The Estimators of the Bent, Shape and Scale Parameters of the Gamma-Exponential Distribution and Their Asymptotic Normality

Abstract

When modeling real phenomena, special cases of the generalized gamma distribution and the generalized beta distribution of the second kind play an important role. The paper discusses the gamma-exponential distribution, which is closely related to the listed ones. The asymptotic normality of the previously obtained strongly consistent estimators for the bent, shape, and scale parameters of the gamma-exponential distribution at fixed concentration parameters is proved. Based on these results, asymptotic confidence intervals for the estimated parameters are constructed. The statements are based on the method of logarithmic cumulants obtained using the Mellin transform of the considered distribution. An algorithm for filtering out unnecessary solutions of the system of equations for logarithmic cumulants and a number of examples illustrating the results obtained using simulated samples are presented. The difficulties arising from the theoretical study of the estimates of concentration parameters associated with the inversion of polygamma functions are also discussed. The results of the paper can be used in the study of probabilistic models based on continuous distributions with unbounded non-negative support.

1. Introduction

Gamma and beta classes of distributions play an important role in applied probability theory and mathematical statistics and have proven to be convenient and effective tools for modeling many real processes. The generalized gamma distribution and generalized beta distribution of the second kind are quite wide classes, including distributions that have such useful properties as, for example, infinite divisibility and stability, which makes it possible to use distributions from these classes as asymptotic approximations in various limit theorems. The article discusses the distribution proposed in the Ref. [1], that is closely related to the listed popular distributions.

Definition 1.

We say that the random variable ζ has the gamma-exponential distribution $GE(r,\nu ,s,t,\delta )$ with the parameters of bent $0\le r<1$, shape $\nu \ne 0$, concentration $s,t>0$, and scale $\delta >0$, if its density at $z>0$ is

$$g_E\left(z\right)=\frac{\left|\nu \right|z^{t\nu -1}}{{\delta }^{t\nu }\mathsf{\Gamma }\left(s\right)\mathsf{\Gamma }\left(t\right)}{\mathsf{Ge}}_{r,tr+s}(-{(z/\delta )}^{\nu }),$$

(1)

where $E=(r,\nu ,s,t,\delta )$ and ${\mathsf{Ge}}_{\alpha ,\beta }\left(z\right)$ are the gamma-exponential function [2]:

$${\mathsf{Ge}}_{\alpha ,\beta }\left(z\right)=\sum _{k=0}^{\infty }\frac{z^k}{k!}\mathsf{\Gamma }(\alpha k+\beta ),z\in \mathbb{R},0\le \alpha <1,\beta >0.$$

(2)

Function (2) generalizes to the case $\beta \ne 1$, which is the transformation introduced by Le Roy [3] to study generating functions of a special form. In addition, Function (2) can be considered (under some assumptions) as a special case of the Srivastava–Tomovski function [4], that generalizes the Mittag–Leffler function [5].

In the Ref. [1] it was shown that the distribution (1) adequately describes Bayesian balance models [6]. This is primarily due to the fact that the distribution with the density (1) can be represented as a scaled mixture of two random variables with generalized gamma distributions.

In turn, the generalized gamma distribution $GG(v,q,\theta )$ with the density

$$f\left(x\right)=\frac{\left|v\right|x^{vq-1}{e}^{-{(x/\theta )}^v}}{{\theta }^{vq}\mathsf{\Gamma }\left(q\right)},v\ne 0,q>0,\theta >0,x>0,$$

(3)

proposed in 1925 by the Italian economist Amoroso [7], has proven its validity in many applied problems that use continuous distributions with an unbounded non-negative support for modeling. The class of distributions (3) is wide enough and includes exponential distribution; ${\chi }^2$-distribution; Erlang distribution; gamma distribution; half-normal distribution (the distribution of the maximum of the Brownian motion process); Rayleigh distribution; Maxwell–Boltzmann distribution; $\chi$-distribution; Nakagami m-distribution; Wilson–Hilferty distribution; Weibull–Gnedenko distribution, and many others, including scaled and inverse analogs of the above.

In addition, it was shown in the Ref. [8] that the distribution (1) when $r\to 1$ gives in, is the limit of the generalized beta distribution of the second kind $GB2(\nu ,s,t,\delta )$ with the density

$$f\left(x\right)=\frac{\left|\nu \right|{\left(x/\delta \right)}^{t\nu -1}}{\delta B(s,t){\left(1+{\left(x/\delta \right)}^{\nu }\right)}^{t+s}},\nu \ne 0,s>0,t>0,\delta >0,x>0,$$

(4)

proposed in 1984 by McDonald [9]. The distribution (4), used primarily in econometrics and regression analysis, includes the Burr distribution (or Singh–Maddala distribution); Dagum distribution; Pearson distribution; Pareto distribution; Lomax distribution; the Fisher–Snedecor F-distribution, and others.

Traditionally, an important place in problems of applied mathematical statistics is occupied by the problem of estimating unknown distribution parameters. At the same time, in order to improve the consistency between mathematical models and analyzed real processes, researchers are considering increasingly complex mathematical abstractions. The relevance of the statistical analysis of the distributions (3) and (4), and their particular types and mixtures is evidenced by a large number of publications on this topic, for example, the Refs. [10,11,12,13,14,15,16,17,18,19].

In the Ref. [1], it was shown that the gamma-exponential distribution has the following properties.

Lemma 1.

1. Let the independent random variables λ and μ have the distributions $GG(v,q,\theta )$ and $GG(u,p,\alpha )$, $uv>0$, respectively. Then the distribution of λ coincides with $GE(0,v,·,q,\theta )$; the distribution of $\lambda /\mu$ for $\left|u\right|>\left|v\right|$ coincides with $GE(v/u,v,p,q,\theta /\alpha )$; and the distribution of $\lambda /\mu$ for $\left|v\right|>\left|u\right|$ coincides with $GE(u/v,-u,q,p,\theta /\alpha )$.

2. For $0<r<1$, the density $g_E\left(x\right)$, $E=(r,\nu ,s,t,\delta )$ coincides with the density of the ratio of independent random variables with generalized gamma distributions $GG(\nu ,t,\delta )$ and $GG(\nu /r,s,1)$.

The possibility of representing the gamma-exponential distribution as a ratio of random variables having a generalized gamma distribution allows it to be used in a wide range of applied problems [6,20]. In addition, the five-parameter gamma-exponential distribution can be used to model a wide range of real phenomena, due to the wide variety of its possible densities [20]. Besides, for a random variable $\zeta$ with a distribution (1), the following representation is valid [8]:

$$\zeta \stackrel{d}{=}\delta {\left(\frac{\lambda }{{\mu }^r}\right)}^{1/\nu },$$

(5)

where the independent random variables $\lambda$ and $\mu$ have gamma distributions $GG(1,t,1)$ and $GG(1,s,1)$, respectively. Moreover, if we put $r=0$, the right-hand part of the ratio (5) will have the distribution $GG(\nu ,t,\delta )$ [8], and for $r=1$, the right-hand part of (5) will have the distribution $GB2(\nu ,s,t,\delta )$ [19]. Consequently, the gamma-exponential distribution can be viewed as the distribution connecting and generalizing distributions from the gamma and beta classes.

In practice, the researcher deals with observable quantities that reflect the evolution of the analyzed real process. In relation to these quantities, some model assumptions are made about the form of their distribution. The problem of estimating unknown parameters from real data also arises in the case of modeling a real process using the gamma-exponential distribution. Due to the representation of the density (1) in terms of a special gamma-exponential Function (2), the maximum likelihood method seems to be too complicated. The same can be said about the direct method of moments, since the moments of distribution (1) can be represented as a product of non-monotone gamma functions [1]:

$$\mathsf{E}{\zeta }^m=\frac{{\delta }^m\mathsf{\Gamma }(t+m/\nu )\mathsf{\Gamma }(s-mr/\nu )}{\mathsf{\Gamma }\left(t\right)\mathsf{\Gamma }\left(s\right)},t+\frac{m}{\nu }>0,s-\frac{mr}{\nu }>0.$$

(6)

For this reason, In the Refs. [20,21] it was proposed to estimate the parameters of the gamma-exponential distribution using a modified method based on logarithmic moments. In this paper, we consider the estimators for three out of five parameters of the gamma-exponential distribution, constructed by the method of logarithmic cumulants.

The paper is organized as follows. Section 2 is devoted to the description of the method based on logarithmic cumulants; it provides an explicit form of theoretical logarithmic cumulants, their connection with logarithmic moments, as well as the form of strongly consistent estimators obtained using this method. Section 3 contains auxiliary relations necessary for formulating the main results. Section 4 contains the main results of the paper on the asymptotic normality of estimators for unknown parameters. In Section 5, a numerical analysis of the obtained results is carried out using the generated samples. The paper also contains the sections of discussions and conclusions.

2. Estimators for the Parameters of the Gamma-Exponential Distribution

This section defines the estimators for the parameters of bent r, shape $\nu$, and scale $\delta$ of the gamma-exponential distribution (1) for fixed values of the concentration parameters s and t. These estimators were obtained by equating the sample and theoretical cumulants of the gamma-exponential distribution.

Let us introduce the polygamma functions

$$\psi \left(z\right)=\frac{d}{dz}ln\mathsf{\Gamma }\left(z\right),{\psi }^{\left(m\right)}\left(z\right)=\frac{d^{m+1}}{d{z}^{m+1}}ln\mathsf{\Gamma }\left(z\right),m=1,2,\dots$$

To obtain an explicit form of theoretical logarithmic cumulants, consider the Mellin transform

$${\mathcal{M}}_{\zeta }\left(z\right)=\underset{0}{\overset{\infty }{\int }}{x}^{z}d{F}_{\zeta }\left(x\right),z\in \mathbb{C}.$$

We use Lemma 1 and the representation $\zeta \stackrel{d}{=}\lambda /\mu$, where the independent random variables $\lambda$ and $\mu$ have distributions $GG(\nu ,t,\delta )$ and $GG(\nu /r,s,1)$, respectively. For $\lambda \sim GG(\nu ,t,\delta )$, the Mellin transform has the form

$${\mathcal{M}}_{\lambda }\left(z\right)=\frac{{\delta }^z}{\mathsf{\Gamma }\left(t\right)}\mathsf{\Gamma }\left(t+\frac{z}{\nu }\right),t+\frac{\mathsf{Re}\left(z\right)}{\nu }>0.$$

Hence, for the ratio of $\lambda \sim GG(\nu ,t,\delta )$ to $\mu \sim GG(\nu /r,s,1)$

$${\mathcal{M}}_{\lambda /\mu }\left(z\right)=\frac{{\delta }^z}{\mathsf{\Gamma }\left(t\right)\mathsf{\Gamma }\left(s\right)}\mathsf{\Gamma }\left(t+\frac{z}{\nu }\right)\mathsf{\Gamma }\left(s-\frac{rz}{\nu }\right),t+\frac{\mathsf{Re}\left(z\right)}{\nu }>0,s-\frac{r\mathsf{Re}\left(z\right)}{\nu }>0,$$

from where we get the characteristic function of the logarithm of $\zeta$:

$$\mathsf{E}{e}^{iyln\zeta }=\frac{{\delta }^{iy}}{\mathsf{\Gamma }\left(t\right)\mathsf{\Gamma }\left(s\right)}\mathsf{\Gamma }\left(t+\frac{iy}{\nu }\right)\mathsf{\Gamma }\left(s-\frac{iry}{\nu }\right),y\in \mathbb{R}.$$

Thus, the cumulants of the random variable $ln\zeta$ for fixed s and t have the form

$${\kappa }_1(r,\nu ,\delta )=\mathsf{E}ln\zeta =\frac{\nu ln\delta +\psi \left(t\right)-r\psi \left(s\right)}{\nu };$$

$${\kappa }_m(r,\nu )={(-i)}^m\frac{d^m}{d{y}^m}ln\mathsf{E}{e}^{iyln\zeta }{|}_{y=0}=\frac{{\psi }^{(m-1)}\left(t\right)+{(-r)}^m{\psi }^{(m-1)}\left(s\right)}{{\nu }^m},m>1.$$

(7)

The moments of the random variable $ln\zeta$ can be represented as [22]

$${\mu }_m(r,\nu ,\delta )\equiv \mathsf{E}{ln}^m\zeta =B_m({\kappa }_1(r,\nu ,\delta ),{\kappa }_2(r,\nu ),\dots ,{\kappa }_m(r,\nu )),$$

(8)

where $B_m$ is a complete exponential Bell polynomial that can be recurrently defined as

$$B_{m+1}(x_1,\dots ,x_{m+1})=\sum _{k=0}^m{C}_m^k{B}_{m-k}(x_1,\dots ,x_{m-k})x_{k+1},B_0=1.$$

An explicit form of the necessary relations connecting moments and cumulants can be found in the Ref. [22].

In addition, we will need the following moment characteristics of the logarithm of a random variable with a gamma-exponential distribution, calculated using the Formula (8):

$${\sigma }_m^2(r,\nu ,\delta )\equiv \mathsf{D}{ln}^m\zeta ={\mu }_{2m}(r,\nu ,\delta )-{\mu }_m^2(r,\nu ,\delta );$$

(9)

$${\sigma }_{ml}(r,\nu ,\delta )\equiv \mathsf{cov}({ln}^m\zeta ,{ln}^l\zeta )={\mu }_{m+l}(r,\nu ,\delta )-{\mu }_m(r,\nu ,\delta ){\mu }_l(r,\nu ,\delta ).$$

(10)

To define the sample logarithmic cumulants, we introduce a notation for the sample logarithmic moments of the random variable $\zeta$:

$$L_m\left(X\right)=\frac{1}{n}\sum _{i=1}^n{ln}^m{X}_i,$$

(11)

where $X=(X_1,\dots ,X_n)$ is a sample from the distribution of $\zeta$.

Let us denote $l=(l_1,l_2,l_3,l_4)$. Consider the functions

$$K_1\left(l\right)\equiv K_1\left(l_1\right)={\left(\psi \left(s\right)\right)}^{-1}{l}_1;$$

$$K_2\left(l\right)\equiv K_2(l_1,l_2)={\left({\psi }^{\prime }\left(s\right)\right)}^{-1}(l_2-l_1^2);$$

$$K_3\left(l\right)\equiv K_3(l_1,l_2,l_3)={\left({\psi }^{″}\left(s\right)\right)}^{-1}(l_3-3l_2{l}_1+2l_1^3);$$

$$K_4\left(l\right)\equiv K_4(l_1,l_2,l_3,l_4)={\left({\psi }^{‴}\left(s\right)\right)}^{-1}(l_4-4l_3{l}_1-3l_2^2+12l_2{l}_1^2-6l_1^4).$$

Consider the statistics

$$K_1\left(X\right)\equiv K_1\left(L_1\left(X\right)\right);$$

$$K_2\left(X\right)\equiv K_2(L_1\left(X\right),L_2\left(X\right));$$

(12)

$$K_3\left(X\right)\equiv K_3(L_1\left(X\right),L_2\left(X\right),L_3\left(X\right));$$

$$K_4\left(X\right)\equiv K_4(L_1\left(X\right),L_2\left(X\right),L_3\left(X\right),L_4\left(X\right));$$

(13)

$$K\left(X\right)=(K_1\left(X\right),K_2\left(X\right),K_3\left(X\right),K_4\left(X\right)).$$

Note that the statistics ${\psi }^{(m-1)}\left(s\right)K_m\left(X\right)$ are the m-th sample logarithmic cumulants of the gamma-exponential distribution.

The method for estimating unknown parameters considered in the paper is based on solving the system for logarithmic cumulants:

$${\kappa }_m(r,\nu ,\delta )={\psi }^{(m-1)}\left(s\right)K_m\left(X\right),m=1,2,3,4.$$

(14)

To describe the solution of this system, we introduce a number of functions of sample logarithmic cumulants with the arguments $k=(k_1,k_2,k_3,k_4)$:

$${\varphi }_m=\frac{{\psi }^{\left(m\right)}\left(t\right)}{{\psi }^{\left(m\right)}\left(s\right)};\tau \left(k\right)\equiv \tau (k_2,k_4)={\varphi }_1^2{k}_4+{\varphi }_3\left(k_4-k_2^2\right);$$

(15)

$$R_{\pm }\left(k\right)\equiv R_{\pm }(k_2,k_4)=\sqrt{\frac{{\varphi }_1{k}_4\pm k_2\sqrt{\tau \left(k\right)}}{k_2^2-k_4}};$$

(16)

$$V_{\pm }\left(k\right)\equiv V_{\pm }(k_2,k_4)=\sqrt{\frac{{\varphi }_1{k}_2\pm \sqrt{\tau \left(k\right)}}{k_2^2-k_4}};$$

(17)

$$D_{\pm }\left(k\right)\equiv D_{\pm }(k_1,k_2,k_4)=exp\left\{\psi \left(s\right)k_1+\frac{\psi \left(s\right)R_{\pm }\left(k\right)-\psi \left(t\right)}{V_{\pm }\left(k\right)}\right\};$$

(18)

$${\Delta }_{\pm }\left(k\right)\equiv {\Delta }_{\pm }(k_2,k_3,k_4)=\left|\frac{{\varphi }_2-R_{\pm }^3\left(k\right)}{V_{\pm }^3\left(k\right)}-k_3\right|;$$

$$\Delta \left(k\right)\equiv \Delta (k_2,k_3,k_4)=min\{{\Delta }_{+}\left(k\right),{\Delta }_{-}\left(k\right)\};$$

$$R_{\Delta }\left(k\right)\equiv R_{\Delta }(k_2,k_3,k_4)=\sqrt{\frac{{\varphi }_1{k}_4-\mathrm{sgn}({\Delta }_{+}\left(k\right)-{\Delta }_{-}\left(k\right))k_2\sqrt{\tau \left(k\right)}}{k_2^2-k_4}};$$

$$V_{\Delta }\left(k\right)\equiv V_{\Delta }(k_2,k_3,k_4)=\sqrt{\frac{{\varphi }_1{k}_2-\mathrm{sgn}({\Delta }_{+}\left(k\right)-{\Delta }_{-}\left(k\right))\sqrt{\tau \left(k\right)}}{k_2^2-k_4}};$$

$$D_{\Delta }\left(k\right)\equiv D_{\Delta }(k_1,k_2,k_3,k_4)=exp\left\{\psi \left(s\right)k_1+\frac{\psi \left(s\right)R_{\Delta }\left(k\right)-\psi \left(t\right)}{V_{\Delta }\left(k\right)}\right\}.$$

The system (14) has several solutions. It was shown in the Ref. [23] that the estimators for the parameters of bent r, shape $\nu$, and scale $\delta$ have the form

$$\widehat{r}\left(X\right)=R_{\Delta }\left(K\left(X\right)\right);$$

(19)

$$\widehat{\nu }\left(X\right)=V_{\Delta }\left(K\left(X\right)\right);$$

(20)

$$\widehat{\delta }\left(X\right)=D_{\Delta }\left(K\left(X\right)\right),$$

(21)

and the following statement holds.

Lemma 2.

For fixed parameters s and t of the distribution $GE(r,\nu ,s,t,\delta )$, the estimators (19)–(21) for the parameters $0\le r<1$, $\nu >0$ and $\delta >0$ are strongly consistent.

Remark 1.

If it is known that $\nu <0$, one should consider the estimator $\widehat{\nu }\left(X\right)=-V_{\Delta }\left(K\left(X\right)\right)$ instead of $V_{\Delta }\left(K\left(X\right)\right)$, and the estimator

$$\widehat{\delta }\left(X\right)=exp\left\{\psi \left(s\right)K_1\left(X\right)-\frac{\psi \left(s\right)R_{\Delta }\left(K\left(X\right)\right)-\psi \left(t\right)}{V_{\Delta }\left(K\left(X\right)\right)}\right\}$$

(22)

instead of $D_{\Delta }\left(K\left(X\right)\right)$.

3. Auxiliary Relations

In what follows, we will need the derivatives of Functions (16)–(18) expressed in terms of the functions ${\varphi }_m$ and $\tau$, defined in (15). Note that

$$\begin{array}{c}\hfill R_{k_2,\pm }\left(k\right)\equiv \frac{\partial R_{\pm }}{\partial k_2}(k_2,k_4)=\mp \frac{k_4\left({\varphi }_1^2{k}_2^2+\tau \left(k\right)\pm 2{\varphi }_1{k}_2\sqrt{\tau \left(k\right)}\right)}{2{\left(k_2^2-k_4\right)}^{3/2}\sqrt{\tau \left(k\right)}\sqrt{{\varphi }_1{k}_4\pm k_2\sqrt{\tau \left(k\right)}}};\\ \hfill R_{k_4,\pm }\left(k\right)\equiv \frac{\partial R_{\pm }}{\partial k_4}(k_2,k_4)=\pm \frac{k_2\left({\varphi }_1^2{k}_2^2+\tau \left(k\right)\pm 2{\varphi }_1{k}_2\sqrt{\tau \left(k\right)}\right)}{4{(k_2^2-k_4)}^{3/2}\sqrt{\tau \left(k\right)}\sqrt{{\varphi }_1{k}_4\pm k_2\sqrt{\tau \left(k\right)}}};\\ \hfill V_{k_2,\pm }\left(k\right)\equiv \frac{\partial V_{\pm }}{\partial k_2}(k_2,k_4)=\mp \frac{k_2\left({\varphi }_1^2{k}_4+\tau \left(k\right)\right)\pm {\varphi }_1(k_2^2+k_4)\sqrt{\tau \left(k\right)}}{2{(k_2^2-k_4)}^{3/2}\sqrt{\tau \left(k\right)}\sqrt{{\varphi }_1{k}_2\pm \sqrt{\tau \left(k\right)}}};\\ \hfill V_{k_4,\pm }\left(k\right)\equiv \frac{\partial V_{\pm }}{\partial k_4}(k_2,k_4)=\pm \frac{{\varphi }_1^2{k}_2^2+\tau \left(k\right)\pm 2{\varphi }_1{k}_2\sqrt{\tau \left(k\right)}}{4{(k_2^2-k_4)}^{3/2}\sqrt{\tau \left(k\right)}\sqrt{{\varphi }_1{k}_2\pm \sqrt{\tau \left(k\right)}}};\\ \hfill D_{k_1,\pm }\left(k\right)\equiv \frac{\partial D_{\pm }}{\partial k_1}(k_1,k_2,k_4)=\psi \left(s\right)exp\left\{\psi \left(s\right)k_1+\frac{\psi \left(s\right)R_{\pm }\left(k\right)-\psi \left(t\right)}{V_{\pm }\left(k\right)}\right\};\\ \hfill D_{k_2,\pm }\left(k\right)\equiv \frac{\partial D_{\pm }}{\partial k_2}(k_1,k_2,k_4)=exp\left\{\psi \left(s\right)k_1+\frac{\psi \left(s\right)R_{\pm }\left(k\right)-\psi \left(t\right)}{V_{\pm }\left(k\right)}\right\}\times \\ \hfill \times \frac{\psi \left(t\right)V_{k_2,\pm }\left(k\right)+\psi \left(s\right)R_{k_2,\pm }\left(k\right)V_{\pm }\left(k\right)-\psi \left(s\right)R_{\pm }\left(k\right)V_{k_2,\pm }\left(k\right)}{V_{\pm }^2\left(k\right)};\\ \hfill D_{k_4,\pm }\left(k\right)\equiv \frac{\partial D_{\pm }}{\partial k_4}(k_1,k_2,k_4)=exp\left\{\psi \left(s\right)k_1+\frac{\psi \left(s\right)R_{\pm }\left(k\right)-\psi \left(t\right)}{V_{\pm }\left(k\right)}\right\}\times \\ \hfill \times \frac{\psi \left(t\right)V_{k_4,\pm }\left(k\right)+\psi \left(s\right)R_{k_4,\pm }\left(k\right)V_{\pm }\left(k\right)-\psi \left(s\right)R_{\pm }\left(k\right)V_{k_4,\pm }\left(k\right)}{V_{\pm }^2\left(k\right)}.\end{array}$$

(23)

Using the formula for the derivative of a composite function, we obtain

$$\begin{array}{c}\hfill \frac{\partial R_{\pm }}{\partial l_1}\left(l\right)=-\frac{2l_1}{{\psi }^{\prime }\left(s\right)}{R}_{k_2,\pm }(K_2\left(l\right),K_4\left(l\right))-\frac{4l_3-24l_2{l}_1+24l_1^3}{{\psi }^{‴}\left(s\right)}{R}_{k_4,\pm }(K_2\left(l\right),K_4\left(l\right));\\ \hfill \frac{\partial R_{\pm }}{\partial l_2}\left(l\right)=\frac{1}{{\psi }^{\prime }\left(s\right)}{R}_{k_2,\pm }(K_2\left(l\right),K_4\left(l\right))-\frac{6l_2-12l_1^2}{{\psi }^{‴}\left(s\right)}{R}_{k_4,\pm }(K_2\left(l\right),K_4\left(l\right));\\ \hfill \frac{\partial R_{\pm }}{\partial l_3}\left(l\right)=-\frac{4l_1}{{\psi }^{‴}\left(s\right)}{R}_{k_4,\pm }(K_2\left(l\right),K_4\left(l\right));\\ \hfill \frac{\partial R_{\pm }}{\partial l_4}\left(l\right)=\frac{1}{{\psi }^{‴}\left(s\right)}{R}_{k_4,\pm }(K_2\left(l\right),K_4\left(l\right));\\ \hfill \frac{\partial V_{\pm }}{\partial l_1}\left(l\right)=-\frac{2l_1}{{\psi }^{\prime }\left(s\right)}{V}_{k_2,\pm }(K_2\left(l\right),K_4\left(l\right))-\frac{4l_3-24l_2{l}_1+24l_1^3}{{\psi }^{‴}\left(s\right)}{V}_{k_4,\pm }(K_2\left(l\right),K_4\left(l\right));\\ \hfill \frac{\partial V_{\pm }}{\partial l_2}\left(l\right)=\frac{1}{{\psi }^{\prime }\left(s\right)}{V}_{k_2,\pm }(K_2\left(l\right),K_4\left(l\right))-\frac{6l_2-12l_1^2}{{\psi }^{‴}\left(s\right)}{V}_{k_4,\pm }(K_2\left(l\right),K_4\left(l\right));\\ \hfill \frac{\partial V_{\pm }}{\partial l_3}\left(l\right)=-\frac{4l_1}{{\psi }^{‴}\left(s\right)}{V}_{k_4,\pm }(K_2\left(l\right),K_4\left(l\right));\\ \hfill \frac{\partial V_{\pm }}{\partial l_4}\left(l\right)=\frac{1}{{\psi }^{‴}\left(s\right)}{V}_{k_4,\pm }(K_2\left(l\right),K_4\left(l\right));\\ \hfill \frac{\partial D_{\pm }}{\partial l_1}\left(l\right)=\frac{1}{\psi \left(s\right)}{D}_{k_1,\pm }(K_1\left(l\right),K_2\left(l\right),K_4\left(l\right))-\frac{2l_1}{{\psi }^{\prime }\left(s\right)}{D}_{k_2,\pm }(K_1\left(l\right),K_2\left(l\right),K_4\left(l\right))-\\ \hfill -\frac{4l_3-24l_2{l}_1+24l_1^3}{{\psi }^{‴}\left(s\right)}{D}_{k_4,\pm }(K_1\left(l\right),K_2\left(l\right),K_4\left(l\right));\\ \hfill \frac{\partial D_{\pm }}{\partial l_2}\left(l\right)=\frac{1}{{\psi }^{\prime }\left(s\right)}{D}_{k_2,\pm }(K_1\left(l\right),K_2\left(l\right),K_4\left(l\right))-\frac{6l_2-12l_1^2}{{\psi }^{‴}\left(s\right)}{D}_{k_4,\pm }(K_1\left(l\right),K_2\left(l\right),K_4\left(l\right));\\ \hfill \frac{\partial D_{\pm }}{\partial l_3}\left(l\right)=-\frac{4l_1}{{\psi }^{‴}\left(s\right)}{D}_{k_4,\pm }(K_1\left(l\right),K_2\left(l\right),K_4\left(l\right));\\ \hfill \frac{\partial D_{\pm }}{\partial l_4}\left(l\right)=\frac{1}{{\psi }^{‴}\left(s\right)}{D}_{k_4,\pm }(K_1\left(l\right),K_2\left(l\right),K_4\left(l\right)),\end{array}$$

(24)

where the partial derivatives of the functions $R_{\pm }\left(k\right)$, $V_{\pm }\left(k\right)$ and $D_{\pm }\left(k\right)$ are defined in the relations (23).

4. Asymptotic Normality of the Estimators for the Parameters of the Gamma-Exponential Distribution

Further arguments are based on the following statements [24].

Lemma 3.

In ${\mathbb{R}}^n$, the random vector $X_n$ converges in distribution to the random vector X if and only if each linear combination of the components of $X_n$ converges in a distribution to the same linear combination of the components of X.

Lemma 4.

Suppose that in ${\mathbb{R}}^m$,

$$\sqrt{n}(T_{n1},\dots ,T_{nm})⟹N\left(\mu ,\Sigma \right),n\to \infty ,$$

with Σ a covariance matrix. Let $g\left(t\right)=g(t_1,\dots ,t_m)$ be a real-valued function with a nonzero differential at $t=\mu$. Put

$$d=\left(\frac{\partial g}{\partial t_1}{|}_{t=\mu },\dots ,\frac{\partial g}{\partial t_m}{|}_{t=\mu }\right).$$

Then $\sqrt{n}g(T_{n1},\dots ,T_{nm})⟹N(g\left(\mu \right),d\Sigma d^T)$.

Let us formulate the statements about the asymptotic normality of the estimators (19)–(21) with fixed concentration parameters s and t.

Denote

$$\Sigma =\left(\begin{array}{cccc}{\sigma }_1^2(r,\nu ,\delta )& {\sigma }_{12}(r,\nu ,\delta )& {\sigma }_{13}(r,\nu ,\delta )& {\sigma }_{14}(r,\nu ,\delta )\\ {\sigma }_{12}(r,\nu ,\delta )& {\sigma }_2^2(r,\nu ,\delta )& {\sigma }_{23}(r,\nu ,\delta )& {\sigma }_{24}(r,\nu ,\delta )\\ {\sigma }_{13}(r,\nu ,\delta )& {\sigma }_{23}(r,\nu ,\delta )& {\sigma }_3^2(r,\nu ,\delta )& {\sigma }_{34}(r,\nu ,\delta )\\ {\sigma }_{14}(r,\nu ,\delta )& {\sigma }_{24}(r,\nu ,\delta )& {\sigma }_{34}(r,\nu ,\delta )& {\sigma }_4^2(r,\nu ,\delta )\end{array}\right);$$

(25)

$$d_{R_{\pm }}=\left(\frac{\partial R_{\pm }}{\partial l_1}\left(l\right){|}_{l=\mu },\frac{\partial R_{\pm }}{\partial l_2}\left(l\right){|}_{l=\mu },\frac{\partial R_{\pm }}{\partial l_3}\left(l\right){|}_{l=\mu },\frac{\partial R_{\pm }}{\partial l_4}\left(l\right){|}_{l=\mu }\right),$$

(26)

where the variances ${\sigma }_m^2(r,\nu ,\delta )$ and the covariances ${\sigma }_{ml}(r,\nu ,\delta )$ are defined in (9) and (10), respectively, and partial derivatives $\partial R_{\pm }/\partial l_k\left(l\right)$ are defined in (24).

Let ${\varphi }_m$, $m=1,3$ be defined in (15); $R_{\pm }(K_2\left(X\right),K_4\left(X\right))$ be defined in (16); $K_m\left(X\right)$, $m=2,4$, be defined in (12) and (13). The following statement holds.

Theorem 1.

Suppose that $r^2\ne ({\varphi }_3-{\varphi }_1^2)/\left(2{\varphi }_1\right)$.

1. Let $r^2>{\varphi }_3/{\varphi }_1$. Then the estimator $\widehat{r}\left(X\right)$ for the unknown parameter r has the form $\widehat{r}\left(X\right)=R_{+}(K_2\left(X\right),K_4\left(X\right))$, and when $n\to \infty$ has the property of asymptotic normality:

$$\sqrt{n}\frac{\widehat{r}\left(X\right)-r}{\sqrt{d_{R_{+}}\Sigma d_{R_{+}}^T}}⟹N(0,1).$$

2. Let $r^2<{\varphi }_3/{\varphi }_1$. Then the estimator $\widehat{r}\left(X\right)$ for the unknown parameter r has the form $\widehat{r}\left(X\right)=R_{-}(K_2\left(X\right),K_4\left(X\right))$ and when $n\to \infty$ has the property of asymptotic normality:

$$\sqrt{n}\frac{\widehat{r}\left(X\right)-r}{\sqrt{d_{R_{-}}\Sigma d_{R_{-}}^T}}⟹N(0,1).$$

Proof of Theorem 1.

The sample logarithmic moments $L_m\left(X\right)$ defined in (11) are sums of independent identically distributed random variables with means ${\mu }_m(r,\nu ,\delta )$ defined in (8) and variances ${\sigma }_m^2(r,\nu ,\delta )/n$ defined in (9). Therefore, when $n\to \infty$, the statistics $L_m\left(X\right)$, $k=1,2,3,4$, together with any of their linear combinations, have the property of asymptotic normality with the corresponding limit means depending on ${\mu }_m(r,\nu ,\delta )$, and variances determined by the covariance matrix $\Sigma$ given in (25).

In addition, under the conditions of the theorem, the components of the vector $d_{R_{\pm }}$ defined in (26) are finite and the function $R_{\pm }(K_2\left(l\right),K_4\left(l\right))$ has a nonzero differential at the point $\mu =({\mu }_1(r,\nu ,\delta ),\dots ,{\mu }_4(r,\nu ,\delta ))$.

Thus, all conditions of Lemmas 3 and 4 are satisfied. Hence,

$$\sqrt{n}{R}_{\pm }(K_2\left(X\right),K_4\left(X\right))⟹N(R_{\pm }(K_2\left(\mu \right),K_4\left(\mu \right)),d_{R_{\pm }}\Sigma d_{R_{\pm }}^T).$$

Consider the limiting mean $R_{\pm }(K_2\left(\mu \right),K_4\left(\mu \right))$. Note that when $n\to \infty$

$$K_2\left(X\right)⟶\frac{{\varphi }_1+r^2}{{\nu }^2}\text{a.s.};$$

$$K_4\left(X\right)⟶\frac{{\varphi }_3+r^4}{{\nu }^4}\text{a.s.}$$

Therefore, for the function $\tau \left(k\right)$ defined in (15),

$$\tau (K_2\left(X\right),K_4\left(X\right))⟶\frac{{({\varphi }_1{r}^2-{\varphi }_3)}^2}{{\nu }^4}\text{a.s.}$$

when $n\to \infty$.

Let $r^2>{\varphi }_3/{\varphi }_1$. Then

$$R_{+}(K_2\left(X\right),K_4\left(X\right))⟶R_{+}(K_2\left(\mu \right),K_4\left(\mu \right))=r\text{a.s.},$$

and the statistic $\widehat{r}\left(X\right)=R_{+}(K_2\left(X\right),K_4\left(X\right))$ is a strongly consistent estimator for r.

Since

$$R_{-}(K_2\left(\mu \right),K_4\left(\mu \right))=\sqrt{\frac{2{\varphi }_1{\varphi }_3+({\varphi }_3-{\varphi }_1^2)r^2}{{\varphi }_1^2-{\varphi }_3+2{\varphi }_1{r}^2}},$$

the statistic $R_{-}(K_2\left(X\right),K_4\left(X\right))$ estimates the function $R_{-}(K_2\left(\mu \right),K_4\left(\mu \right))\ne r$ and does not satisfy the statement of Lemma 2.

If $r^2<{\varphi }_3/{\varphi }_1$, we similarly conclude that the statistic $\widehat{r}\left(X\right)=R_{-}(K_2\left(X\right),K_4\left(X\right))$ is a strongly consistent estimator for r and the statistic $R_{+}(K_2\left(X\right),K_4\left(X\right))$ does not satisfy the statement of Lemma 2. □

Let

$$d_{V_{\pm }}=\left(\frac{\partial V_{\pm }}{\partial l_1}\left(l\right){|}_{l=\mu },\frac{\partial V_{\pm }}{\partial l_2}\left(l\right){|}_{l=\mu },\frac{\partial V_{\pm }}{\partial l_3}\left(l\right){|}_{l=\mu },\frac{\partial V_{\pm }}{\partial l_4}\left(l\right){|}_{l=\mu }\right);$$

$$d_{D_{\pm }}=\left(\frac{\partial D_{\pm }}{\partial l_1}\left(l\right){|}_{l=\mu },\frac{\partial D_{\pm }}{\partial l_2}\left(l\right){|}_{l=\mu },\frac{\partial D_{\pm }}{\partial l_3}\left(l\right){|}_{l=\mu },\frac{\partial D_{\pm }}{\partial l_4}\left(l\right){|}_{l=\mu }\right),$$

where the partial derivatives $\partial V_{\pm }/\partial l_k\left(l\right)$ and $\partial D_{\pm }/\partial l_k\left(l\right)$ are defined in (24).

Let ${\varphi }_m$, $m=1,3$ be defined in (15); $V_{\pm }(K_2\left(X\right),K_4\left(X\right))$ be defined in (17); $D_{\pm }(K_2\left(X\right)$, $K_4\left(X\right))$ be defined in (18); $K_m\left(X\right)$, $m=2,4$, be defined in (12) and (13); and the matrix $\Sigma$ be defined in (25).

Theorems 2 and 3 are proved in a completely similar way to Theorem 1.

Theorem 2.

Suppose that $r^2\ne ({\varphi }_3-{\varphi }_1^2)/\left(2{\varphi }_1\right)$, $\nu >0$.

1. Let $r^2>{\varphi }_3/{\varphi }_1$. Then the estimator $\widehat{\nu }\left(X\right)$ for the unknown parameter ν has the form $\widehat{\nu }\left(X\right)=V_{+}(K_2\left(X\right),K_4\left(X\right))$, and when $n\to \infty$ has the property of asymptotic normality:

$$\sqrt{n}\frac{\widehat{\nu }\left(X\right)-\nu }{\sqrt{d_{V_{+}}\Sigma d_{V_{+}}^T}}⟹N(0,1).$$

2. Let $r^2<{\varphi }_3/{\varphi }_1$. Then the estimator $\widehat{\nu }\left(X\right)$ for the unknown parameter ν has the form $\widehat{\nu }\left(X\right)=V_{-}(K_2\left(X\right),K_4\left(X\right))$, and when $n\to \infty$ has the property of asymptotic normality:

$$\sqrt{n}\frac{\widehat{\nu }\left(X\right)-\nu }{\sqrt{d_{V_{-}}\Sigma d_{V_{-}}^T}}⟹N(0,1).$$

Theorem 3.

Suppose that $r^2\ne ({\varphi }_3-{\varphi }_1^2)/\left(2{\varphi }_1\right)$, $\nu >0$.

1. Let $r^2>{\varphi }_3/{\varphi }_1$. Then the estimator $\widehat{\delta }\left(X\right)$ for the unknown parameter δ has the form $\widehat{\delta }\left(X\right)=D_{+}(K_2\left(X\right),K_4\left(X\right))$, and when $n\to \infty$ has the property of asymptotic normality:

$$\sqrt{n}\frac{\widehat{\delta }\left(X\right)-\delta }{\sqrt{d_{D_{+}}\Sigma d_{D_{+}}^T}}⟹N(0,1).$$

2. Let $r^2<{\varphi }_3/{\varphi }_1$. Then the estimator $\widehat{\delta }\left(X\right)$ for the unknown parameter δ has the form $\widehat{\delta }\left(X\right)=D_{-}(K_2\left(X\right),K_4\left(X\right))$ and when $n\to \infty$ has the property of asymptotic normality:

$$\sqrt{n}\frac{\widehat{\delta }\left(X\right)-\delta }{\sqrt{d_{D_{-}}\Sigma d_{D_{-}}^T}}⟹N(0,1).$$

Remark 2.

By analogy with the arguments of the Ref. [23] concerning the statement of Lemma 2, if it is known that $\nu <0$, it is easy to show that the statements of the Theorems 2 and 3 hold for the statistics $\widehat{\nu }\left(X\right)=-V_{\Delta }\left(K\left(X\right)\right)$ and $\widehat{\delta }\left(X\right)$ defined in (22) with the corresponding modification of the vectors of derivatives $d_{V_{\pm }}$ and $d_{D_{\pm }}$.

Denote

$$s_{mm}\left(X\right)\equiv {\sigma }_m^2(\widehat{r}\left(X\right),\widehat{\nu }\left(X\right),\widehat{\delta }\left(X\right));$$

(27)

$$s_{ml}\left(X\right)=s_{lm}\left(X\right)\equiv {\sigma }_{ml}(\widehat{r}\left(X\right),\widehat{\nu }\left(X\right),\widehat{\delta }\left(X\right));$$

(28)

$$d_r^{\left[m\right]}\left(X\right)\equiv \frac{\partial \widehat{r}\left(X\right)}{\partial l_m};d_{\nu }^{\left[m\right]}\left(X\right)\equiv \frac{\partial \widehat{\nu }\left(X\right)}{\partial l_m};d_{\delta }^{\left[m\right]}\left(X\right)\equiv \frac{\partial \widehat{\delta }\left(X\right)}{\partial l_m},$$

(29)

where ${\sigma }_m^2(r,\nu ,\delta )$ and ${\sigma }_{ml}(r,\nu ,\delta )$ are defined in (9) and (10), respectively, and the estimators $\widehat{r}\left(X\right)$, $\widehat{\nu }\left(X\right)$ and $\widehat{\delta }\left(X\right)$ satisfy the conditions of Theorems 1–3.

Corollary 1.

Suppose that the conditions of Theorems 1–3 are met; then,

$$\sqrt{n}\frac{\widehat{r}\left(X\right)-r}{\sqrt{{\sum }_{m=1}^4{\sum }_{l=1}^4{d}_r^{\left[m\right]}\left(X\right)s_{ml}\left(X\right)d_r^{\left[l\right]}\left(X\right)}}⟹N(0,1);$$

$$\sqrt{n}\frac{\widehat{\nu }\left(X\right)-\nu }{\sqrt{{\sum }_{m=1}^4{\sum }_{l=1}^4{d}_{\nu }^{\left[m\right]}\left(X\right)s_{ml}\left(X\right)d_{\nu }^{\left[l\right]}\left(X\right)}}⟹N(0,1);$$

$$\sqrt{n}\frac{\widehat{\delta }\left(X\right)-\delta }{\sqrt{{\sum }_{m=1}^4{\sum }_{l=1}^4{d}_{\delta }^{\left[m\right]}\left(X\right)s_{ml}\left(X\right)d_{\delta }^{\left[l\right]}\left(X\right)}}⟹N(0,1),$$

when $n\to \infty$, where $s_{ml}\left(X\right)$, $d_r^{\left[m\right]}\left(X\right)$, $d_{\nu }^{\left[m\right]}\left(X\right)$, $d_{\delta }^{\left[m\right]}\left(X\right)$ are defined in (27)–(29).

Proof of Corollary 2.

Due to the strong consistency of the estimators $\widehat{r}\left(X\right)$, $\widehat{\nu }\left(X\right)$, and $\widehat{\delta }\left(X\right)$, the quadratic form

$$\sum _{m=1}^4\sum _{l=1}^4{d}_r^{\left[m\right]}\left(X\right)s_{ml}\left(X\right)d_r^{\left[l\right]}\left(X\right)$$

converges almost surely to the normalizing function from the Theorem 1. Therefore, by Slutsky’s theorem, we obtain the statement of the Corollary 1 for the estimator of the parameter r. Similarly, we obtain statements for estimators of the parameters $\nu$ and $\delta$. □

Based on Corollary 1, it is possible to construct asymptotic confidence intervals for unknown parameters of the gamma-exponential distribution.

By $u_{\gamma }$, we denote the $(1+\gamma )/2$-quantile of the standard normal distribution.

Corollary 2.

Suppose that the conditions of Theorems 1–3 are met; then, asymptotic confidence intervals with the confidence level γ based on the estimators $\widehat{r}\left(X\right)$, $\widehat{\nu }\left(X\right)$, $\widehat{\delta }\left(X\right)$ for the unknown parameters r, ν, δ have the form

$$(A_r\left(X\right),B_r\left(X\right))=\left(\widehat{r}\left(X\right)-\frac{u_{\gamma }}{\sqrt{n}}{C}_r\left(X\right),\widehat{r}\left(X\right)+\frac{u_{\gamma }}{\sqrt{n}}{C}_r\left(X\right)\right);$$

$$(A_{\nu }\left(X\right),B_{\nu }\left(X\right))=\left(\widehat{\nu }\left(X\right)-\frac{u_{\gamma }}{\sqrt{n}}{C}_{\nu }\left(X\right),\widehat{\nu }\left(X\right)+\frac{u_{\gamma }}{\sqrt{n}}{C}_{\nu }\left(X\right)\right);$$

$$(A_{\delta }\left(X\right),B_{\delta }\left(X\right))=\left(\widehat{\delta }\left(X\right)-\frac{u_{\gamma }}{\sqrt{n}}{C}_{\delta }\left(X\right),\widehat{\delta }\left(X\right)+\frac{u_{\gamma }}{\sqrt{n}}{C}_{\delta }\left(X\right)\right),$$

where

$$C_r\left(X\right)=\sqrt{\sum _{m=1}^4\sum _{l=1}^4{d}_r^{\left[m\right]}\left(X\right)s_{ml}\left(X\right)d_r^{\left[l\right]}\left(X\right)};$$

$$C_{\nu }\left(X\right)=\sqrt{\sum _{m=1}^4\sum _{l=1}^4{d}_{\nu }^{\left[m\right]}\left(X\right)s_{ml}\left(X\right)d_{\nu }^{\left[l\right]}\left(X\right)};$$

$$C_{\delta }\left(X\right)=\sqrt{\sum _{m=1}^4\sum _{l=1}^4{d}_{\delta }^{\left[m\right]}\left(X\right)s_{ml}\left(X\right)d_{\delta }^{\left[l\right]}\left(X\right)},$$

and $s_{ml}\left(X\right)$, $d_r^{\left[m\right]}\left(X\right)$, $d_{\nu }^{\left[m\right]}\left(X\right)$, $d_{\delta }^{\left[m\right]}\left(X\right)$ are defined in (27)–(29).

Proof of Corollary 2.

The proof is based on the relation

$$\mathsf{P}\left(\left|\widehat{r}\left(X\right)-r\right|<\frac{u_{\gamma }}{\sqrt{n}}{C}_r\left(X\right)\right)=\mathsf{P}\left(\frac{\sqrt{n}}{C_r\left(X\right)}\left|\widehat{r}\left(X\right)-r\right|<u_{\gamma }\right)\asymp 2\mathsf{\Phi }\left(u_{\gamma }\right)-1=\gamma ,$$

from which we obtain the form of the confidence interval $(A_r\left(X\right),B_r\left(X\right))$. Similarly, asymptotic confidence intervals for the parameters $\nu$ and $\delta$ are obtained. □

5. Numerical Analysis of Theoretical Results

Let us consider the problem of obtaining numerical values of estimates for the parameters of bent r, shape $\nu$, and scale $\delta$ of the gamma-exponential distribution $GE(r,\nu ,s,t,\delta )$ for fixed values of concentration parameters s and t.

The method for obtaining estimators for the parameters r and $\nu$ is based on solving the system of equations [23], where the theoretical logarithmic cumulants (7) of the second and fourth orders are equated to their sample counterparts:

$$\frac{{\varphi }_1+r^2}{{\nu }^2}=K_2\left(X\right);$$

(30)

$$\frac{{\varphi }_3+r^4}{{\nu }^4}=K_4\left(X\right).$$

(31)

Note that the solutions [23]

$${\widehat{r}}_{\pm }^2\left(X\right)=\frac{{\varphi }_1{K}_4\left(X\right)\pm K_2\left(X\right)\sqrt{\tau \left(K\right(X\left)\right)}}{K_2^2\left(X\right)-K_4\left(X\right)};$$

(32)

$${\widehat{\nu }}_{\pm }^2\left(X\right)=\frac{{\varphi }_1{K}_2\left(X\right)\pm \sqrt{\tau \left(K\right(X\left)\right)}}{K_2^2\left(X\right)-K_4\left(X\right)}$$

do not uniquely determine the estimators for the parameters r and $\nu$, and whereas the sign of r is always known, $\nu$ can be either positive or negative. In addition, numerical experiments show that for a fixed sample, the expression (32) can give non-controversial estimates for any sign before the radical. For this reason, when processing real data, one should use the algorithm for filtering out unnecessary system solutions.

The algorithm for choosing the “correct” solution $(\widehat{r}\left(X\right),\widehat{\nu }\left(X\right))$ of the system is as follows.

At the first stage, one should try to determine the sign before the radical in the relation (32), using the domain of the parameter $r\in [0,1)$. According to the condition of Theorem 1, it is necessary to compare the value of ${\varphi }_3/{\varphi }_1$ with one. If ${\varphi }_3/{\varphi }_1\ge 1$, one should choose $\widehat{r}\left(X\right)=R_{-}\left(K\left(X\right)\right)$, where the function $R_{-}\left(k\right)$ is defined in (16). If ${\varphi }_3/{\varphi }_1<1$, the values of the right-hand sides of (32) are calculated for the given sample. If the value of ${\widehat{r}}_{\pm }^2\left(X\right)$ for some signs before the radical does not belong to the interval $[0,1)$, the corresponding solution is eliminated and the solution with the opposite sign before the radical is chosen as the estimate of r.

At the second stage, one should additionally use an equation similar to (30) and (31):

$$\frac{{\varphi }_2-r^3}{{\nu }^3}=K_3\left(X\right).$$

Since the estimators $\widehat{r}\left(X\right)$ and $\widehat{\nu }\left(X\right)$, along with the statistics $K_3\left(X\right)$, are continuous functions of sample logarithmic moments,

$$\left|\frac{{\varphi }_2-{\widehat{r}}^3\left(X\right)}{{\widehat{\nu }}^3\left(X\right)}-K_3\left(X\right)\right|⟶0\text{a.s.}$$

when $n\to \infty$.

For a fixed sample size n, this relation makes it possible to determine the “correct” solution of the system using the values

$${\Delta }_{\pm ,+}=\left|\frac{{\varphi }_2-R_{\pm }^3\left(K\left(X\right)\right)}{V_{\pm }^3\left(K\left(X\right)\right)}-K_3\left(X\right)\right|\mathrm{and}{\Delta }_{\pm ,-}=\left|\frac{{\varphi }_2-R_{\pm }^3\left(K\left(X\right)\right)}{-V_{\pm }^3\left(K\left(X\right)\right)}-K_3\left(X\right)\right|$$

based on the following criteria:

• If the result $\widehat{r}\left(X\right)={\widehat{r}}_{+}\left(X\right)$ is obtained at the first stage, one should choose the pair $(R_{+}\left(K\left(X\right)\right),V_{+}\left(K\left(X\right)\right))$ as the solution $(\widehat{r}\left(X\right),\widehat{\nu }\left(X\right))$ if ${\Delta }_{+,+}<{\Delta }_{+,-}$, and the pair $(R_{+}\left(K\left(X\right)\right),-V_{+}\left(K\left(X\right)\right))$ if ${\Delta }_{+,-}<{\Delta }_{+,+}$;
• If the result $\widehat{r}\left(X\right)={\widehat{r}}_{-}\left(X\right)$ is obtained at the first stage, one should choose the pair $(R_{-}\left(K\left(X\right)\right),V_{-}\left(K\left(X\right)\right))$ as the solution $(\widehat{r}\left(X\right),\widehat{\nu }\left(X\right))$ if ${\Delta }_{-,+}<{\Delta }_{-,-}$, and the pair $(R_{-}\left(K\left(X\right)\right),-V_{-}\left(K\left(X\right)\right))$ if ${\Delta }_{-,-}<{\Delta }_{-,+}$;
• If the values (32) obtained at the first stage are non-controversial for any sign before the radical, the solution $(\widehat{r}\left(X\right),\widehat{\nu }\left(X\right))$ should be chosen from four possible pairs $(R_{\pm }\left(K\left(X\right)\right),\pm V_{\pm }\left(K\left(X\right)\right))$, using the minimum of the values ${\Delta }_{+,+}$, ${\Delta }_{+,-}$, ${\Delta }_{-,+}$, ${\Delta }_{-,-}$ to determine the “correct” set of signs.

The estimate $\widehat{\delta }\left(X\right)$ of the scale parameter, $\delta$ is found from the equation for the first logarithmic cumulant

$$ln\delta +\frac{\psi \left(t\right)-r\psi \left(s\right)}{\nu }=L_1\left(X\right),$$

substituting the solution $(\widehat{r}\left(X\right),\widehat{\nu }\left(X\right))$ is found by the above algorithm instead of $(r,\nu )$, and has the form

$$\widehat{\delta }\left(X\right)=exp\left\{L_1\left(X\right)+\frac{\psi \left(s\right)\widehat{r}\left(X\right)-\psi \left(t\right)}{\widehat{\nu }\left(X\right)}\right\}.$$

(33)

Let us present some numerical results illustrating the method of choosing the “correct” estimates for the parameters of bent r, shape $\nu$, and scale $\delta$ at fixed concentration parameters s and t of the gamma-exponential distribution (1).

Table 1. Examples of estimates for the parameters of the model distribution $GE(0.5;2.5;2.4;1.9;1.0)$.

n	${\mathit{R}}_{-}\left(\mathit{K}\left(\mathit{X}\right)\right)$	${\mathbf{\Delta }}_{-,+}$	${\mathbf{\Delta }}_{-,-}$	$\pm {\mathit{V}}_{-}\left(\mathit{K}\left(\mathit{X}\right)\right)$	$\widehat{\mathit{\delta }}\left(\mathit{X}\right)$
${10}^5$	$0.5096$	$0.0008$	$0.2048$	$\pm 2.5179$	$1.0018$
$5\times {10}^5$	$0.5040$	$0.0009$	$0.2089$	$\pm 2.5038$	$1.0005$
${10}^6$	$0.4980$	$0.0001$	$0.2101$	$\pm 2.4985$	$0.9990$
$5\times {10}^6$	$0.5008$	$0.0001$	$0.2091$	$\pm 2.5013$	$1.0002$

,

Table 2. Examples of estimates for the parameters and boundaries of confidence intervals for a model distribution $GE(0.5;2.5;2.4;1.9;1.0)$.

n	$\widehat{\mathit{r}}\left(\mathit{X}\right)$	${\mathit{A}}_{\mathit{r}}\left(\mathit{X}\right)$	${\mathit{B}}_{\mathit{r}}\left(\mathit{X}\right)$	$\widehat{\mathit{\nu }}\left(\mathit{X}\right)$	${\mathit{A}}_{\mathit{\nu }}\left(\mathit{X}\right)$	${\mathit{B}}_{\mathit{\nu }}\left(\mathit{X}\right)$	$\widehat{\mathit{\delta }}\left(\mathit{X}\right)$	${\mathit{A}}_{\mathit{\delta }}\left(\mathit{X}\right)$	${\mathit{B}}_{\mathit{\delta }}\left(\mathit{X}\right)$
${10}^5$	$0.5096$	$0.3935$	$0.6258$	$2.5179$	$2.4197$	$2.6161$	$1.0018$	$0.9711$	$1.0325$
$5\times {10}^5$	$0.5040$	$0.4519$	$0.5560$	$2.5038$	$2.4603$	$2.5472$	$1.0005$	$0.9866$	$1.0143$
${10}^6$	$0.4980$	$0.4611$	$0.5349$	$2.4985$	$2.4680$	$2.5290$	$0.9990$	$0.9891$	$1.0088$
$5\times {10}^6$	$0.5008$	$0.4843$	$0.5173$	$2.5013$	$2.4877$	$2.5150$	$1.0002$	$0.9958$	$1.0046$

,

Table 3. Examples of estimates for the parameters of the model distribution $GE(0.7;-1.8;3.6;3.9;1.5)$.

n	${\mathit{R}}_{+}\left(\mathit{K}\left(\mathit{X}\right)\right)$	${\mathit{R}}_{-}\left(\mathit{K}\left(\mathit{X}\right)\right)$	${\mathbf{\Delta }}_{-,+}$	${\mathbf{\Delta }}_{-,-}$	$\pm {\mathit{V}}_{-}\left(\mathit{K}\left(\mathit{X}\right)\right)$	$\widehat{\mathit{\delta }}\left(\mathit{X}\right)$
${10}^5$	$1.1037$	$0.7556$	$0.1394$	$0.0132$	$\pm 1.8551$	$1.4415$
$5\times {10}^5$	$1.1254$	$0.7405$	$0.1526$	$0.0138$	$\pm 1.8347$	$1.4579$
${10}^6$	$1.1940$	$0.6962$	$0.1696$	$0.0013$	$\pm 1.7980$	$1.5036$
$5\times {10}^6$	$1.1828$	$0.7031$	$0.1681$	$0.0016$	$\pm 1.8018$	$1.4965$

,

Table 4. Examples of estimates for the parameters and boundaries of confidence intervals for a model distribution $GE(0.7;-1.8;3.6;3.9;1.5)$.

n	$\widehat{\mathit{r}}\left(\mathit{X}\right)$	${\mathit{A}}_{\mathit{r}}\left(\mathit{X}\right)$	${\mathit{B}}_{\mathit{r}}\left(\mathit{X}\right)$	$\widehat{\mathit{\nu }}\left(\mathit{X}\right)$	${\mathit{A}}_{\mathit{\nu }}\left(\mathit{X}\right)$	${\mathit{B}}_{\mathit{\nu }}\left(\mathit{X}\right)$	$\widehat{\mathit{\delta }}\left(\mathit{X}\right)$	${\mathit{A}}_{\mathit{\delta }}\left(\mathit{X}\right)$	${\mathit{B}}_{\mathit{\delta }}\left(\mathit{X}\right)$
${10}^5$	$0.7556$	$0.4082$	$1.1031$	$-1.8551$	$-2.1847$	$-1.5255$	$1.4415$	$1.0839$	$1.7991$
$5\times {10}^5$	$0.7405$	$0.5998$	$0.8813$	$-1.8347$	$-1.9662$	$-1.7033$	$1.4579$	$1.3090$	$1.6069$
${10}^6$	$0.6962$	$0.6182$	$0.7742$	$-1.7980$	$-1.8683$	$-1.7277$	$1.5036$	$1.4152$	$1.5920$
$5\times {10}^6$	$0.7031$	$0.6670$	$0.7392$	$-1.8018$	$-1.8345$	$-1.7691$	$1.4965$	$1.4560$	$1.5370$

,

Table 5. Examples of estimates for the parameters of the model distribution $GE(0.8;1.3;0.3;1.4;2.5)$.

n	${\mathit{R}}_{+}\left(\mathit{K}\left(\mathit{X}\right)\right)$	${\mathbf{\Delta }}_{+,+}$	${\mathbf{\Delta }}_{+,-}$	$\pm {\mathit{V}}_{+}\left(\mathit{K}\left(\mathit{X}\right)\right)$	$\widehat{\mathit{\delta }}\left(\mathit{X}\right)$
${10}^5$	$0.8254$	$0.0006$	$0.4652$	$\pm 1.3310$	$2.4735$
$5\times {10}^5$	$0.8146$	$0.0007$	$0.4565$	$\pm 1.3213$	$2.4893$
${10}^6$	$0.7958$	$0.0002$	$0.4525$	$\pm 1.2944$	$2.4996$
$5\times {10}^6$	$0.8009$	$0.0001$	$0.4548$	$\pm 1.3007$	$2.5035$

,

Table 6. Examples of estimates for the parameters and boundaries of confidence intervals for a model distribution $GE(0.8;1.3;0.3;1.4;2.5)$.

n	$\widehat{\mathit{r}}\left(\mathit{X}\right)$	${\mathit{A}}_{\mathit{r}}\left(\mathit{X}\right)$	${\mathit{B}}_{\mathit{r}}\left(\mathit{X}\right)$	$\widehat{\mathit{\nu }}\left(\mathit{X}\right)$	${\mathit{A}}_{\mathit{\nu }}\left(\mathit{X}\right)$	${\mathit{B}}_{\mathit{\nu }}\left(\mathit{X}\right)$	$\widehat{\mathit{\delta }}\left(\mathit{X}\right)$	${\mathit{A}}_{\mathit{\delta }}\left(\mathit{X}\right)$	${\mathit{B}}_{\mathit{\delta }}\left(\mathit{X}\right)$
${10}^5$	$0.8254$	$0.6302$	$1.0207$	$1.3310$	$1.0545$	$1.6076$	$2.4735$	$2.2936$	$2.6535$
$5\times {10}^5$	$0.8146$	$0.7302$	$0.8989$	$1.3213$	$1.2015$	$1.4411$	$2.4893$	$2.4088$	$2.5698$
${10}^6$	$0.7958$	$0.7397$	$0.8519$	$1.2944$	$1.2149$	$1.3738$	$2.4996$	$2.4426$	$2.5566$
$5\times {10}^6$	$0.8009$	$0.7754$	$0.8264$	$1.3007$	$1.2646$	$1.3369$	$2.5035$	$2.4779$	$2.5290$

,

Table 7. Examples of estimates for the parameters of the model distribution $GE(0.6;-2.9;2.1;3.9;0.5)$.

n	${\mathit{R}}_{+}\left(\mathit{K}\left(\mathit{X}\right)\right)$	${\mathbf{\Delta }}_{+,+}$	${\mathbf{\Delta }}_{+,-}$	$\pm {\mathit{V}}_{+}\left(\mathit{K}\left(\mathit{X}\right)\right)$	$\widehat{\mathit{\delta }}\left(\mathit{X}\right)$
	${\mathit{R}}_{-}\left(\mathit{K}\left(\mathit{X}\right)\right)$	${\mathbf{\Delta }}_{-,+}$	${\mathbf{\Delta }}_{-,-}$	$\pm {\mathit{V}}_{-}\left(\mathit{K}\left(\mathit{X}\right)\right)$
${10}^5$	$0.6125$	$0.0009$	$0.0004$	$\pm 2.9217$	$0.4979$
	$0.3854$	$0.0120$	$0.0106$	$\pm 2.5055$
$5\times {10}^5$	$0.5871$	$0.0021$	$0.0006$	$\pm 2.8789$	$0.5023$
	$0.4075$	$0.0109$	$0.0094$	$\pm 2.5487$
${10}^6$	$0.6047$	$0.0013$	$0.0001$	$\pm 2.9081$	$0.4991$
	$0.3921$	$0.0117$	$0.0103$	$\pm 2.5181$
$5\times {10}^6$	$0.6017$	$0.0015$	$0.0001$	$\pm 2.9047$	$0.4995$
	$0.3948$	$0.0116$	$0.0100$	$\pm 2.5249$

and

Table 8. Examples of estimates for the parameters and boundaries of confidence intervals for a model distribution $GE(0.6;-2.9;2.1;3.9;0.5)$.

n	$\widehat{\mathit{r}}\left(\mathit{X}\right)$	${\mathit{A}}_{\mathit{r}}\left(\mathit{X}\right)$	${\mathit{B}}_{\mathit{r}}\left(\mathit{X}\right)$	$\widehat{\mathit{\nu }}\left(\mathit{X}\right)$	${\mathit{A}}_{\mathit{\nu }}\left(\mathit{X}\right)$	${\mathit{B}}_{\mathit{\nu }}\left(\mathit{X}\right)$	$\widehat{\mathit{\delta }}\left(\mathit{X}\right)$	${\mathit{A}}_{\mathit{\delta }}\left(\mathit{X}\right)$	${\mathit{B}}_{\mathit{\delta }}\left(\mathit{X}\right)$
${10}^5$	$0.6125$	$0.4204$	$0.8046$	$-2.9217$	$-3.3210$	$-2.5224$	$0.4979$	$0.4604$	$0.5355$
$5\times {10}^5$	$0.5871$	$0.4833$	$0.6909$	$-2.8789$	$-3.0904$	$-2.6674$	$0.5023$	$0.4814$	$0.5232$
${10}^6$	$0.6047$	$0.5408$	$0.6686$	$-2.9081$	$-3.0401$	$-2.7760$	$0.4991$	$0.4865$	$0.5117$
$5\times {10}^6$	$0.6017$	$0.5724$	$0.6309$	$-2.9047$	$-2.9650$	$-2.8445$	$0.4995$	$0.4938$	$0.5053$

provide examples of numerical values of parameter estimates obtained using the algorithm for eliminating unnecessary solutions and constructed from the samples of the size n from the model distribution (1) with a set of parameters $E=(r,\nu ,s,t,\delta )$ and examples of the boundary values of asymptotic confidence intervals with a confidence level $\gamma =0.95$ for these estimates.

A simulation of pseudo-random samples from the gamma-exponential distribution is based on the relation (5) and is carried out using standard tools in any programming language that has the ability to generate samples from the gamma distribution.

Table 1 shows the values of the estimates of the parameters r, $\nu$ and $\delta$, obtained from the sample of the size n from a model distribution with a set of parameters $E=(0.5;2.5;2.4;1.9;1.0)$. For this set of parameters, the inequality ${\varphi }_3/{\varphi }_1\ge 1$ holds, and therefore the first stage of the algorithm for eliminating unnecessary solutions gives the estimate $\widehat{r}\left(X\right)=R_{-}\left(K\left(X\right)\right)$. At the second stage, since ${\Delta }_{-,+}<{\Delta }_{-,-}$, the pair $(R_{-}\left(K\left(X\right)\right),V_{-}\left(K\left(X\right)\right))$ is selected as a solution $(\widehat{r}\left(X\right),\widehat{\nu }\left(X\right))$. The estimate $\widehat{\delta }\left(X\right)$ is obtained from the relation (33). Table 2 shows the values of the estimates $(\widehat{r}\left(X\right),\widehat{\nu }\left(X\right),\widehat{\delta }\left(X\right))$ obtained in the previous step and the boundaries of corresponding confidence intervals.

Table 3 shows the values of the estimates of the parameters r, $\nu$ and $\delta$ obtained from the sample of the size n from a model distribution with a set of parameters $E=(0.7;-1.8;3.6;3.9;1.5)$. For this set of parameters, the inequality ${\varphi }_3/{\varphi }_1<1$ holds, while the values of the statistics $R_{+}\left(K\left(X\right)\right)$ are outside the domain of the parameter r, and therefore the first stage of the algorithm for eliminating unnecessary solutions gives the estimate $\widehat{r}\left(X\right)=R_{-}\left(K\left(X\right)\right)$. At the second stage, since ${\Delta }_{-,-}<{\Delta }_{-,+}$, the pair $(R_{-}\left(K\left(X\right)\right),-V_{-}\left(K\left(X\right)\right))$ is selected as a solution $(\widehat{r}\left(X\right),\widehat{\nu }\left(X\right))$. The estimate $\widehat{\delta }\left(X\right)$ is obtained from the relation (33). Table 4 shows the values of the estimates $(\widehat{r}\left(X\right),\widehat{\nu }\left(X\right),\widehat{\delta }\left(X\right))$ obtained in the previous step, and the boundaries of corresponding confidence intervals.

Table 5 shows the values of the estimates of the parameters r, $\nu$ and $\delta$ obtained from the sample of the size n from a model distribution with a set of parameters $E=(0.8;1.3;0.3;1.4;2.5)$. For this set of parameters, the expression under the outer radical in $R_{-}\left(K\left(X\right)\right)$ is negative, so the first stage of the algorithm for eliminating unnecessary solutions gives the estimate $\widehat{r}\left(X\right)=R_{+}\left(K\left(X\right)\right)$. At the second stage, since ${\Delta }_{+,+}<{\Delta }_{+,-}$, the pair $(R_{+}\left(K\left(X\right)\right),V_{+}\left(K\left(X\right)\right))$ is selected as a solution $(\widehat{r}\left(X\right),\widehat{\nu }\left(X\right))$. The estimate $\widehat{\delta }\left(X\right)$ is obtained from the relation (33). Table 6 shows the values of the estimates $(\widehat{r}\left(X\right),\widehat{\nu }\left(X\right),\widehat{\delta }\left(X\right))$ obtained in the previous step, and the boundaries of corresponding confidence intervals.

Table 7 shows the values of the estimates of the parameters r, $\nu$ and $\delta$ obtained from the sample of the size n from a model distribution with a set of parameters $E=(0.6;-2.9;2.1;3.9;0.5)$. For this set of parameters, the inequality ${\varphi }_3/{\varphi }_1<1$ holds, while the values of both statistics $R_{+}\left(K\left(X\right)\right)$ and $R_{-}\left(K\left(X\right)\right)$ lie in the interval $[0,1)$. Therefore, at the second stage, since ${\Delta }_{+,-}=min\{{\Delta }_{+,+},{\Delta }_{+,-},{\Delta }_{-,+},{\Delta }_{-,-}\}$, the pair $(R_{+}\left(K\left(X\right)\right),-V_{+}\left(K\left(X\right)\right))$ is selected as a solution $(\widehat{r}\left(X\right),\widehat{\nu }\left(X\right))$. The estimate $\widehat{\delta }\left(X\right)$ is obtained from the relation (33). Table 8 shows the values of the estimates $(\widehat{r}\left(X\right),\widehat{\nu }\left(X\right),\widehat{\delta }\left(X\right))$ obtained in the previous step, and the boundaries of corresponding confidence intervals.

Remark 3.

In some cases, when processing real data using the above methods, the lengths of confidence intervals may unboundedly increase. This indicates that the conditions of Theorems 1–3 are violated, that is, either $r^2=({\varphi }_3-{\varphi }_1^2)/\left(2{\varphi }_1\right)$ or $r^2={\varphi }_3/{\varphi }_1$.

6. Discussion

The majority of models of real processes using continuous distributions with unbounded non-negative support operate with special cases of the generalized gamma distribution, proposed in the 1920s by the Italian economist Amoroso in the framework of the study of the dynamic equilibrium theory [7], and special cases of the generalized beta distribution of the second kind, proposed in the 1980s by McDonald as a generalization of the well-known beta-type distributions used to model profitability [9].

The study of probabilistic and statistical properties of distributions from the gamma and beta classes is very important. For example, in the Refs. [10,11,12,13,14], it was proposed to use the generalized gamma distribution and its particular cases in problems of processing radar signals and images, evaluating the concentration of harmful gases in industrial areas, studying the periods of remission of cancer patients, analyzing neurotransmission and anorexia. The results of the Refs. [15,16,17,18,19] concerning the generalized beta distribution of the second kind and its representatives are used for meteorological research, analysis of infectious diseases, climatic phenomena and profitability, the study of physiological characteristics, and consumer price indexes, and can also be used in the theory of reliability when modeling the time of failure.

This article considers the gamma-exponential distribution, a generalization of the Amoroso distribution that gives the McDonald distribution in the limit. Thus, it can be argued that the results of the article will be in demand when studying various models that allow descriptions of real processes using continuous distributions with non-negative unbounded support.

The main problem in the study of the gamma-exponential distribution is the representation of the density (1) in terms of a special gamma-exponential function (2). This fact makes it difficult to study the probabilistic and statistical properties of the distribution using classical methods such as, for example, the maximum likelihood method. In addition, the moments (6) of the gamma-exponential distribution may not exist for some parameter values and are the products of nonmonotonic gamma functions, with arguments depending on several parameters at once. It significantly complicates not only the application of the method of moments, but also the interpretation of the parameters as characteristics of the mean, spread, asymmetry, and so forth. The latter cannot be considered as a disadvantage of this distribution, since in practice, the value of each characteristic is influenced by many factors of a different nature.

The results of this paper concern the estimation of the bent, shape and scale parameters of the gamma-exponential distribution under the assumption that the concentration parameters are known and fixed. This formulation naturally arises in the case of using the gamma-exponential distribution to study scale mixtures of Rayleigh, Maxwell–Boltzmann, Fréchet (Weibull–Gnedenko), Lévy (with zero bias) distributions, and some others. However, a natural question arises about the form of statistical estimates in the case when all five parameters are unknown.

The difficulty in applying the considered method, based on equating theoretical logarithmic cumulants to their sample counterparts, lies in the fact that the concentration parameters enter the equations as arguments of polygamma functions.

There are many papers related to the study of polygamma functions. For example, the Refs. [25,26] provide the estimates of polygamma functions and inverse polygamma functions in terms of elementary functions, Riemann and Hurwitz zeta functions, and Bernoulli numbers, and also investigate the monotonicity properties of expressions associated with polygamma functions. However, the usefulness of these results for the statistical estimation of the arguments of polygamma functions is not obvious yet.

Thus, the problem of developing effective theoretical methods of inverting polygamma functions is an urgent and, apparently, unsolved problem. However, due to the strict monotonicity and continuity of the polygamma functions, the possibility of numerical inversion is obvious, which will allow for an estimation of all five parameters of the gamma-exponential distribution using computer technologies. The solution to this problem is a direction of further research of the authors.

7. Conclusions

The paper considers the problem of estimating the parameters of the gamma-exponential distribution, which is a generalization and an intermediate link between the generalized gamma distribution and the generalized beta distribution of the second kind. A method for estimating unknown parameters based on logarithmic cumulants is discussed. An algorithm for eliminating unnecessary solutions obtained by solving a system based on logarithmic cumulants is described. The asymptotic normality of the strongly consistent estimators for the bent, shape and scale parameters of the gamma-exponential distribution at fixed concentration parameters is proved. Based on this result, asymptotic confidence intervals for the estimated parameters are constructed. The results are illustrated by numerical examples constructed on the basis of model samples from the gamma-exponential distribution, implemented using the representation of the gamma-exponential distribution as a fractional-scale mixture of gamma distributions. Possible applications of the results in the analysis of processes using continuous distributions with a non-negative unbounded support are discussed.