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. 2024 Nov 5;2024:9423417. doi: 10.1155/2024/9423417

A Test for Discriminating Between Members of the Odd Weibull-G Family of Distributions

Boikanyo Makubate 1,
PMCID: PMC11557169  PMID: 39534790

Abstract

The Odd Weibull-G (OWG) family of distributions has been discussed earlier in the literature. This family of distributions provides a “better fit” in certain practical situations. In a similar fashion, the OWG family of distributions is defined in this article. A method of moments estimator based on the maximum entropy principle is proposed for the discrimination of two members of the OWG family of distributions.

Keywords: discrimination between distributions, maximum entropy principle, odd Weibull-G family of distributions, Shannon entropy

MSC2020 Classification: 62E10, 62F30

1. Introduction

There have recently been attempts to create new families of probability distributions to represent and understand real-world phenomena [17]. Generating new distributions allows researchers and practitioners to develop more accurate models that can capture the underlying characteristics of different datasets. One such illustration is a large family of univariate distributions derived from the Weibull distribution that Bourguignon, Silva, and Cordeiro [8] proposed. Bourguignon, Silva, and Cordeiro [8] extend any continuous baseline distribution by two additional parameters using the T-X technique [9]. The two additional parameters control the skewness and the kurtosis of the distribution. For any baseline cumulative distribution function (cdf), G(x; ζ), which depends on a parameter vector ζ, following the notation of Bourguignon, Silva, and Cordeiro [8], the cdf and probability density function (pdf) of the Odd Weibull-G (OWG) family of distributions is defined by

Fx;β1,β2,ζ=1expβ1Gx;ζG¯x;ζβ2 (1)

and

fx;β1,β2,ζ=β1β2gx;ζGx;ζβ21G¯x;ζβ2+1expβ1Gx;ζG¯x;ζβ2, (2)

respectively, where G¯x;ζ=1Gx;ζ, g(x; ζ) = dG(x; ζ)/dx, x > 0, and β1 and β2 are positive parameters. Model (2) has the advantage that a vast family of distributions can be formed from any continuous distribution, G(x; ζ), with the parameters β1 and β2 regulating the skewness and the kurtosis. In their article, Bourguignon, Silva, and Cordeiro [8] showed that this family of distributions provides a better fit than other commonly used distributions. In the literature, families of Weibull related distributions have also been addressed, for example, the new Weibull generalized-G by Oluyede, Sengweni, and Makubate [10]; the Weibull normal distribution by Famoye, Akarawak, and Ekum [11]; the Weibull exponential distribution by Oguntunde et al. [12] and the Weibull Dagum by Tahir et al. [13]; to mention a few. The class of distributions for the special case of β2 = 1 is referred to as the Odd Exponential-G (OEG) family of distributions with cdf and pdf given by

FOWGx;β1,ζ=1expβ1Gx;ζG¯x;ζ (3)

and

fOWGx;β1,ζ=β1gx;ζ1G¯x;ζ2expβ1Gx;ζG¯x;ζ, (4)

respectively.

Determining whether specific data can be presumed to have come from one of the two provided arbitrary probability distributions has been a long-standing problem in statistics. Atkinson [14, 15], Chambers and Cox [16], Chen [17], Cox [18, 19], Dumonceaux and Antle [20], Dyer [21], Gupta and Kundu [22, 23], Jackson [24], Kundu et al. [25], Lee and Max [26], and Raqab, Al-Awadhi, and Kundu [27] all had earlier discussed and provided respective solutions for this problem. In particular, Dumonceaux and Antle [20], Lee and Max [26], and Gupta and Kundu [22] derived methods of choosing between the Weibull and the log-normal distributions, the Weibull and the gamma distributions, and the Weibull and the generalized exponential distributions, respectively.

Raqab, Al-Awadhi, and Kundu [27] considered discriminating among three positively skewed models being Weibull, log-normal and log-logistic distributions. In this article, a method of moments estimator based on the maximum entropy principle [28] is proposed for the discrimination of two members of the OWG family of distributions. A not very dissimilar idea was earlier proposed by Zografos and Balakrishnan [29] in discriminating between beta generated models and gamma generated models, respectively.

Due to the increasing applications of the OWG family of distributions [1113], in this article, special attention is given to developing a test that would allow mathematicians to determine whether a sample taken at random from (4) is coming from a specific G(x; ζ) distribution. In this research, it is observed that the difference of the Shannon entropy [30] of any two given members of the OWG can be used to discriminate between the members of the family in (4). In addition, it is possible to construct an analytical form for the Shannon entropy of some members of the OWG family. Earlier, Huang et al. [31], although on a different problem, discussed in their article some entropy based methods.

The paper is outlined as follows. The Shannon entropy of the OWG, with pdf in (4) is presented in Section 2. The Shannon entropies of three univariate distributions produced by model (4) will be determined in a closed form and provided as examples. In Section 3, a procedure for discriminating between members of the OWG family is proposed, followed by conclusions.

2. Shannon Entropy

Shannon entropy, named after the mathematician and information theorist Claude Shannon [30], is a measure of information content in a given set of data. It provides a quantitative measure of the average amount of information needed to specify an outcome from a set of possibilities. Shannon entropy has applications in various fields, including information theory, data compression, cryptography, and machine learning [29]. The Shannon entropy of a continuous distribution with density, let us say f(x), is defined by

Sf=fxlnfxdx. (5)

Thus, the Shannon entropy of the OWG family of distributions, with pdf in (4), is given by

SfOWG=lnβ1EOWGlngx;ζ+2EOWGlnG¯x;ζ+β1EOWGGx;ζG¯x;ζ, (6)

where EOWG denotes expectation under the pdf in (4).

Lemma 1 . —

Let the random variable X be described by the OWG family in (4). Then, the random variable 0<Z=Gx;ζ/G¯x;ζ< has an exponential distribution with pdf

fz;β1=β1expβ1z. (7)

Proof —

We let Z = G(x; ζ)/1 − G(x; ζ). Thus, the Jacobian is given by dx/dz = (1 − G(x; ζ))2/g(x; ζ). Substituting Z into (4) and multiplying by the Jacobian yields the pdf of the random variable Z as

fz;β1=β1expβ1z. (8)

Lemma 2 . —

If G(x; ζ) and g(x; ζ) are any arbitrary continuous cdf and pdf, respectively, then

  • a. EOWGGx;ζ/G¯x;ζ=1/β1;

  • b. EOWGlnGx;ζ/G¯x;ζ=γlnβ1;

  • c. EOWGlnG¯x;ζ=eβ1Ei1,β1;

  • d. EOWG[ln g(x; ζ)] = EZ[ln g(G−1[z/(1 + z)])],

where Z has an exponential distribution with the rate parameter β1, EOWG denotes expectation under the pdf in (4), γ ≈ 0.5772 is the Euler–Mascheoni constant, and

Ei1,β1=β1lnzβ1ezdz<. (9)

Proof —

To verify Parts (a), (b), (c), and (d), we let Z = G(x; ζ)/1 − G(x; ζ) to obtain

EOWGGx;ζG¯x;ζ=EZz=1β1, (10)
EOWGlnGx;ζG¯x;ζ=β10lnzeβ1zdz=γlnβ1, (11)
EOWGlnG¯x;ζ=β10ln11+zeβ1zdz=eβ1Ei1,β1, (12)
EOWGlngx;ζ=0lngG1z1+zβ1expβ1dz (13)

Lemma 3 . —

The Shannon entropy of the OWG distribution with pdf in (4) is given by

SfOWG=1lnβ12eβ1Ei1,β1EZlngG1z1+z. (14)

where Z has an exponential distribution with the rate parameter β1, and

Ei1,β=β1lnz/β1ezdz. (15)

Proof —

It is readily obtained by applying Lemma 2 into equation (6).

Lemma 4 . —

The pdf of OWG defined in (4) is the unique solution of the optimization problem

fOWGx=argmaxfSf, (16)

under the constraints

  • 1. EfGx;ζ/G¯x;ζ=1/β1;

  • 2. EflnG¯x;ζ=eβ1Ei1,β1;

  • 3. Ef[ln g(x; ζ)] = EZ[ln g(G−1[z/(1 + z)])],

where Z has an exponential distribution with the rate parameter β1, and

Ei1,β1=β1lnz/β1ezdz. (17)

Proof —

Let f be a pdf satisfying the requirements 1 − 3. The Kullback–Leibler divergence between f and fOWG is given by

0Df,fOWG=fxlnfxfOWGxdx=fxlnfxdxfxlnfOWGxdx=SffxlnfOWGxdx. (18)

For more details regarding Kullback–Leibler divergence between two arbitrary distributions, the reader is referred to Zografos and Balakrishnan [29] and references contained in it. Using the definition of the fOWG as given in (4) and based on the constraints (1)–(3) yields

fxlnfOWGxdx=lnβ1+Eflngx;ζ2EflnG¯x;ζβ1EfGx;ζG¯x;ζ=lnβ1+EZlngG1z/1+z+2eβ1Ei1,β11=SfOWG. (19)

Substituting (19) into (18) yields

0Df,fOWG=Sf+SfOWG, (20)

with equality if and only if D(f, fOWG) = 0, that is, if f = fOWG, which was to be proved.

As demonstrated by Lemma 3, the Shannon entropy of the OWG family in (4) is divided into two components. The first component is tied to the parameter β1 of the Weibull distribution, whereas the second part is entirely related to the arbitrary distribution G(x; ζ). Moreover, all members of the family in (4) share the first component and they are distinguished from each other solely by EZ[ln g(G−1[z/(1 + z)])]. Hence, the term EZ[ln g(G−1[z/(1 + z)])] can be used to distinguish between the members of the family in (4). It is possible, in some cases, to obtain an analytic form for the Shannon entropy of the family in (4), as shown in the following examples.

Example 1 . —

Consider the odd Weibull uniform (OWU). The cdf and pdf of the uniform distribution is G(x; η) = x/η and g(x; η) = 1/η, respectively, where 0 < x < η. As a result EZ[ln g(G−1[z/(1 + z)])] = −ln η. Thus, using Lemma 3, The Shannon entropy of the OWU is given by

SfOWU=1lnβ12eβ1Ei1,β1+lnη. (21)

Example 2 . —

As a second example, let us consider the odd Weibull exponential (OWE). The cdf and pdf of the exponential distribution is G(x; λ) = 1 − eλx, and g(x; λ) = λeλx, respectively, for 0 < x < and λ > 0. Consequently, Ez[ln g(G−1[z/(1 + z)])] = ln λeβ1Ei(1, β1). Thus using Lemma 3, the Shannon entropy of the OWE is given by

SfOWE=1lnβ1eβ1Ei1,β1lnλ. (22)

Example 3 . —

Consider the odd Weibull logistic (OWL). This is obtained from (4) when the baseline distribution is the logistic distribution with cdf and pdf G(x) = 1/1 + eλx and g(x) = λeλx/(1 + eλx)2, for x, λ > 0, respectively. Then, G−1[z/(1 + z)] = ln(z)/λ and g(G−1(z/(1 + z)) = λz/(1 + z)2. Consequently,

EZlngG1z1+z=lnλγlnβ12eβ1Ei1,β1. (23)

Thus, the Shannon entropy of the OWL distribution is simply

Sf=1+γlnλ. (24)

Similarly, the Shannon entropy of other members of the family described in (4), including but not limited to, odd Weibull Pareto (OWP), odd Weibull half logistic (OWHL), odd Weibull power function (OWPF), and Odd Weibull Weibull (OWW) could be easily obtained via Lemma 3.

3. Discrimination Process

Consider a random sample X1, X2,…, Xn of size n from the OWG distribution with pdf in (4). The goal is to determine which model from the OWG family of distributions best fits a given dataset. To achieve this, we require a method for differentiating between various models within the OWG family. According to the maximum entropy principle, the most appropriate model to describe the data is the one with the distribution function G = G(x; ζ) that maximizes the corresponding Shannon entropy. In view of Lemma 3, to discriminate between two candidate models, with arbitrary cdfs G1 and G2 and respective pdfs g1 and g2, we use the following test statistic:

M1,2=1ni=1nlng2G21zi2/1+zi2g1G11zi1/1+zi1, (25)

where

  • Zi = Gj(xi)/1 − Gj(xi), for i = 1, 2,…, n, and for arbitrary cdf Gj (where j = 1, 2).

3.1. Interpretation of Results

  • • If 1,2 is positive, the test supports the model OWG1.

  • • If 1,2 is negative, the test supports the model OWG2.

This procedure effectively discriminates between two models within the OWG family based on their respective cdfs G1 and G2.

3.2. Hypothesis Testing

To decide between the hypotheses:

  • H0: parent distribution is G2

  • H1: parent distribution is G1,

We use the statistic 1,2. The null hypothesis H0 is rejected at a significance level α if

M1,2yα, (26)

where yα represents the upper 100 × α% point of the distribution of 1,2, under the null hypothesis H0.

4. Conclusions

In this article, the problem of discriminating between two members of the OWG family was considered. This discriminating process, using the method based on the maximum entropy principle, allows us to objectively decide which distribution from the OWG family best represents the data. It provides a robust statistical method for comparing two potential parent distributions by evaluating the entropy associated with each candidate.

Data Availability Statement

No data were used in the development of this study.

Conflicts of Interest

The author declares no conflicts of interest.

Funding

The author received no specific funding for this work.

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Data Availability Statement

No data were used in the development of this study.


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