Abstract
Some characterizations and entropy measures of the Exponentiated Generalized Fréchet Geometric (EGFG) distribution are studied in this paper. Firstly, characterizations of the EGFG distribution based on five different approaches are discussed. The submodels for the EGFG distribution with their characterization expressions formed on the ratio of two truncated moments are also presented. Secondly, four different entropy measures are considered and expressed analytically via the incomplete gamma function. The behavior of all these entropy measures is discussed by performing a numerical study.
1. Introduction
Characterization is a distributional property of statistics that determines the associated stochastic model. In designing a stochastic model for a particular modeling problem, it is crucial to validate whether the specified probability distribution fulfills the underlying conditions by its characterization before a specified probability model is used to fit data applications. Thus, it will rely on the characterization of the chosen distribution. It plays an essential role in different fields of mathematical and statistical sciences.
Although in several statistical applications, an increase in the number of parameters gives a more appropriate model, and a smaller number of parameters (without affecting the model suitability) is mathematically more attractive in characterization complications. Over the years, many researchers have studied the different techniques of characterization of continuous probability distributions such as Glänzel [1]; Glänzel [2]; Hamedani [3]; Hamedani [4]; Hamedani [5]; Hamedani and Ahsanullah [6]; Hamedani [7]; Bhatti et al. [8]; and Rafique and Saud [9]. Consequently, numerous characterization techniques have been studied in this paper.
Entropy is used to measure the amount of uncertainty, randomness, and disorderness in the system. These essential mathematical techniques are used to evaluate the uncertainty of the stochastic variable. To be more specific, the distribution of a random variable is related to some level of uncertainty, which is quantified by entropy.
In any stochastic process, the probability distribution varies over time; hence, it is clear that the entropy or uncertainty of a probability distribution changes over time as well. It becomes knowing how the uncertainty changes over time. The application of entropy measures is useful in many branches of science and disciplines. Several examples from different areas where different entropy concepts were used, e.g., topological entropy, Kolmogorov–Sinai entropy, topological order, can be found in the following references proposed by Caneco et al. [10], Rocha and Aleixo [11], and Rocha and Carvalho [12].
The essential references in this regard are briefly presented below. Shannon [13] introduced the concept of entropy. Cho et al. [14] estimated the entropy of a Weibull distribution under generalized progressive hybrid censoring. The comparison of different entropy measures was discussed by Dey et al. [15]. Basit et al. [16] studied the entropy measures for weighted and truncated weighted exponential distributions. The entropy measures for the uncertainty quantification of stochastic processes were investigated by Namdari and Li [17], while Ijaz et al. [18] compared different entropy measures for Lomax distribution using relative loss approach.
This present work deals with the characterization and entropy measures of the univariate continuous distribution named as “Exponentiated Generalized Fréchet Geometric (EGFG) distribution.” The EGFG distribution was proposed by Rafique and Saud [19]. They studied the mathematical properties of the suggested model and also provided its application study.
The probability density function (pdf) and cumulative distribution function (cdf) of the Exponentiated Generalized Fréchet Geometric distribution with parameters , , , , and are defined, respectively, as follows:and
given by [19].
2. Structural Properties
According to Rafique and Saud [19], the hazard rate, reverse hazard rate, and survival function of the EGFG are defined, respectively, by
The mills ratio and elasticity function of any probability distribution can be defined as
[20] The mills ratio and elasticity function of the EGFG distribution are given by
3. Materials and Methods
3.1. Characterization via Ratio of Two Truncated Moments
The EGFG distribution is characterized through ratio of two truncated moments. Glanzel [21] stated that this characterization is stable in the sense of weak convergence.
Proposition 1. Suppose be a continuous random variable with probability density function defined in equation (1) and letand then, the function which is presented in theorem (Glänzel [1, 2]) has the following expression:
Proof. Suppose has pdf in equation (1), thenand soAsthe proof is as follows:
Conversely, if is defined as above, thenand hence,andTherefore, in the light of theorem, given in Glänzel [1] and Glanzel [21], has pdf in equation (1).
Corollary 1. Suppose that the continuous random and let as given in proposition equation (1). The pdf of Y in equation (1) provided functions and defined in the theorem (Glänzel [1]). The theorem is based on truncated moments specified in the ratio form.
Remark 1. The solution of the equation in corollary equation (1) iswhere D is the constant.
3.1.1. Special Cases of the Exponentiated Generalized Fréchet Geometric Distribution
The expressions of , , and to characterize the submodels of the EGFG distribution are given as follows:
(1) Generalized Fréchet Geometric Distribution. Putting , suppose thatso that follows GFG distribution if and only if the function obtained in Proposition 1 has the following expression:
(2) Fréchet Geometric Distribution. For , letand if obtained in Proposition 1 has the expression, then Y follows Fréchet Geometric distribution as follows:
(3) Inverse Rayleigh Geometric Distribution. With , , and, we getand then, r.v. Y follows IRG distribution if the function has the following expression:
(4) Inverse Exponential Distribution. For , we haveand
3.2. Characterization of EGFG under Hazard Rate Function
In this subsection, the characterization of the EGFG distribution is obtained by hazard rate function.
Definition 1. Suppose the absolutely continuous random variable “” with pdf and cdf. The hazard function to F is symbolized as and is obtained bywhere F represents twice “differentiable distribution function.” The hazard function satisfies the following differential equation:where is an appropriate integrable function. The purpose here is to define a differential equation that has a simple form as possible and a nontrivial form as in equation (30). However, in some continuous distributions, this may be impossible.
Proposition 2. Let the random variable follows the EGFG distribution. The density function of “” is equation (1) if and only if its hazard rate function satisfies the following differential equation:
Proof. If the r.v has the hrf given in equation (3), thenand simplification results in equation (3).
3.3. Characterization of EGFG under Reverse Hazard Rate Function
In this subsection, the EGFG distribution is characterized via the reverse hazard rate function.
Definition 2. Suppose the absolutely continuous random variable “Y” with pdf equation (1) and cdf equation (2). The reverse hazard rate function to F is symbolized as and is given bywhere F represents the twice “differentiable distribution function.” The reverse hazard function satisfies the following differential equation:
Proposition 3. Let the random variable follows the EGFG distribution. The density function of is equation (1) if and only if its reverse hazard function satisfies the following differential equation:
Proof. If the r.v has density function in equation (1), then surely equation (35) holds.and simplification results in equation (4).
3.4. Characterization of EGFG under Mills Ratio
In this subsection, the EGFG distribution is characterized via mills ratio.
Definition 3. Suppose the absolutely continuous random variable “Y” with pdf f(y) and cdf F(y). The mills ratio to F is symbolized as and is obtained bywhere F represents the twice “differentiable distribution function.” The mills ratio satisfies the following differential equation:where is an appropriate integrable function.
Proposition 4. Let the random variable follows the EGFG distribution. The density function of is equation (1) if and only if its mills ratio defined in equation (7) satisfies the following differential equation:
Proof. If the r.v has density function equation (1), then surely equation (7) holds.and simplification results in equation (7).
3.5. Characterization of EGFG under Elasticity Function
In this subsection, the EGFG distribution is characterized through the elasticity function.
Definition 4. Suppose the absolutely continuous random variable “” with pdf f(y) and cdf F(y). The elasticity function to F is symbolized as and is obtained bywhere F represents the twice “differentiable distribution function.” The elasticity function satisfies the following equation:
Proposition 5. Let the random variable follows the EGFG distribution. The density function of Y is equation (1) if and only if its elasticity function satisfies the following differential equation:
Proof. If the r.v has density function equation (1), then surely equation (43) holds.Now, the result is as follows: conversely if equation (43) holds, thenwhich is the elasticity function of the EGFG distribution.
4. Entropy of the Exponentiated Generalized Fréchet Geometric Distribution
Entropy measures have a strength of prediction. Different studies of time series may influence the predictive capacity of entropy measures. According to Liang et al. [22], time series analysis affects many entropy methods. Different entropy methods for measuring the complexity of time series were studied by Chen et al. [23]. According to Yin and Shang [24], entropy is a useful tool for examining time series because it does not include any constraints on the probability distribution. Solís-Montufar et al. [25] stated that several entropy measures can be used for complex time series. Its value may be positive, negative, or zero. A higher value shows randomness, while zero shows the certainty of information.
4.1. An Integral Result
The following results indicate that some integrals, including the density of the EGFG distribution, may be defined in terms of the incomplete gamma function.
Proposition 6. Let , , be specified by equation (1) andThen, exists if and only if , and it is defined aswhere .
Proof. Owing to equation (1), we haveFor any positive real number b and for , we have the binomial and generalized binomial expansion:Applying equations (50) and (49), we getIn this study, the interest of Proposition 6 is that is the major ingredient in the definitions of various entropy measures of the EGFG distribution as derived in the next part.
4.2. Various Entropy Measures
In this subsection, some important entropy measures are studied using different techniques. The literature contains many entropies that are defined in Table 1 for the general distribution with pdf. Suppose that and as basic assumptions in this general case.
From Table 2, we see that the integral is central to obtain the considered entropy measures. Now, we study the entropy measures of the EGFG distribution based on Proposition 6.
4.2.1. Rényi Entropy
Based on Table 2, equation (1), and Proposition 6, the Rényi entropy of the EGFG distribution can be defined as
4.2.2. Havrda and Charvát Entropy
From Table 2, equation (1), and Proposition 6, the Havrda and Charvát entropy of the EGFG distribution is specified by
4.2.3. Arimoto Entropy
Again, from Table 2, equation (1), and Proposition 6, the Arimoto entropy of the EGFG distribution is expressed by
4.2.4. Tsallis Entropy
Based on Table 2, equation (1), and Proposition 6, the Tsallis entropy of the EGFG distribution can be presented as
4.3. Numerical Results
In this section, the idea of following Al-Babtain et al.’s [30] numerical values for the four different entropy measures of the EGFG distribution is calculated. The findings of all considered entropy measures are presented in Tables 3–7.
Tables 3–7 show the numerical values of the entropy measures for the different sets of the parameters. It can be noticed that,(i)Rényi, Havrda and Charvát, Arimoto, and Tsallis entropy measures are increasing when and are increasing.(ii)Rényi, Havrda and Charvát, Arimoto, and Tsallis entropy measures are increasing when and are increasing.(iii)Rényi, Havrda and Charvát, Arimoto, and Tsallis entropy measures are decreasing when and are increasing.(iv)Rényi, Havrda and Charvát, Arimoto, and Tsallis entropy measures are decreasing when and are increasing.(v)Rényi, Havrda and Charvát, Arimoto, and Tsallis entropy measures are increasing when all parameter values are decreasing.
5. Conclusion
In this article, the characterization results for the Exponentiated Generalized Fréchet Geometric distribution through five approaches are presented. In the previous section, the analytical expressions of four entropy measures for the EGFG distribution are investigated. All entropy measures are compared numerically by considering the various combination of the parametric values. We hope these expressions of entropies’ measures by various techniques may be helpful in uncertainty measures of a random variable of related fields.
Data Availability
The data is generated by using the idea of Al-Babtain et al. [30].
Conflicts of Interest
The authors declare that they have no conflicts of interest.