980, 2461.5000)
$\beta$ | 2.2279 | (0.4887, 3.9670) | 1.9793 | (1.5386, 2.4238)
$\theta$ | 1.1717 | (0.6660, 1.6775) | 1.2513 | (0.8811, 1.6287)

Figure 6: (a) Empirical CDF, (b) P-P plot, (c) Q-Q plot, (d) boxplot, (e)
histogram with theoretical density and (f) TTT plot for the IGLFR distribution
under the flood data.

###  5.2 Covid-19 data set

A Covid-19 data set from Liu et al. (2021) is considered to check the
applicability of the proposed IGLFR distribution model. This data set contains
mortality rates of Covid-19 patients of Canada which is given by
————————————————————————————————————————— 1.5157, 1.5806, 1.9048, 2.1901,
2.4141, 2.4946, 2.5261, 2.6029, 2.7704, 2.7957, 2.8349, 2.8636, 2.9078,
3.0914, 3.1091, 3.1091, 3.1444, 3.1348, 3.2110, 3.2135, 3.2218, 3.2823,
3.3592, 3.3769, 3.3825, 3.5146, 3.6346, 3.6426, 3.8594, 4.0480, 4.1685,
4.2202, 4.2781, 4.9274, 4.9378, 6.8686.
—————————————————————————————————————————

Table 9: Goodness-of-fit test for the proposed IGLFR distribution with some
other distributions (Dist) models for Covid-19 data.

| Estimates |
---|---|---
Dist | $\alpha$ | $\beta$ | $\theta$ | K-S |  $p$-value
IGLFR | 11.7507 | 1.7358 | 30.6509 | 0.1073 | 0.7614
GLFR | 0.6272 | 0.1843 | 12.4830 | 0.1081 | 0.7544
IW | 23.4053 | 3.1691 |  | 0.1737 | 0.2023
GIW | 1.7154 | 3.1692 | 4.2315 | 0.1738 | 0.2018
GIR | 1.2174 | 2.6005 |  | 0.2789 | 0.0057
IG | 0.0908 | 6.5107 |  | 0.2006 | 0.0961

To fit this data set with the proposed IGLFR distribution, we compare it with
some well-known inverse distributions such as generalized linear failure rate
(GLFR), inverse Weibull (IW), generalized inverse Rayleigh (GIR), inverse
Gompertz (IG) and generalized inverse Weibull (GIW) distributions. The
goodness-of-fit test has been compared of the above mentioned distribution
models by using Kolmogorov-Smirnov (K-S) distance and corresponding $p$-value.
Table 9 represents the MLEs of the parameters and the corresponding K-S
distance and associated $p$-values of the competing model distributions. From
Table 9, the smallest K-S distance and the largest $p$-value yields that IGLFR
distribution fits the given covid-19 data better than the other mentioned
distributions. Figure $7$ contains the empirical CDF (ECDF) plot, probability-
probability (P-P) plot, quantile-quantile (Q-Q) plot, boxplot, histogram with
density plot and TTT plot for IGLFR distribution under the given covid-19
data. This shows that our proposed IGLFR distribution is a better fit to the
covid-19 data than the other above mentioned inverse distributions. The point
and interval estimates of the model parameters of IGLFR distribution for the
covid-19 data have been tabulated in Table 10. From Table 10, it has been
observed that BCIs perform better than ACIs in terms of interval length.

Figure 7: (a) Empirical CDF, (b) P-P, (c) Q-Q plot, (d) boxplot, (e) histogram
with theoretical density and (f) TTT plotfor the IGLFR distribution under the
covid-19 data. Table 10: Point and interval estimates of the parameters of
IGLFR distribution for flood level data.

Parameter | MLE | ACI | Bayes | BCI
---|---|---|---|---
$\alpha$ | 11.7507 | (0.1104, 23.3909) | 10.9631 | (9.9737, 12.0084)
$\beta$ | 1.7315 | (0.1053, 3.2439) | 1.3119 | (0.7948, 1.8901)
$\theta$ | 30.6446 | (0.8895, 68.4806) | 24.9705 | (20.8091, 30.0026)

##  6 Conclusions

In this communication, a new statistical model, called as IGLFR distribution
has been introduced along with its statistical properties. Point as well as
interval estimates of the parameters have been obtained. We have obtained
MLEs, MPS and Bayes estimates. For Bayes estimates, SELL is used. A detailed
simulation study has been performed using various packages in $R$ software to
see the performance of the proposed estimates. From the simulated tables, it
has been noticed that Bayesian estimates are more efficient than classical
estimates based on average biaes and MSEs for point estimates and ALs and CPs
for interval estimates. Finally, two data sets are considered and analyzed.

Disclosure Statement: Both the authors state that they do not have any
conflict of interest.

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