# Image Denoising with Control over Deep Network Hallucination

Qiyuan Liang, Florian Cassayre, Haley Owsianko, Majed El Helou, Sabine
Süsstrunk
School of Computer and Communication Sciences, EPFL, Lausanne, Switzerland

###### Abstract

Deep image denoisers achieve state-of-the-art results but with a hidden cost.
As witnessed in recent literature, these deep networks are capable of
overfitting their training distributions, causing inaccurate hallucinations to
be added to the output and generalizing poorly to varying data. For better
control and interpretability over a deep denoiser, we propose a novel
framework exploiting a denoising network. We call it controllable confidence-
based image denoising (CCID). In this framework, we exploit the outputs of a
deep denoising network alongside an image convolved with a reliable filter.
Such a filter can be a simple convolution kernel which does not risk adding
hallucinated information. We propose to fuse the two components with a
frequency-domain approach that takes into account the reliability of the deep
network outputs. With our framework, the user can control the fusion of the
two components in the frequency domain. We also provide a user-friendly map
estimating spatially the confidence in the output that potentially contains
network hallucination. Results show that our CCID not only provides more
interpretability and control, but can even outperform both the quantitative
performance of the deep denoiser and that of the reliable filter, especially
when the test data diverge from the training data.

## Introduction

††footnotetext: Our code and models are publicly available at
https://github.com/IVRL/CCID

Additive Gaussian image denoising is one of the most fundamental tasks in
image restoration due to its numerous theoretical and practical uses. Various
denoising problems can be reduced to Additive White Gaussian Noise (AWGN)
removal with variance stabilization transforms, and AWGN denoisers can also
serve as regularizers [1, 2]. Although denoisers are omnipresent in imaging
pipelines, the denoising problem remains inherently challenging due to its
ill-posed nature, and the difficulty of image quality assessment itself [3].

Classic denoisers proposed various image priors, adding constraints to reduce
the ill-posed nature of the image denoising problem [4, 5]. With the emergence
of deep learning, novel solutions were developed, outperforming the classic
methods both quantitatively and qualitatively [6, 7, 8]. However, deep
learning methods have their own drawbacks, such as poor generalization on
unseen images from another domain and images with different noise levels [9].
Deep networks are also inherently a black box, making it difficult to
interpret their outputs. When deep networks make mistakes, having control over
the system and the ability to reason about the output becomes necessary [10].

To address the aforementioned problems, multiple approaches were proposed,
including increasing the overall robustness [11], modeling the confidence
value [10], detecting out-of-distribution (OOD) data [12], or improving
generalization [9, 13]. Here, we present a controllable confidence-based image
denoising (CCID) method. It is a novel framework addressing generalization and
interpretability in image denoising by exploiting reliable convolution filters
and giving control and insight to the user. As shown in Figure 1, CCID
denoises images with a convolution filter in parallel to the deep network.
Users fuse the two outputs to produce the final denoised image by exploiting
the image denoised by convolution, the deep network denoised image, and a
predicted confidence map.

Our contributions can be summarized as follows. We propose and evaluate
frequency-domain methods to fuse an image denoised by convolution and the deep
network denoised image. We provide a confidence prediction that reveals the
regions where the deep denoiser is likely to produce an error. We thus give
users the flexibility to smoothly fuse the image denoised by convolution and
the deep denoised image in the frequency domain based on confidence
predictions. This framework enables users to exploit the convolution denoising
when the deep network is likely inaccurate, hence safeguarding against
incorrect network hallucinated information. Our experimental results show that
beyond providing this control to users, our CCID outperforms both of its
underlying denoisers quantitatively. Such is the case with data from a
different domain, noise level, or noise type than that of the training data.

Figure 1: Our controllable confidence-based image denoising (CCID). The input
image is denoised in parallel with a convolution filter and with a deep neural
network, and the final output is obtained by fusing the two results. The deep
network output is further fed into the confidence prediction network to obtain
a confidence map. Users can visualize this map to interpret the network’s
result, or additionally apply it to steer the overall fusion.

## Related Work

Image denoising is a widely studied problem in the literature, particularly
the fundamental AWGN removal. Classic image denoising methods make various
assumptions to improve denoising. These assumptions form a prior that can
improve over the maximum likelihood solution for this ill-posed problem [14].
For instance, BM3D makes the assumption that there are similar patches within
an image [4] and applies collaborative filters to groups of similar patches.
WNNM assumes that images can be represented as low-rank matrices [5] and
exploits this attribute to filter out noise components.

With the advent of deep learning methods, richer priors could be learned over
training image distributions. Deep learning thus achieved superior denoising
performance, notably with DnCNN [6], where a residual connection is used to
directly predict the noise map rather than the denoised image, thus
alleviating the learning burden [6]. MWCNN introduces a new form of
downsampling and upsampling layers for building deep networks in the discrete
wavelet domain [7]. Feeding the network with differentiated low- and high-
frequency data guides the learning process and hence improves the performance.
DDFN integrates a dilated convolutional network with the deep boosting
framework to further boost the denoising performance [15]. To deal with blind
noise settings, FFDNet employs a customizable noise level map as input [16],
which, although not practical in test settings, can improve the results. BUIFD
presents a blind and universal denoiser for AWGN removal [14] improving the
generalization strength. This is achieved by exploiting the maximum a priori
solution under a theoretical prior and integrating it into the network’s
architecture along with an internal noise map estimation.

Although deep networks significantly improve the denoising results, they do so
by hallucinating data based on their rich learned priors. This reliance on
learned priors can cause inaccurate hallucinations, particularly when the test
image lies outside the training distribution. What we call hallucination is
the information derived from prior information and not directly inferred from
observed data. Unlike previous methods, we propose a framework to wrap deep
denoisers with reliable filtering techniques, and thus provide control over
the network hallucination.

Confidence in deep learning is a critical problem. While neural networks
achieve high accuracy, they can also provide unreliable outputs [17]. The
issue is that these unreliable outputs can have high confidence. Due to the
black-box nature of such networks, the lack of interpretability does not
enable any safe-guarding against such errors that can be critical in certain
applications like surveillance [18]. This problem is aggravated in the case of
test data that are Out Of Distribution (OOD) relative to the training set. To
address this issue, a baseline for assessing network confidence was introduced
in classification. It is based on the insight that successfully classified
examples have higher maximum softmax probabilities than incorrectly classified
OOD examples [12, 19].

Confidence has thus been explored in classification [20], but in the field of
image restoration, it has received less attention. Two recent papers developed
an architecture that incorporates confidence within the neural network to
guide the deraining process, with the confidence map calculated at multiple
levels to obtain higher accuracy [21, 22]. The epistemic uncertainty of deep
denoisers is exploited in [23] to create a virtual self ensemble and improve
denoising results. In contrast, our method estimates confidence to provide
more control and interpretability to the user of a denoising method. This
confidence can be exploited to guide the fusion of the reliable denoising
outputs and the deep denoiser outputs. The closest work to ours is the
BIGPrior framework [24]. It decouples the deep network hallucination from data
fidelity by exploiting a pretrained generative network projection prior. This
framework provides more interpretability to the user over deep hallucination.
In contrast, we additionally provide control to the user. We enable the user
to control the contribution of deep network outputs with a carefully designed
frequency fusion, while providing an estimated confidence map.

Image fusion is the process of combining complementary images from multiple
sensors, temporal domains, viewpoints, etc. [25]. We focus on the fusion of
the same image but restored using complementary methods. Classic image fusion
techniques can be divided into those carried out in the spatial or in the
frequency domain. Direct fusion in the spatial domain can cause undesired
distortions in image details [26]. As a result, it produces inferior results
compared to fusion in the frequency domain. To work in the frequency domain,
the discrete cosine transform (DCT) and the discrete wavelet transform (DWT)
are often exploited by fusion algorithms [27, 28, 29]. We make use of both
transforms in our experimental evaluation.

## Method

### Denoising

We term reliable denoisers the methods that rely on minimal priors to produce
a predictable output, with practically no hallucination. These approaches,
such as convolution filters, are robust against various noise settings or data
distributions. This is due to their simple prior assumptions that can
generalize well rather than overfit. The Gaussian filter is one such example.
It acts as a low-pass filter and, as a result, causes a certain loss of
sharpness but ensures no hallucinated information is added. We use the
Gaussian filter in our experiments for its consistency across images, its
computational efficiency, and as it requires no settings adjustments besides
the choice of the kernel. Other methods such as bilateral filtering and non-
local means (NLM) are also potential alternatives.

In parallel, we also exploit deep neural network denoisers in our CCID
framework. Deep learning approaches outperform traditional methods, however,
their rich priors are dependent on the training data and can cause incorrect
hallucinations. Hence, their strength can also be their main weakness when
test data differ from the training conditions (data distribution, degradation
model, noise level, etc.). This can be detrimental in practical settings
especially when users are not aware of the training conditions of a pretrained
denoiser. In our experiments, we rely on the widely used denoising model DnCNN
[6]. However, any other deep denoiser can be used to replace it in CCID.

We explain in what follows the two key components in our CCID framework. They
enable the fusion of both denoising techniques while providing user control
over network hallucinations and interpretability.

### Confidence prediction

We design a confidence prediction network that takes the initial noisy image,
the image denoised with the convolution filter, and the noise map predicted
from the deep denoising network as inputs. The objective is to determine the
reliability of the noise map prediction. We observed empirically that the
additional information and structural features provided by the initial noisy
image and the filtered image reduce the convergence time and improve the
accuracy of our confidence predictor. Our model predicts a confidence value
for each $8\times 8$ region in the image. Predicting the exact error of the
deep denoising network would imply that our confidence prediction network is
able to outperform the underlying denoising network by simply correcting its
errors. Instead, our objective is to estimate the extent of potential error
rather than the exact error itself. To that end, we define our target ground-
truth confidence $c_{GT}$ to be

$c_{GT}=1-\frac{\left\lVert
y_{GT}-y_{DNN}\right\lVert_{1}\downarrow_{8}}{\sigma_{max}},$ (1)

where $y_{GT}$ is the ground-truth noise-free image, $y_{DNN}$ is the denoised
image predicted by the deep neural network, $\left\lVert\cdot\right\lVert_{1}$
is the $\ell_{1}$ norm, $\downarrow_{8}$ is a downsampling by a factor of $8$
carried out with an average pooling operation, and the error is normalized to
$\sigma_{max}$ that we define next. In our experiments, we consider noise
levels between $\sigma=0$ and $100$, as is common in the literature.
Therefore, $\sigma_{max}$ is set to be $100$ for normalization, which is the
highest test noise level. Using the $8\times 8$ region provides a good balance
between spatial precision and network accuracy. This region-based technique
generates reasonably accurate predictions while also offering useful
confidence information that enables users to identify regions that may contain
incorrect hallucinations. We thus train a neural network to predict a
confidence $c$ (visualized in Figure 4), which is an estimate for $c_{GT}$.
This network is trained with the setup provided in the experimental evaluation
section.

### Proposed fusion

We propose two modes of fusion to combine images denoised by a reliable filter
and a deep denoiser. One uses a scalar parameter for the fusion, the other
exploits our confidence map. For the first type, we carry out the fusion in
the frequency domain to smoothly blend the structure of both images. We
considered two domain transforms, the DWT and the DCT for the fusion. The
second mode builds on our DWT fusion.

#### Wavelet and cosine fusion

With the DWT, the fusion is computed as the weighted average of the
transformed images, level by level. Two images of the same size go through the
same decomposition levels of the DWT, and the coefficients at each level are
retrieved accordingly. The fusion is then computed as the weighted average
between the two. For DCT fusion, the spectral power across is usually modeled
as a probability density function following a half-Gaussian distribution [30].
We propose a mask-based fusion approach in the DCT domain based on this
observation and empirical assessments. The mask is given by

$\displaystyle
M_{w}(\omega_{x},\omega_{y})=\exp(-(\omega_{x}^{2}+\omega_{y}^{2})/(2s)),$
$\displaystyle\text{where }s=a\left(\frac{1}{1-w+\epsilon}-1\right),$ (2)

and where $\omega_{x}$ and $\omega_{y}$ are the DCT frequency coefficient
indices; $w$ is the user-controlled fusion weight; $a$ is a scale factor
($0.1$ in practice); and $\epsilon$ is a small positive offset
($1\mathrm{e}{-3}$ in practice) to maintain the numerical stability.

The mask is used to perform the interpolation between the values of each
frequency, with the output matching the deep network denoised image when $M=1$
and the reliable image when $M=0$. The low frequencies of the deep denoised
image are fused first, followed by higher frequencies as the fusion weight
increases. This is because low frequencies are easier to reliably predict for
most image restoration tasks, hence we integrate them into our output before
higher frequencies.

#### Confidence-aware fusion

Our second, confidence-aware, fusion builds on the previous functions but
additionally relies on the confidence map. The latter contains spatial
reliability information that enables us to perform a fine-grained fusion. The
goal is to preserve the details when the risk of error is low, and otherwise
average them with the reliable filter to mitigate that risk. Note that the
confidence map is only an estimation and can itself be prone to errors.
However, as demonstrated in our experimental evaluation, the addition of this
component is still beneficial (Figure 7). With the DWT, we introduce a patch-
wise fusion, applying the DWT on each $8\times 8$ region and stitching back
the fusion results. Each $8\times 8$ region has a confidence value, therefore,
we use the confidence value per region to guide the fusion. We modulate the
fusion weight on each region such that

$w_{region}=w\cdot(1+c-t),$ (3)

where $w$ is again the user-controlled fusion weight, $c$ is the region’s
confidence value, and $t$ is a fixed threshold. If the confidence value is
higher than the threshold, we give the deep denoised image more weight to
obtain a better denoising result. The threshold is set to $0.8$ in practice
based on the in-distribution and OOD data, as discussed in the following
section.

## Experimental Evaluation

We use DnCNN [6] (trained on Gaussian noise $\sigma=25$) as the deep network
and a Gaussian kernel as the reliable denoising filter in our experiments. As
mentioned earlier, other methods can replace these denoisers in a straight-
forward manner in our CCID.

### Controllable fusion

(a) Gaussian

(b) $w=0.2$

(c) $w=0.4$

(d) $w=0.6$

(e) $w=0.8$

(f) Original

Figure 2: Visualization of our fusion techniques with different fusion weights
$w$. The Gaussian filter is a kernel with $\sigma=4$. The deep denoised image
is replaced by the ground-truth for comparison purposes. The first two rows
show the DCT fusion results, and the last two show the DWT ones (with zoom-in
crops). Observe that the edges gradually become sharper with increasing $w$.

$\displaystyle{0.0}$$\displaystyle{0.2}$$\displaystyle{0.4}$$\displaystyle{0.6}$$\displaystyle{0.8}$$\displaystyle{1.0}$Fusion
weight$\displaystyle{20}$$\displaystyle{30}$$\displaystyle{40}$$\displaystyle{50}$$\displaystyle{60}$Average
PSNR
(dB)DWTDCT$\displaystyle{0.0}$$\displaystyle{0.2}$$\displaystyle{0.4}$$\displaystyle{0.6}$$\displaystyle{0.8}$$\displaystyle{1.0}$Fusion
weight$\displaystyle{0.8}$$\displaystyle{0.9}$$\displaystyle{1.0}$Average SSIM
DWTDCT

Figure 3: Quantitative evaluation of the DCT and DWT fusion on the BSD68 [31]
image dataset. The experimental settings are identical to those in Figure 2.
Both approaches provide similar fusion performance, with an advantage to DWT
on larger weights.

Figure 2 shows the qualitative results of the proposed fusions, with DWT and
with DCT. We replace the deep denoised images with clean images for
illustrative purposes, and the Gaussian filter is applied to the clean images.
As can be observed, the fusion progresses smoothly from the Gaussian-filtered
to the clean image, with the increasing $w$ weight. In the zoom-in crops, we
see the edges and textures monotonically appearing in the second and fourth
row of Figure 2. We also evaluate the results using quantitative measurements,
with the peak-signal-to-noise ratio (PSNR) and the structural similarity index
measure (SSIM). The corresponding results are shown in Figure 3. The
progressive increase in PSNR and SSIM is in accordance with our qualitative
observations, with added high frequencies as the weight increases. For the
given metrics, the DWT and DCT fusion algorithms produce relatively similar
results. This smooth fusion in the frequency domain enables the CCID users to
control the contribution of reliable denoising and deep denoising in their
final output.

### Confidence prediction

Figure 4: Visualization of the confidence maps for noisy images ($\sigma=25$).
The threshold is $0.95$ for better visualization, and all the values above it
are green, while those below it are purple. The color intensity represents the
distance to the threshold. The confidence map is $8$ times smaller than the
input stack.

The training data of our confidence prediction network comprise varying noise
levels. It is trained on the same data generated by the DnCNN network trained
for Gaussian noise removal. The data could potentially be further extended to
cover different noise types and image domains. Our network produces meaningful
and precise confidence maps on held out test sets. Figure 4 shows the
predicted confidence map for each corresponding image. The confidence value is
high for low-frequency regions, such as the sky, walls, and plain T-shirts.
The confidence value is relatively low for high-frequency regions, such as
meadows, table cloth, and hair. Indeed, high-frequency component corruption is
harder to correct. As a result, high-frequency regions are areas where deep
networks are most likely to produce hallucinated errors [9], resulting in a
generally lower confidence score.

### Fusion with confidence

$\displaystyle{0.0}$$\displaystyle{0.2}$$\displaystyle{0.4}$$\displaystyle{0.6}$$\displaystyle{0.8}$$\displaystyle{1.0}$Confidence
valueProbability
distributionGaussianDnCNN$\displaystyle{0}$$\displaystyle{20}$$\displaystyle{40}$$\displaystyle{60}$$\displaystyle{80}$$\displaystyle{100}$Noise
sigma$\displaystyle{0.0}$$\displaystyle{0.2}$$\displaystyle{0.4}$$\displaystyle{0.6}$$\displaystyle{0.8}$$\displaystyle{1.0}$Average
confidenceGaussianDnCNN

Figure 5: Confidence distribution (left) for the Gaussian filter and DnCNN,
for noise levels $\sigma=0$ to $100$. Average confidence value (right) as a
function of the noise level. DnCNN confidence values decrease for increasing
noise levels after 0.4-0.5, which is corresponding to its training noise
level. As the noise sigma increases, the average confidence of the Gaussian
filter drops slowly, while that of DnCNN drops quickly after sigma exceeds 25.

(a) Ground-truth

(b) Noisy

(c) Gaussian

(d) DnCNN

(e) Confidence fusion

(f) Confidence

Figure 6: Confidence-aware DWT fusion for noise level $\sigma=25$. On the
zoomed-in images (bottom row), the left part has higher confidence than the
right part of the crop. The fused output is both sharp and reliable for the
left part. The right part is mostly based on the reliable-denoiser output and
is thus blurry. Note that we aim to be safe rather than overconfident, which
pays off in terms of quantitative performance for out-of-distribution data.

$\displaystyle{0.0}$$\displaystyle{0.2}$$\displaystyle{0.4}$$\displaystyle{0.6}$$\displaystyle{0.8}$$\displaystyle{1.0}$Fusion
weight$\displaystyle{22}$$\displaystyle{24}$$\displaystyle{26}$$\displaystyle{28}$$\displaystyle{30}$Average
PSNR
(dB)UnguidedGuided$\displaystyle{0.0}$$\displaystyle{0.2}$$\displaystyle{0.4}$$\displaystyle{0.6}$$\displaystyle{0.8}$$\displaystyle{1.0}$Fusion
weight$\displaystyle{0.75}$$\displaystyle{0.80}$$\displaystyle{0.85}$$\displaystyle{0.90}$Average
SSIM UnguidedGuided

Figure 7: Quantitative evaluation of the unguided and confidence-guided fusion
on BSD68 [31] with varying fusion weight.

(a) Ground-truth

(b) Noisy

(c) Gaussian

(d) DnCNN

(e) CCID

Figure 8: Sample zoomed-in fusion outputs that optimize PSNR and SSIM on OOD
data. The corresponding fusion weights are $0.25$, $0.2$, and $0.4$. Row 1 is
from the FMD dataset [32] with noise level $25$. Row 2 comes from the BSD
dataset with Gaussian noise $\sigma=35$. Row 3 is from the BSD dataset with
Poisson noise. In our DWT fused output, the incorrect hallucinations
introduced by DnCNN are mitigated, at the cost of sharpness.

To determine the appropriate threshold for Equation (3), the results of the
reliable filter and DnCNN need to be differentiated. We plot the confidence
value distribution for the two methods in Figure 5. When varying the noise
level from $\sigma=0$ to $100$ in our experiments, the results for the
Gaussian filter have an exponential distribution, whereas DnCNN’s distribution
is bi-modal. DnCNN is only trained on noise level $\sigma=25$, and does not
generalize very well. The peak of confidence values around 0.4-0.5 are due to
its better-than-average performance on noise level 25. For the other noise
levels, the confidence drops on average with increasing noise levels for both
denoisers, albeit faster for the deep network. To differentiate the two
approaches, we fix our threshold at $0.8$ in Equation (3) for our experiments.

Figure 6 shows the fusion results using a confidence guide. The zoomed-in
regions show two areas with disparate confidence values; the simple fabric on
the left has a higher confidence value than the threshold, while the right
textured part has a lower confidence value. For the confidence-aware fusion,
we intentionally put a higher weight on the confident region by scaling the
global fusion weight using Equation (3). The confidence-aware method performs
slightly better than the fusion method without confidence by preserving the
confident component from the deep denoiser. The proposed guide does not
significantly improve the overall quantitative performance for the in-
distribution denoising results, as shown in Figure 7. However, it gives more
control and interpretability to the users. When the test set is similar to the
training set, we almost always achieve the best performance at $w=1$ (deep
denoised image). However, this is not the case with OOD data. In real-life
scenarios, it is common to deal with data that is not similar to the training
set. We study this configuration in the next section.

### Out-of-distribution data

We consider three types of OOD data, namely, images from another domain,
images with noise levels different from the ones in the training set, and
images with a different noise type. As shown in Figure 8 and Table 1, the
fusion no longer achieves the best performance with $w=1$. Instead, in certain
cases, the results obtained with a Gaussian filter are already better than
those of DnCNN. The DnCNN network achieved a better performance than the
Gaussian filter only on the dataset from another domain, but not for varying
noise. These results highlight the generalizability problem of the deep
learning network, a problem that is not faced by the Gaussian filters. For the
microscopy data in Figure 8, we can observe strange line patterns near the
cell in DnCNN’s results that do not exist in the original noisy image.
Furthermore, as the noise level increases, DnCNN can no longer remove the
noise effectively. The training set covers only the noise level $\sigma=25$,
therefore, the model is not capable of predicting a noise map with a larger
standard deviation. When the input is deteriorated with a different noise
type, black dots appear in the denoised output of DnCNN, while the background
is left noisy. Our proposed fusion algorithm employs both the reliable filter
to maintain the basic structural information and the deep learning method to
obtain sharper, although potentially incorrect, details. Enabling users to
adjust the fusion weight offers another layer of protection against deep
network hallucination. Table 1 shows that CCID outperforms both of its
underlying denoisers. We show for reference the results of CCIDd that uses a
fixed default fusion weight, which already outperforms both denoisers on data
from a different noise level or noise type. The CCID results tweaked by a user
for best reconstruction outperform on all OOD types.

OOD Type | Gaussian | DnCNN | CCIDd | CCID
---|---|---|---|---
Data Domain | 32.48/0.91 | 35.17/0.95 | 34.41/0.94 | 35.20/0.95
Noise Level | 23.80/0.79 | 20.02/0.48 | 24.45/0.78 | 24.55/0.83
Noise Type | 23.92/0.80 | 21.60/0.62 | 24.69/0.72 | 25.01/0.81

Table 1: Average PSNR and SSIM, shown in this order, on out-of-distribution
(OOD) data. We vary the data domain distribution, the noise level, and the
type of noise of the test images. The experimental settings are identical to
those in Figure 8. The fusion weight in CCID is tweaked by the user for best
restoration. For reference, we also include for reference the results of CCIDd
that fixes a default invariable weight $w=0.5$.

## Conclusion

We present a novel denoising framework, CCID, where a reliable filter is
exploited alongside a deep network to provide control over network
hallucination. Our framework improves the overall interpretability and the
generalizability of the deep network by merging it with the reliable filter.
We further provide a confidence map that can either be integrated into our
fusion or visualized by the user. Our results show that our fusion provides
flexibility to the user without artifacts. It also improves the robustness
against out-of-distribution data, on which it outperforms both underlying
denoisers. Future work can study the generalization of our framework to other
restoration tasks.

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NEW CHARACTERIZATIONS OF THE GAMMA DISTRIBUTION VIA INDEPENDENCE OF TWO
STATISTICS BY USING
ANOSOV’S THEOREM

GWO DONG LIN 111Institute of Statistical Science, Academia Sinica, Nankang,
Taipei 11529, Taiwan (ROC); e-mail<EMAIL_ADDRESS>JORDAN M.
STOYANOV 222Institute of Mathematics and Informatics, Bulgarian Academy of
Sciences, 1113 Sofia, Bulgaria; e-mail<EMAIL_ADDRESS>
Abstract. Available in the literature are properties which characterize the
gamma distribution via independence of two appropriately chosen statistics.
Well-known is the classical result when one of the statistics is the sample
mean and the other one the sample coefficient of variation. In this paper, we
elaborate on a version of Anosov’s theorem which allows to establish a general
result, Theorem 1, and a series of seven corollaries providing new
characterization results for gamma distributions. We keep the sample mean as
one of involved statistics, while now the second one can be taken from a quite
large class of homogeneous feasible definite statistics. It is relevant to
mention that there is an interesting parallel between the new characterization
results for gamma distributions and recent characterization results for the
normal distribution.

Key words: characterization of distributions, order statistics, gamma
distribution, sample mean, sample coefficient of variation, Anosov’s theorem,
feasible definite statistics, normal distribution.

Mathematics Subject Classification 2020: 60E05, 62E10, 62G30

1\. Introduction. As usual we write $X\sim\gamma(a,b)$ and say that $X$ has a
gamma distribution with parameters $a>0$ and $b>0$ if $X$ is absolutely
continuous and has density

$f(x)=\frac{1}{\Gamma(a)\,b^{a}}\,x^{a-1}{\rm e}^{-x/b},\ \ x>0;\quad f(x)=0,\
x\leq 0.$

This distribution is well studied and widely used in both theory and
applications. Our main interests are properties which uniquely characterize
this distribution.

We start with a general situation. Suppose that $X\sim F$ is a non-degenerate
and positive random variable, so the distribution function $F$ (we use just ‘a
distribution’) has a support contained in ${\mathbb{R}}_{+}:=(0,\infty).$ Let
$X_{1},X_{2},\ldots,X_{n}$ be a random (independent) sample of size $n\geq 2$
from $F.$ Consider the sample mean
$\overline{X}_{n}:=\frac{1}{n}\sum_{i=1}^{n}X_{i}$, the sample variance
$S_{n}^{2}:=\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\overline{X}_{n})^{2},$ and the
sample deviation $S_{n}={\sqrt{S_{n}^{2}}}$. Well-known is the following
characterization property.

Classical result: Under the above conditions and notations, the sample mean
$\overline{X}_{n}$ and the sample coefficient of variation
$S_{n}/\overline{X}_{n}$ are independent _if and only if_ $F$ is a gamma
distribution.

Notice that for this general result there are no a priori moment or smoothness
conditions. We mention briefly that the proof of the sufficiency part is based
on the fact that if $F$ is a gamma distribution, then the sample mean
$\overline{X}_{n}$ is independent of the random vector
$(X_{1}/\overline{X}_{n},X_{2}/\overline{X}_{n},\ldots,X_{n}/\overline{X}_{n})$
(see [10], [9]). Hence $\overline{X}_{n}$ is independent of
$S_{n}^{2}/\overline{X}_{n}^{2},$ because the latter is a function of the
random vector. This in turn implies that $\overline{X}_{n}$ and
$S_{n}/\overline{X}_{n}$ are independent. For the proof of the necessity part
(see [6], p. 753).

The paper is organized as follows. In Section 2, we recall the definition of
feasible definite statistics as given and exploited in [3]. In Section 3 we
formulate the main results, Theorem 1 and a series of seven specific and
interesting corollaries. The needed tools and lemmas are given in Section 4.
In Section 5, we prove the main theorem. Finally, in Section 6 we make some
comments and outline two conjectures.

In what follows, we use the symbol $\perp\\!\\!\\!\perp$ to indicate that some
random quantities are ‘independent’. The abbreviation ‘iff’ is used for ‘if
and only if’, while ‘a.s.’ stands for ‘almost surely’.

2\. Feasible definite statistics. First we introduce a _base function_ ,
$U(\lambda),\lambda\in{\bf A}$, where ${\bf A}\subset\mathbb{R}^{n}$ is the
ordered set:

$\displaystyle{\bf
A}=\Big{\\{}\lambda:=(\lambda_{1},\lambda_{2},\ldots,\lambda_{n}):\
\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n},\
\Sigma_{i=1}^{n}\lambda_{i}=0\Big{\\}}.$ (1)

We say that $U$ is a _feasible definite function_ of positive degree $p$ of
homogeneity (briefly, of h-degree $p$), if it satisfies the following
conditions:
(i) $U$ is nonnegative and continuous on ${\bf A}$ (nonnegativity and
continuity);
(ii) $U(\lambda)=0$ iff $\lambda=(0,0,\ldots,0)$ (definiteness);
(iii) $U(s\lambda)=s^{p}\,U(\lambda)$ for all $s>0$ and all
$\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in{\bf A}$ (homogeneity

of degree $p$).

Suppose that $U(\lambda),\lambda\in{\bf A},$ is a feasible definite function
of h-degree $p$ and let $X_{(1)}\leq X_{(2)}\leq\cdots\leq X_{(n)}$ be the
order statistics of the random sample $X_{1},X_{2},\ldots,X_{n}$ of size
$n\geq 2$ from $F.$ We define

$\displaystyle
Z_{n}=U(X_{(1)}-\overline{X}_{n},X_{(2)}-\overline{X}_{n},\ldots,X_{(n)}-\overline{X}_{n}),$
(2)

and say that $Z_{n}$ is a _feasible definite statistic_ on ${\bf A}$ of
positive degree $p$ of homogeneity, or, simply, that $Z_{n}$ is a feasible
definite statistic of h-degree $p.$

It is easy to see that if $Z_{n}$ is a feasible definite statistic of h-degree
$p,$ then $(Z_{n})^{1/p}$ is a feasible definite statistic of h-degree $1.$
Notice that the sample mean $\overline{X}_{n}$ is however not a feasible
definite statistic because it is not of the form (2).

3\. Main results: Theorem 1 and Corollaries. We deal with a random variable
$X\sim F$ having positive continuous density $f$ on $(0,\infty).$ Let
$X_{(1)}\leq X_{(2)}\leq\cdots\leq X_{(n)}$ be the order statistics of a
random sample $X_{1},X_{2},\ldots,X_{n}$ of size $n\geq 3$ from $F$ and
$\overline{X}_{n}$ its sample mean. We formulate now our main result.

Theorem 1. Suppose that $U(\lambda),\lambda\in{\bf A},$ is a feasible definite
function of h-degree 1. Define
$Z_{n}=U(X_{(1)}-\overline{X}_{n},X_{(2)}-\overline{X}_{n},\ldots,X_{(n)}-\overline{X}_{n}),$
a feasible definite statistic of h-degree $1$. Then the sample mean
$\overline{X}_{n}$ and the quotient $V_{n}:=Z_{n}/\overline{X}_{n}$ are
independent iff $F$ is a gamma distribution. Symbolically:

$\overline{X}_{n}\ \perp\\!\\!\\!\perp\
V_{n}:=Z_{n}/\,\overline{X}_{n}\quad\Longleftrightarrow\quad F\mbox{ is a
gamma distribution}.$

Remark 1\. Notice that in the classical result given in the Introduction,
involved are two statistics, the sample mean $\overline{X}_{n}$ and the sample
coefficient of variation $S_{n}/\overline{X}_{n}.$ In Theorem 1, while keeping
one of the statistics, $\overline{X}_{n},$ we enjoy the freedom of choosing
the second statistic from a large class. This fact will be well illustrated by
formulating seven specific and diverse corollaries followed by useful brief
comments. For each corollary it will be clear which base function $U$ is used.

Remark 2\. Note also that if instead of ${\mathbb{R}}_{+}$ we assume that
$X\sim F$ has a positive continuous density $f$ on the whole real line
$\mathbb{R}=(-\infty,\infty),$ then the independence of the sample mean
$\overline{X}_{n}$ and the homogeneous feasible definite statistic $Z_{n}$
(rather than $V_{n}$), defined in (2) or in Theorem 1, will characterize $F$
as being a normal distribution. Therefore, our results for gamma distributions
can be considered in a sense as counterparts of the normal characterizations
derived recently in [3].

The statements in Corollaries 1 and 5 are known. They are obtained by a
different approach in [7] and are listed here for completeness of the general
picture and for comparison with the other corollaries which are new. In fact,
the known Corollaries 1 and 5 are special cases of our Corollaries 2 and 4, as
explained below.

Corollary 1. Let $R_{n}:=X_{(n)}-X_{(1)}$ be the range of the random sample.
Then the sample mean $\overline{X}_{n}$ and the quotient
$R_{n}/\overline{X}_{n}$ are independent iff $F$ is a gamma distribution,
i.e.,

$\overline{X}_{n}\ \perp\\!\\!\\!\perp\
R_{n}/\,\overline{X}_{n}\quad\Longleftrightarrow\quad F\mbox{ is a gamma
distribution}.$

Indeed, note that
$R_{n}=X_{(n)}-X_{(1)}=(X_{(n)}-\overline{X}_{n})-(X_{(1)}-\overline{X}_{n})$
is a feasible definite statistic of h-degree 1, so the conditions in Theorem 1
are satisfied.

Corollary 2. Let $a_{1},a_{2},\ldots,a_{n}$ be arbitrary real numbers, not all
coinciding, and such that $a_{1}\leq a_{2}\leq\cdots\leq a_{n}$. Define the
quantity

$Z_{n}=\sum_{i=1}^{n}a_{i}(X_{(i)}-\overline{X}_{n}).$ (3)

Then the sample mean $\overline{X}_{n}$ and the quotient
$Z_{n}/\overline{X}_{n}$ are independent iff $F$ is a gamma distribution,
i.e.,

$\overline{X}_{n}\ \perp\\!\\!\\!\perp\
Z_{n}/\,\overline{X}_{n}\quad\Longleftrightarrow\quad F\mbox{ is a gamma
distribution}.$

It is enough to see that $Z_{n}$ in (3) is a proper feasible definite
statistic of h-degree $1$. If $a_{1}=a_{2}=\cdots=a_{n}=a$, we have
$Z_{n}\equiv 0$, a degenerate case, which is not interesting.

Moreover, when $a_{1}=-1,\,a_{2}=\cdots=a_{n-1}=0$ and $a_{n}=1$, the quantity
$Z_{n}$ in (3) reduces to the sample range $R_{n}.$ Thus Corollary 2 is an
extension of Corollary 1.

Corollary 3. Given are $n+1$ real numbers,
$p,a_{1},a_{2},\ldots,a_{n-1},a_{n}$, such that $p>0,a_{1}>0,a_{n}>0,$ while
$a_{i}\geq 0,$ for $i=2,3,\ldots,n-1.$ Consider the quantity

$\displaystyle Z_{n}=\sum_{i=1}^{n}a_{i}|X_{(i)}-\overline{X}_{n}|^{p}.$ (4)

Then the sample mean $\overline{X}_{n}$ and the quotient
$(Z_{n})^{1/p}/\overline{X}_{n}$ are independent iff $F$ is a gamma
distribution, i.e.,

$\overline{X}_{n}\ \perp\\!\\!\\!\perp\
(Z_{n})^{1/p}/\,\overline{X}_{n}\quad\Longleftrightarrow\quad F\mbox{ is a
gamma distribution}.$

A little algebra shows that $Z_{n}$ in (4) is a feasible definite statistic of
h-degree $p.$ Hence $(Z_{n})^{1/p}$ is a feasible definite statistic of
h-order 1, just as required in Theorem 1.

It is worth mentioning that if $p=2$ and $a_{i}=1/(n-1)$ for all $i,$ the
quantity $Z_{n}$ in (4) reduces to the sample variance $S_{n}^{2}.$ Hence this
corollary includes the classical characterization result as a special case.

Corollary 4. Suppose that the real numbers $p$ and $a_{i,j}\geq 0,$ where
$1\leq i,\ j\leq n,$ are such that $p>0$ and $a_{1,n}+a_{n,1}>0.$ Consider the
quantity

$\displaystyle
Z_{n}=\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i,j}|X_{(i)}-X_{(j)}|^{p}.$ (5)

Then the sample mean $\overline{X}_{n}$ and the quotient
$(Z_{n})^{1/p}/\overline{X}_{n}$ are independent iff $F$ is a gamma
distribution, i.e.,

$\overline{X}_{n}\ \perp\\!\\!\\!\perp\
(Z_{n})^{1/p}/\,\overline{X}_{n}\quad\Longleftrightarrow\quad F\mbox{ is a
gamma distribution}.$

We easily check that $Z_{n}$ is a feasible definite statistic of h-degree $p,$
and as mentioned above, $(Z_{n})^{1/p}$ is such a statistic of h-degree 1. It
remains to refer to Theorem 1.

Corollary 5. Consider the well-known _Gini’s mean difference_

$G_{n}:=\frac{1}{n(n-1)}\sum_{i=1}^{n}\sum_{j=1}^{n}|X_{i}-X_{j}|.$ (6)

Then the sample mean $\overline{X}_{n}$ and the quotient
$G_{n}/\overline{X}_{n}$ are independent iff $F$ is a gamma distribution,
i.e.,

$\overline{X}_{n}\ \perp\\!\\!\\!\perp\
G_{n}/\,\overline{X}_{n}\quad\Longleftrightarrow\quad F\mbox{ is a gamma
distribution}.$

It is not immediately clear that $G_{n}$ is a feasible definite statistic. To
see this, we first check that $G_{n}$ in (6) can be transformed as follows:

$G_{n}=\frac{1}{n(n-1)}\sum_{i=1}^{n}\sum_{j=1}^{n}|X_{(i)}-X_{(j)}|=\frac{4}{n(n-1)}\sum_{i=1}^{n}\left(i-\frac{n+1}{2}\right)\,X_{(i)}.$

For the last representation, see [2], pp. 249, 279. It shows that $G_{n}$ is
indeed a feasible definite statistic of h-degree 1, and this is what we need
in Theorem 1.

Moreover, we easily observe that the quantity $Z_{n}$ in (5) reduces to
$G_{n}$ in (6) when $p=1$ and $a_{i,j}=\frac{1}{n(n-1)},$ thus Corollary 4 is
an extension of Corollary 5. Besides, Corollary 5 also follows from Corollary
2 due to the last identity for $G_{n}$ and the fact that
$\sum_{i=1}^{n}i=\frac{1}{2}n(n+1).$
Corollary 6. Let $(a_{ij})_{i,j=1}^{n}$ be a positive definite $($real$)$
matrix. Consider the quantity

$\displaystyle
Z_{n}=\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}(X_{(i)}-\overline{X}_{n})(X_{(j)}-\overline{X}_{n}).$
(7)

Then the sample mean $\overline{X}_{n}$ and the quotient
$(Z_{n})^{1/2}/\overline{X}_{n}$ are independent iff $F$ is a gamma
distribution, i.e.,

$\overline{X}_{n}\ \perp\\!\\!\\!\perp\
(Z_{n})^{1/2}/\,\overline{X}_{n}\quad\Longleftrightarrow\quad F\mbox{ is a
gamma distribution}.$

Notice that $Z_{n}$ in (7) is a feasible definite statistic of h-degree $2,$
hence $(Z_{n})^{1/2}$ is a feasible definite statistic of h-degree 1.

Corollary 7. For arbitrary real numbers $p>0,q>0,a_{ij}\geq 0$ for
$i,j=1,\ldots,n$ with $a_{11}>0$ and $a_{nn}>0,$ consider the quantity

$\displaystyle
Z_{n}=\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}\,|X_{(i)}-\overline{X}_{n}|^{p}\,|X_{(j)}-\overline{X}_{n}|^{q}.$
(8)

Then the sample mean $\overline{X}_{n}$ and the quotient
$(Z_{n})^{1/(p+q)}/\overline{X}_{n}$ are independent iff $F$ is a gamma
distribution, i.e.,

$\overline{X}_{n}\ \perp\\!\\!\\!\perp\
(Z_{n})^{1/(p+q)}/\,\overline{X}_{n}\quad\Longleftrightarrow\quad F\mbox{ is a
gamma distribution}.$

It follows from (8) that $Z_{n}$ is a feasible definite statistic of h-degree
$p+q,$ which implies that $(Z_{n})^{1/(p+q)}$ is a feasible definite statistic
of h-degree 1.

4\. Anosov’s theorem, its ramifications and auxiliary lemmas. In order to
prove Theorem 1 and see not only the parallel but also the difference between
characterizations of normal and gamma distributions, we recall first Anosov’s
theorem [1] for normal distributions (see [8], Chapter 4). We also provide two
ramifications of Anosov’s theorem, Theorem A and Theorem B below, and three
auxiliary lemmas.

Anosov’s Theorem. Let $n\geq 3$ be an integer and let $X\sim F$ have a
positive continuous density $f$ on $\mathbb{R}.$ Define the
$(n-2)$-dimensional torus

$\displaystyle\Phi=\\{\phi=(\phi_{1},\ldots,\phi_{n-2}):\phi_{j}\in[0,\pi],\
j=1,2,\ldots,n-3;\ \phi_{n-2}\in[0,2\pi]\\}.$ (9)

Let $T$ be a nonnegative continuous function on $\Phi$ such that
$\int_{\Phi}T(\phi)\,{\rm d}\phi\in(0,\infty).$ Assume further that
$\sigma_{j}(\phi),\ \phi\in\Phi,\ j=1,\ldots,n,$ are continuous functions
satisfying the following two conditions:

$\displaystyle\sum_{j=1}^{n}\sigma_{j}(\phi)=0\ \quad
and\quad\sum_{j=1}^{n}\sigma_{j}^{2}(\phi)\ \in(0,\infty)\ \ for\ all\
\phi\in\Phi.$

If the density $f$ satisfies the integro-functional equation

$\displaystyle\int_{\Phi}\Pi_{j=1}^{n}\,f(x+s\sigma_{j}(\phi))\,T(\phi)\,{\rm
d}\phi=c\,(f(x))^{n}\\!\\!\int_{\Phi}\Pi_{j=1}^{n}f(s\sigma_{j}(\phi))\,T(\phi)\,{\rm
d}\phi$ (10)

for all $x\in{\mathbb{R}}$ and all $s\geq 0$, where $c>0$ is a constant, then
$f$ is a normal density function.

In order to make one step more, we start with $n-2$ real numbers
$t_{1},\ldots,t_{n-2}$ and introduce $n-2$ functions
$g_{1},g_{2},\ldots,g_{n-2},$ where $g_{k}$ depends on the first $k$ arguments
$t_{1},\ldots,t_{k},$ as follows:

$g_{k}(t_{1},t_{2},\ldots,t_{k})=1-t_{1}^{2}-t_{2}^{2}-\cdots-t_{k}^{2},\quad
k=1,2,\ldots,n-2.$

The real vector $(t_{1},\ldots,t_{n-2})$ belongs to a set denoted by ${\bf
B}_{n-2},$ a subset of ${\mathbb{R}}^{n-2},$ and we follow a step-by-step
procedure to determine the range of the variables $t_{1},t_{2},$ etc., up to
$t_{n-2}.$ We start with $t_{1}$ and its range $[-1,-\frac{1}{n-1}],$
depending on $n$, hence $g_{1}(t_{1})$ is nonnegative. We continue and find
the range for $t_{2}$, via $n,t_{1}$ and $g_{1}^{1/2}$. Formally we define
${\bf B}_{n-2}$ as follows:

$\displaystyle{\bf B}_{n-2}$ $\displaystyle=$
$\displaystyle\Big{\\{}(t_{1},t_{2},\ldots,t_{n-2}):\ -1\leq
t_{1}\leq-\frac{1}{n-1},$
$\displaystyle\max\Big{\\{}\Big{[}\frac{n-k+2}{n-k}\Big{]}^{1/2}\cdot
t_{k-1},\ -g_{k-1}^{1/2}\Big{\\}}\leq
t_{k}\leq\frac{-g^{1/2}_{k-1}}{n-k},\quad k=2,\ldots,n-2\Big{\\}}.$

It can be checked that the function $g_{n-2}(t),\
t=(t_{1},\ldots,t_{n-2})\in{\bf B}_{n-2},$ is nonnegative and that the
function ${g^{-1/2}_{n-2}}$ is integrable (see Lemma 2 below).

In their analysis of the normal distribution, the authors of [3] applied the
following result which is a ramification of the above Anosov’s
characterization.

Theorem A. If the compact set $\Phi$ in $(9)$ is replaced by the set ${\bf
B}_{n-2}$ in $(11),$ then the conclusion of Anosov’s theorem remains true.

Let us focus now on tools and ideas allowing us to study gamma distributions.
The first one is an important result from [5], [6]; see Theorem B below. This
result is another ramification of Anosov’s theorem, covering a distribution
which is not normal. We use the notation $t:=(t_{1},t_{2},\ldots,t_{n-2})$ for
the points in the domain ${\bf B}_{n-2}.$

Theorem B. Let $n\geq 3$ be an integer, and let the random variable $X$,
$0<X\sim F,$ have a positive continuous density $f$ on $(0,\infty).$ Let $G$
be a distribution on the set ${\bf B}_{n-2}$ given in $(11).$ Further, assume
that $\sigma_{j}(t),j=1,\ldots,n,$ are continuous functions on ${\bf B}_{n-2}$
satisfying two conditions:

$\displaystyle\sum_{j=1}^{n}\sigma_{j}(t)=0\quad{\rm
and}\quad\sum_{j=1}^{n}\sigma_{j}^{2}(t)\ \in(0,\infty)\quad{for\ all}\
t\in{\bf B}_{n-2}.$

Suppose that the density $f$ satisfies the following integro-functional
equation:

$\displaystyle\int_{{\bf B}_{n-2}}\Pi_{j=1}^{n}f(x(s\sigma_{j}(t)+1))\,{\rm
d}G(t)$ (12) $\displaystyle=$
$\displaystyle{\tilde{c}}\,(f(x))^{n}\\!\\!\int_{{\bf
B}_{n-2}}\Pi_{j=1}^{n}f(s\sigma_{j}(t)+1)\,{\rm d}G(t)\quad for\ all\ x>0\
and\ all\ s\in[0,s_{0}),~{}$

where $\tilde{c}$ and $s_{0}$ are two positive constants. Then $f$ is a gamma
density.

Remark 3\. Comparing (10) and (12), we can see the difference between the
integro-functional equations in the normal and the gamma cases. The unknown
density $f$ depends on the parameters in quite a different way. Let us mention
that if we further assume in Theorem B that the density function $f$ is
continuously twice-differentiable on $(0,\infty),$ then the proof can be
essentially simplified by just differentiating the equation (12) twice (with
respect to $s$) at $s=0,$ and then solving the obtained ordinary differential
equation:

$[f^{\prime\prime}(x)f(x)-(f^{\prime}(x))^{2}]x^{2}=c^{\prime}f^{2}(x),\quad
x>0,$

where $c^{\prime}\in\mathbb{R}$ is a constant. It is interesting to note that
in such a derivation procedure, the functions $\sigma_{j}$ and the integral
over the set ${\bf B}_{n-2}$ including the distribution $G$ in (12) can be
canceled out. Thus, they eventually play no roles in the final equation to be
solved. Important is that under the smoothness and positivity conditions on
the density function $f$ on $(0,\infty),$ the solution $f$ of the above
equation is indeed a gamma density.

We need also two lemmas, given below; their proofs are given in [3].

Lemma 1. Let $S_{n}$ be the sample deviation of a sample from $F$ of size
$n\geq 3$ and let $Z_{n}$, defined in $(2)$, be a feasible definite statistic
on ${\bf A}$ of h-degree $1.$ Then there exist two positive constants
$c_{*}<c^{*}$ such that

$0<c_{*}\leq\frac{Z_{n}}{S_{n}}\leq c^{*}<\infty\ \ a.s.$

Lemma 2. Consider the function $g_{n-2}(t)=1-t_{1}^{2}-t_{2}^{2}-\cdots-
t_{n-2}^{2}$ introduced above. Then

$I_{n-2}:=\int_{{\bf B}_{n-2}}\frac{{\rm d}t_{1}\cdots{\rm
d}t_{n-2}}{\sqrt{g_{n-2}(t)}}=\frac{2\,\pi^{(n-1)/2}}{n!\Gamma((n-1)/2)}.$

Since $g_{n-2}$ is nonnegative and $I_{n-2}<\infty$, the normalized function

$\tilde{g}(t)=\frac{1}{I_{n-2}\,\sqrt{g_{n-2}(t)}},\quad t\in{\bf B}_{n-2},$

is a proper density function on the set ${\bf B}_{n-2}$ (see [4]). Similarly,
Lemmas 1 and 2 together will be used to define another distribution $G$
required in Theorem B when proving Theorem 1 (see relation (26) below).

As in the beginning of Section 3, let $X_{(1)}\leq X_{(2)}\leq\cdots\leq
X_{(n)}$ be the order statistics of a random sample $X_{1},X_{2},\ldots,X_{n}$
of size $n\geq 3$ from a distribution $F$ which has positive continuous
density $f$ on $(0,\infty).$ The corresponding realized ordered values are
$x_{(1)}\leq x_{(2)}\leq\cdots\leq x_{(n)},$ with their sample mean and sample
variance:

$\overline{x}_{n}=\frac{1}{n}\sum_{i=1}^{n}x_{(i)},\quad
s_{n}^{2}=\frac{1}{n-1}\sum_{i=1}^{n}(x_{(i)}-\overline{x}_{n})^{2}.$

Following [4], we define the transformation

$\displaystyle T^{*}_{1}:(x_{(1)},x_{(2)},\ldots,x_{(n)})\ \mapsto\
(t_{1},t_{2},\ldots,t_{n-2},w_{1},w_{2})$

as follows:

$\displaystyle\begin{cases}\displaystyle
t_{i}=\Big{[}\frac{n-i+1}{(n-1)(n-i)}\Big{]}^{1/2}\Big{[}\frac{x_{(i)}-\overline{x}_{n}}{s_{n}}+\frac{1}{n-i+1}\sum_{k=1}^{i-1}\frac{x_{(k)}-\overline{x}_{n}}{s_{n}}\Big{]},\
\ 1\leq i\leq n-2,\vspace{0.2cm}\\\ w_{1}=\overline{x}_{n},\vspace{0.1cm}\\\
w_{2}=s_{n}.\\\ \end{cases}$

The summation above in $t_{i}$ is taken to be zero if $i=1.$ Then we use the
functions $g_{k}(t_{1},\ldots,t_{k})=1-\sum_{i=1}^{k}t_{i}^{2},\ 1\leq k\leq
n-2,$ described before, and denote by $(T_{1}^{*})^{-1}$ the inverse
transformation to $T_{1}^{*}$:

$\displaystyle(T_{1}^{*})^{-1}:(t_{1},t_{2},\ldots,t_{n-2},w_{1},w_{2})\
\mapsto\ (x_{(1)},x_{(2)},\ldots,x_{(n)}).$

$(T_{1}^{*})^{-1}$ is defined through the following relations:

$\displaystyle\begin{cases}\displaystyle\frac{x_{(i)}-w_{1}}{w_{2}\sqrt{n-1}}=\Big{[}\frac{n-i}{n-i+1}\Big{]}^{1/2}\cdot
t_{i}-\sum_{k=1}^{i-1}\frac{t_{k}}{[(n-k)(n-k+1)]^{1/2}},\ \ 1\leq i\leq
n-2,\vspace{0.2cm}\\\
\displaystyle\frac{x_{(n-1)}-w_{1}}{w_{2}\sqrt{n-1}}=-\sum_{k=1}^{n-2}\frac{t_{k}}{[(n-k)(n-k+1)]^{1/2}}-[g_{n-2}/2]^{1/2},\vspace{0.1cm}\\\
\displaystyle\frac{x_{(n)}-w_{1}}{w_{2}\sqrt{n-1}}=-\sum_{k=1}^{n-2}\frac{t_{k}}{[(n-k)(n-k+1)]^{1/2}}+[g_{n-2}/2]^{1/2},\\\
\end{cases}$ (13)

where $\overline{x}_{n}=w_{1}$ and $s_{n}=w_{2}.$

Further, consider also another transformation, namely,

$\displaystyle T_{2}^{*}:(x_{(1)},x_{(2)},\ldots,x_{(n)})\ \mapsto\
(t_{1},t_{2},\ldots,t_{n-2},w_{1},w_{3}),$ (14)

where $w_{3}=w_{2}/w_{1},$ and for $n\geq 3,$ define two more subsets ${\bf
D}^{*}_{n}$ and ${\bf R}^{*}_{n}$ of ${\mathbb{R}}^{n}$:

$\displaystyle{\bf D}^{*}_{n}$ $\displaystyle=$
$\displaystyle\\{(x_{(1)},x_{(2)},\ldots,x_{(n)}):\ 0<x_{(1)}\leq
x_{(2)}\leq\cdots\leq x_{(n)}\\};$ (15) $\displaystyle{\bf R}^{*}_{n}$
$\displaystyle=$
$\displaystyle\Big{\\{}(t_{1},t_{2},\ldots,t_{n-2},w_{1},w_{3}):\
\max\Big{\\{}-1,\ -\frac{\sqrt{n}}{(n-1)w_{3}}\Big{\\}}\leq
t_{1}\leq-\frac{1}{n-1},$ (16)
$\displaystyle~{}~{}~{}\max\Big{\\{}\Big{[}\frac{n-k+2}{n-k}\Big{]}^{1/2}\cdot
t_{k-1},\ -g_{k-1}^{1/2}\Big{\\}}\leq t_{k}\leq-\frac{g^{1/2}_{k-1}}{n-k},$
$\displaystyle~{}~{}~{}\ 2\leq k\leq n-2,\ \ w_{1}>0,\ \
0<w_{3}\leq\sqrt{n}\Big{\\}}.$

We need one result more, Lemma 3 below, which is Theorem 2.2 in Hwang and Hu
(1994), rewritten here in terms of the above notations.

Lemma 3. For $n\geq 3,$ let $T_{2}^{*}$ be the transformation defined in
$(14)$. Then $T_{2}^{*}$ establishes a one-to-one correspondence between the
domain ${\bf D}^{*}_{n}$ and the range ${\bf R}^{*}_{n},$ defined in $(15)$
and $(16)$, respectively, except for a set of $n$-dimensional Lebesgue measure
zero. Furthermore, the absolute value of the Jacobian of this transformation
is

$\displaystyle|J_{2}^{*}|=\sqrt{n}\,(n-1)^{(n-1)/2}\cdot
w_{1}^{n-1}w_{3}^{n-2}\cdot g_{n-2}^{-1/2}.$

5\. Proof of Theorem 1.

Proof of Theorem 1. Step 1. Sufficiency. Since the degree of homogeneity of
the feasible definite statistic $Z_{n}$ is 1, it follows that:

$\displaystyle Z_{n}$ $\displaystyle=$ $\displaystyle
U(X_{(1)}-\overline{X}_{n},X_{(2)}-\overline{X}_{n},\ldots,X_{(n)}-\overline{X}_{n})$
$\displaystyle=$
$\displaystyle\overline{X}_{n}U((X_{(1)}-\overline{X}_{n})/\overline{X}_{n},(X_{(2)}-\overline{X}_{n})/\overline{X}_{n},\ldots,(X_{(n)}-\overline{X}_{n})/\overline{X}_{n})$
$\displaystyle=$
$\displaystyle\overline{X}_{n}U(X_{(1)}/\overline{X}_{n}-1,X_{(2)}/\overline{X}_{n}-1,\ldots,X_{(n)}/\overline{X}_{n}-1).$

Hence,

$\displaystyle\frac{Z_{n}}{\overline{X}_{n}}=U\left(\frac{X_{(1)}}{\overline{X}_{n}}-1,\frac{X_{(2)}}{\overline{X}_{n}}-1,\ldots,\frac{X_{(n)}}{\overline{X}_{n}}-1\right).$
(17)

The RHS of relation (17) is a function of the random vector
$\left(\frac{X_{(1)}}{\overline{X}_{n}},\frac{X_{(2)}}{\overline{X}_{n}},\ldots,\frac{X_{(n)}}{\overline{X}_{n}}\right).$
If $F$ is a gamma distribution, then $\overline{X}_{n}$ is independent of
$\left(\frac{X_{(1)}}{\overline{X}_{n}},\frac{X_{(2)}}{\overline{X}_{n}},\ldots,\frac{X_{(n)}}{\overline{X}_{n}}\right)$
and hence $\overline{X}_{n}$ is independent of ${Z_{n}}/{\overline{X}_{n}}.$
See also [10] or [9].

This proves the sufficiency part of the theorem.

Step 2. Necessity. Suppose that the sample mean $\overline{X}_{n}$ of a sample
of size $n$ coming from a distribution $F$ is independent of the quotient
$V_{n}=Z_{n}/\overline{X}_{n},$ where $Z_{n}$ is a homogeneous feasible
definite statistic of degree 1. Then we want to show that $F$ is a gamma
distribution. By the homogeneity property again, rewrite

$\displaystyle Z_{n}$ $\displaystyle=$ $\displaystyle
U(X_{(1)}-\overline{X}_{n},X_{(2)}-\overline{X}_{n},\ldots,X_{(n)}-\overline{X}_{n})=S_{n}\,U(\Lambda_{1},\Lambda_{2},\ldots,\Lambda_{n}),$

where

$\displaystyle\Lambda_{i}=\frac{X_{(i)}-\overline{X}_{n}}{S_{n}},\quad\
i=1,2,\ldots,n,$ (18)

and $(\Lambda_{1},\Lambda_{2},\ldots,\Lambda_{n})$ takes a.s. values
$(\tilde{\lambda}_{1},\tilde{\lambda}_{2},\ldots,\tilde{\lambda}_{n})$ in the
compact subset ${\bf A}_{n}\subset{\bf A}$,

$\displaystyle{\bf
A}_{n}=\Big{\\{}\lambda=(\lambda_{1},\ldots,\lambda_{n}):\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n},\
\small\sum_{i=1}^{n}\lambda_{i}=0,\
\sum_{i=1}^{n}\lambda_{i}^{2}=n-1\Big{\\}}.$

It then follows from Lemma 1 that there exist two positive constants
$c_{*}<c^{*}$ such that

$\displaystyle
0<c_{*}\leq\frac{Z_{n}}{S_{n}}=U(\Lambda_{1},\Lambda_{2},\ldots,\Lambda_{n})\leq
c^{*}<\infty\ \mbox{ a.s.}$ (19)

Write the realized values of

$S_{n}=\frac{Z_{n}}{U(\Lambda_{1},\Lambda_{2},\ldots,\Lambda_{n})}$

as

$\displaystyle
s_{n}=\frac{z_{n}}{U(\tilde{\lambda}_{1},\tilde{\lambda}_{2},\ldots,\tilde{\lambda}_{n})}:=\frac{z_{n}}{U(\tilde{\lambda})},\quad\tilde{\lambda}=(\tilde{\lambda}_{1},\ldots,\tilde{\lambda}_{n}),$
(20)

where each $\tilde{\lambda}_{i}$ is a function of
$t=(t_{1},t_{2},\ldots,t_{n-2})$ defined through (13) and (18).

Recall now that the joint density of the order statistics
$(X_{(1)},X_{(2)},\ldots,X_{(n)}),$ denoted by $h$, is expressed via the
1-dimensional density $f$ of $X$:

$h(x_{(1)},x_{(2)},\ldots,x_{(n)})=n!\prod_{i=1}^{n}f(x_{(i)}),\ \
0<x_{(1)}\leq x_{(2)}\leq\cdots\leq x_{(n)}.$

Next, consider the composition of two transformations, $T_{2}^{*}$ from (14)
and a new one, $T^{*},$ defined by

$T^{*}:\ (t_{1},t_{2},\ldots,t_{n-2},w_{1},w_{3})\ \mapsto\
(t_{1},t_{2},\ldots,t_{n-2},w_{1},v_{n}).$

Here, $w_{1}=\overline{x}_{n},\ w_{3}=s_{n}/w_{1},$
$v_{n}=z_{n}/w_{1}=U(\tilde{\lambda})w_{3}$ and $z_{n}$ is defined in (20).
The transformation $T^{*}$ has a Jacobian $J^{*}=1/U(\tilde{\lambda})$ which
is positive. Hence, the composition $T_{2}^{*}\circ T^{*}$ has a Jacobian with
absolute value (by Lemma 3):

$\displaystyle|J_{2}^{*}|J^{*}$ $\displaystyle=$
$\displaystyle\sqrt{n}\,(n-1)^{(n-1)/2}\cdot w_{1}^{n-1}w_{3}^{n-2}\cdot
g_{n-2}^{-1/2}(t)\cdot\frac{1}{U(\tilde{\lambda})}$ $\displaystyle=$
$\displaystyle\sqrt{n}\,(n-1)^{(n-1)/2}\cdot(\overline{x}_{n})^{n-1}v_{n}^{n-2}\Big{(}\frac{1}{U(\tilde{\lambda})}\Big{)}^{n-1}\cdot
g_{n-2}^{-1/2}(t),$

in which, to recall that $g_{n-2}(t)=1-\sum_{i=1}^{n-2}t_{i}^{2}.$ Therefore,
the joint density of the statistics $(T_{1},T_{2},\ldots,T_{n-2},$
$\overline{X}_{n},V_{n})$ is of the form:

$\tilde{h}(t_{1},t_{2},\ldots,t_{n-2},\overline{x}_{n},v_{n})=n!\prod_{i=1}^{n}f(x_{(i)})|J_{2}^{*}|J^{*},$

where each $T_{i}$, a function of order statistics $X_{(1)},\ldots,X_{(n)}$
(or of statistics $\Lambda_{1},\ldots,\Lambda_{n-2}$ in (18)), takes values
$t_{i}$ defined in the transformation $T_{1}^{*}.$

Notice now that by (13), (18) and (20), we have, for $i=1,2,\ldots,n,$ that

$x_{(i)}=\overline{x}_{n}+\frac{v_{n}\overline{x}_{n}}{U(\tilde{\lambda})}\cdot\tilde{\lambda}_{i}(t_{1},t_{2},\ldots,t_{n-2})=\overline{x}_{n}\Big{[}1+\frac{v_{n}}{U(\tilde{\lambda})}\cdot\tilde{\lambda}_{i}(t_{1},t_{2},\ldots,t_{n-2})\Big{]}.\quad$

Denote $v_{0}={c_{*}\sqrt{n}}/{(n-1)}.$ If $v_{n}\in(0,v_{0}),$ then we have,
by (19) and (20), that

$w_{3}\leq\frac{c_{*}}{U(\tilde{\lambda})}\frac{\sqrt{n}}{n-1}\leq\frac{\sqrt{n}}{n-1},$

and hence $(t_{1},t_{2},\ldots,t_{n-2})\in{\bf B}_{n-2}$, provided that
$(t_{1},t_{2},\ldots,t_{n-2},w_{1},w_{3})\in{\bf R}^{*}_{n}.$ This helps us to
simplify the domain of the joint density
$\tilde{h}(t_{1},t_{2},\ldots,t_{n-2},\overline{x}_{n},v_{n})$ when
$v_{n}\in(0,v_{0}).$ Thus, for $\overline{x}_{n}>0$ and $v_{n}\in(0,v_{0}),$
the 2-dimensional marginal joint density of $(\overline{X}_{n},V_{n})$ has the
following form (for simplicity, we use the notations
$\overline{x}=\overline{x}_{n},\ v=v_{n}$):

$\displaystyle\tilde{h}(\overline{x},v)$ $\displaystyle=$ $\displaystyle
n!\sqrt{n}\,(n-1)^{(n-1)/2}\cdot(\overline{x})^{n-1}v^{n-2}$ (21)
$\displaystyle\times\int_{{\bf
B}_{n-2}}\Big{(}\frac{1}{U(\tilde{\lambda})}\Big{)}^{n-1}\,g_{n-2}^{-1/2}(t)\prod_{i=1}^{n}f\Big{(}\overline{x}\big{[}1+\frac{v}{U(\tilde{\lambda})}\,\tilde{\lambda}_{i}(t)\big{]}\Big{)}\,{\rm
d}t_{1}\cdots{\rm d}t_{n-2},~{}~{}$

where $t=(t_{1},t_{2},\ldots,t_{n-2})\in{\bf
B}_{n-2}\subset{\mathbb{R}}^{n-2}$ defined in (11).

Now we apply the independence condition on $\overline{X}_{n}$ and $V_{n},$ and
write the joint density in (21) as the product of the densities of
$\overline{X}_{n}$ and $V_{n}:$

$\displaystyle\tilde{h}(\overline{x},v)=\tilde{h}_{\overline{X}_{n}}(\overline{x})\,\tilde{h}_{V_{n}}(v),\quad{\rm
if}\quad\overline{x}>0\quad{\rm and}\quad v\in(0,v_{0}).$ (22)

Letting $\overline{x}=1$ in (21) and (22), we obtain the density of $V_{n}:$

$\displaystyle\tilde{h}_{V_{n}}(v)$ $\displaystyle=$
$\displaystyle\frac{1}{\tilde{h}_{\overline{X}_{n}}(1)}\,\tilde{h}(1,v)$ (23)
$\displaystyle=$
$\displaystyle\frac{1}{\tilde{h}_{\overline{X}_{n}}(1)}\,n!\sqrt{n}\,(n-1)^{(n-1)/2}\cdot
v^{n-2}\int_{{\bf B}_{n-2}}\Big{(}\frac{1}{U(\tilde{\lambda})}\Big{)}^{n-1}$
$\displaystyle\times\,g_{n-2}^{-1/2}(t)\prod_{i=1}^{n}f\Big{(}1+\frac{v}{U(\tilde{\lambda})}\cdot\tilde{\lambda}_{i}(t)\Big{)}\,{\rm
d}t_{1}\cdots{\rm d}t_{n-2},\quad{\rm if}\ v\in(0,v_{0}).$

Plugging (21) and (23) in (22), canceling the common term
$n!\sqrt{n}\,(n-1)^{(n-1)/2}\cdot v^{n-2}$, then letting $v\to 0$ and
canceling the common integral term, we find the density of $\overline{X}_{n}:$

$\displaystyle\tilde{h}_{\overline{X}_{n}}(\overline{x})={\tilde{c}}\,(\overline{x})^{n-1}\,(f(\overline{x}))^{n},\quad\overline{x}>0;\quad\tilde{h}_{\overline{X}_{n}}(\overline{x})=0,\
\mbox{ otherwise,}$ (24)

where ${\tilde{c}}=\tilde{h}_{\overline{X}_{n}}(1)/(f(1))^{n}>0$ is a
constant.

Combining (21) through (24), we finally have the integro-functional equation :

$\displaystyle\int_{{\bf
B}_{n-2}}\Big{(}\frac{1}{U(\tilde{\lambda})}\Big{)}^{n-1}\cdot
g_{n-2}^{-1/2}(t)\prod_{i=1}^{n}f\Big{(}\overline{x}\big{(}1+\frac{v}{U(\tilde{\lambda})}\cdot\tilde{\lambda}_{i}(t)\big{)}\Big{)}\,{\rm
d}t_{1}\cdots{\rm d}t_{n-2}$ (25) $\displaystyle=$
$\displaystyle\\!\\!C\,(f(\overline{x}))^{n}\\!\\!\int_{{\bf
B}_{n-2}}\\!\Big{(}\frac{1}{U(\tilde{\lambda})}\Big{)}^{n-1}\\!\cdot
g_{n-2}^{-1/2}(t)\prod_{i=1}^{n}f\Big{(}1+\frac{v}{U(\tilde{\lambda})}\cdot\tilde{\lambda}_{i}(t)\Big{)}\,{\rm
d}t_{1}\cdots{\rm d}t_{n-2},~{}~{}~{}$

for all $\overline{x}>0$ and all $v\in[0,v_{0}),$ where $C=1/(f(1))^{n}>0$ is
a constant.

It is seen that (25) is of the form (12) in Theorem B, because, by Lemmas 1
and 2 and Eq. (18), we have

$M:=\int_{{\bf B}_{n-2}}\Big{(}\frac{1}{U(\tilde{\lambda})}\Big{)}^{n-1}\cdot
g_{n-2}^{-1/2}(t)\,{\rm d}t_{1}\cdots{\rm d}t_{n-2}\ \in\ (0,\infty),$
$\sum_{i=1}^{n}\sigma_{i}(t):=\sum_{i=1}^{n}\frac{\tilde{\lambda}_{i}(t)}{U(\tilde{\lambda})}=0\
\ \mbox{ for all }\ t\in{\bf B}_{n-2},$
$\sum_{i=1}^{n}\sigma_{i}^{2}(t)=\sum_{i=1}^{n}\Big{[}\frac{\tilde{\lambda}_{i}(t)}{U(\tilde{\lambda})}\Big{]}^{2}=\frac{n-1}{U^{2}(\tilde{\lambda})}\
\ \in(0,\infty)\ \ \mbox{ for all }\ t\in{\bf B}_{n-2}.$

Here, the last two conditions are required in Theorem B, and we may define the
distribution $G(t)$ for $t=(t_{1},t_{2},\ldots,t_{n-2})\in{\bf B}_{n-2},$ as
follows:

${\rm d}G(t):=\frac{1}{M}\Big{(}\frac{1}{U(\tilde{\lambda})}\Big{)}^{n-1}\cdot
g_{n-2}^{-1/2}(t)\,{\rm d}t_{1}\,\cdots{\rm d}t_{n-2}.$ (26)

Notice that here, $\overline{x}$ and $v$ play the roles of $x$ and $s$ in Eq.
(12), respectively. Therefore, by Theorem B, $f$ is a gamma density. This
completes the proof of Theorem 1. ∎

6\. Final comments and two conjectures. In our main results, Theorem 1 and
Corollaries 1 – 7, we assume that the sample size is $n\geq 3$ and impose
smoothness condition on the underlying distribution $F$ of $X.$ A natural
question arises:

Question. Is it possible to relax, or dispose, the smoothness condition on
$F\,?$

Here are some motivating arguments when looking closely to the case of a
sample size $n=2.$ In this case we have a simple relationship between the
sample variance $S_{2}^{2}$ and the sample range $R_{2},$ namely
$S_{2}^{2}=\frac{1}{2}R_{2}^{2}.$ Hence, if the sample mean $\overline{X}_{2}$
and the quotient $R_{2}/\overline{X}_{2}$ are independent, then so are
$\overline{X}_{2}$ and $S_{2}^{2}/\overline{X}_{2}^{2}.$ This in turn implies
that $\overline{X}_{2}$ and $S_{2}/\overline{X}_{2}$ are independent. The
latter property, according to the classical result, characterizes the
underlying distribution $F$ as being a gamma distribution. Notice that this
conclusion holds without any smoothness condition on $F$. Similarly, for
Gini’s mean difference in (6), we find that $G_{2}$ exactly equals $R_{2}.$
Therefore, if $\overline{X}_{2}$ and $G_{2}/\overline{X}_{2}$ are independent,
then $F$ is a gamma distribution.

These observations lead to the idea of eventually extending some of our
results, in particular, the statements in Corollary 1 and Corollary 5, without
assuming any smoothness condition on the underlying distribution $F.$

Conjecture 1. Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample of size $n\geq
3$ from _any_ nondegenerate distribution $F$ on $(0,\infty).$ If the sample
mean $\overline{X}_{n}$ and the quotient $R_{n}/\overline{X}_{n}$ are
independent, then $F$ is a gamma distribution.

Conjecture 2. Let $X_{1},X_{2},\ldots,X_{n}$ be a random sample of size $n\geq
3$ from _any_ nondegenerate distribution $F$ on $(0,\infty).$ If the sample
mean $\overline{X}_{n}$ and the quotient $G_{n}/\overline{X}_{n}$ are
independent, then $F$ is a gamma distribution.

Acknowledgment. We are grateful to Prof. Chin-Yuan Hu (National Changhua
University of Education, Taiwan) for the fruitful discussions and
collaboration over the years, and in particular for his useful comments when
preparing this paper.

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