Generalized Weighted Chris-Jerry Distribution: Properties and ApplicationsO. IbeakuziePreciousC. AnabikeIfeanyiM. NwankwoPeaceQ. ChineduEbereChukwuN. EkemezieDivine-FavourIgbokweChinyere P.Department of Statistics, Faculty of Physical Sciences, Nnamdi Azikiwe University, Awka, Anambra State, Nigeria.Department of Statistics, Faculty of Physical Sciences, Nnamdi Azikiwe University, Awka, Anambra State, NigeriaDepartment of Statistics, School of Science and Industrial Technology, Abia State Polytechnic, Aba, Abia State, Nigeria.
This study is a generalization of the length-biased and weighted Chris-Jerry distribution. It introduces an additional scale parameter to make the distribution more flexible. The functional form of the distribution which includes the density function, the distribution function, the survival function and the hazard function together with their plots were presented. The study also encapsulates the characteristics of the model with the estimation of the parameters using the maximum likelihood method. The applicability was demonstrated using data on remission times of a sample of 128 bladder cancer patients, mortality rate of children in Japan and Ireland under the age of five. The results validate that the model is apt in describing real life events.
Generalized Weighted Chris-Jerry distributionLength-Biased Chris-Jerry distributionCancerMortality RateGoodness of FitIntroduction
No model possesses an absolute fit feature. What researchers accept to be the best fit is a relative attribute. Therefore, there is a continuous search for improved models, especially as industries and the complexities of human engagements produce big data with inherent characteristics. Another related reason for the improvement in models is the flexibility of use. No single model can suit all situations, and only a handful of models have wider applicability. It is on these premises that researchers develop generalizations of probability models.
Some of these generalizations introduce new parameters, such as scale parameters, which in turn improve model performance. Other generalizations focus on tractability, providing easy ways of deriving properties, a concept known as parsimony in the statistical literature.
The literature is rich in probability distribution models. Related works include: Kharazmi et al. [3] introduced the generalized weighted exponential distribution, which is an extension of the standard exponential distribution. This generalization includes additional parameters that allow for greater flexibility in modeling data with varying hazard rates. The inclusion of weight functions helps in adjusting the shape and scale of the distribution, making it more adaptable to different types of data. Ramos and Louzada [4] extends the Lindley distribution by incorporating weight functions. This generalization improves the model’s ability to fit data with diverse characteristics, particularly in scenarios involving skewed data or varying hazard rates. The added parameters allow for better control over the distribution’s shape and scale, enhancing its applicability in real-world situations. Kharazmi [9] introduced the generalized weighted Weibull distribution, which is an extension of the Weibull distribution. This model includes additional parameters to adjust the weight function, allowing for more flexible modeling of life data and reliability analysis. The generalized version can accommodate different shapes of hazard functions, including increasing, decreasing, and bathtub-shaped hazard rates. Abbas et al. [8] developed a new generalized weighted Weibull distribution, which further extends the flexibility of the Weibull distribution. This model incorporates new parameters to adjust both the scale and shape of the distribution, providing a better fit for complex data patterns. The generalized weighted Weibull distribution is particularly useful in survival analysis and reliability engineering. Beaulieu [10] introduced the generalized multinomial distribution, which extends the standard multinomial distribution by incorporating additional parameters. This generalization allows for more flexible modeling of categorical data with complex dependencies. The model is particularly useful in situations where the probabilities of different categories are not fixed but vary according to some underlying factors. Domma [11] proposed a new generalized weighted Weibull distribution that can model different hazard rate shapes, including decreasing, increasing, upside-down bathtub, N-shape, and M-shape hazard rates. This model provides greater flexibility in fitting data with varying hazard rate patterns, making it suitable for diverse applications in reliability and survival analysis. The extension of the generalized half-normal distribution by Acitas [12] introduces weighting mechanisms that improve its applicability to real-world data. This new weighted distribution can better capture the characteristics of skewed data, making it useful in environmental studies, quality control, and other fields where data may not follow symmetric patterns. Generalized weighted exponential-Gompertz distribution by Teamah et al. [13]. The generalized weighted exponential-Gompertz (GWE-G) distribution combines the properties of the exponential and Gompertz distributions with additional weighting parameters. This model is particularly effective in describing data with monotonically increasing or decreasing hazard rates, commonly found in demographic studies and reliability engineering. Generalized probability weighted moments: application to the generalized Pareto distribution by Rasmussen [14]. Rasmussen’s work provides a framework for estimating parameters of generalized distributions. GPWM is particularly useful for modeling extreme values and tail behavior of distributions, which is crucial in fields such as finance, hydrology, and environmental studies.
These generalizations highlight the ongoing efforts to create more flexible and adaptable models that can handle diverse data characteristics. By introducing new parameters and refining existing ones, researchers can develop models that provide better fits and more accurate predictions across various fields. Other useful models include [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41].
The remainder of the work is organized as follows; section is on model specification. In section the properties are studied. In section , the parameters are estimated using maximum likelihood method. In numerical analysis are presented in section . The article is concluded in section .
Model Specification
Chris-Jerry distribution due to Onyekwere and Obulezi [37] is a one-parameter lifetime model with probability density function (PDF) and cumulative distribution function (CDF) given respectively as
θ 2 −θx z (x, θ) = 1 + θx e , θ > 0, x > 0 (1) θ + 2 and θx (θx + 2) −θx Z (x, θ) = 1 − 1 + e , θ > 0, x > 0 (2) θ + 2 proposed a two-parameter Chris-Jerry distribution with p.d.f and CDF given as
θ 2 −θx g (x, θ, λ) = λ + θx e , θ > 0, λ > 0, x > 0 (3) λθ + 2 and 1 2 2 −θx G (x, θ, λ) = 1 − θ x + 2θx + θλ + 2 e (4) λθ + 2
The PDF of the Generalized Weighted Chris-Jerry (GWCJ) distribution with parameters θ, λ, and β is given by
β+1 β−1
θ x 2 −θx f (x, θ, λ, β) = λ + θx e , θ, λ, β > 0, x > 0 (5) θλ + β (β + 1) Γ(β) where θ is the scale parameter, λ and β are the shape parameters
Proof : Tesfalem and Shanker [1] provided a basis for this proof. Having f (x, θ, λ, β) = kw(x, θ, λ, β)g (x, θ, λ) (6) where k is the normalizing constant, w (x, β) = x is the weight function and g (x, θ, λ) is the PDF of the two-parameter Chris-Jerry distribution. We require to prove that equation (6) is a proper PDF by deriving the value of k. Hence, it is easy to see that for Z ∞ Z ∞
(θλ + 2) θ (θλ + β(β + 1)) θΓ (β) Therefore, making appropriate substitutions in equation (6) completes the proof. □
Notice that if X ∼ GW CJ (θ, λ, β), the PDF which is in equation (5) is a two-component mixture of Gamma (θ, β)
θλ
and Gamma (θ, β + 2) with mixing proportion in the form
θλ+β(β+1)
pGamma (θ, β) + (1 − p)Gamma (θ, β + 2) The CDF of the GWCJ distribution with parameters θ, λ and β is given by Γ(β, θx) 1 β β −θx F (x, θ, λ, β) = 1 − + (1 + β + θx) θ x e , x, θ, λ, β > 0 (8) Γ(β) θλ + β (β + 1)
β−1 −y
where Γ(β, θx) = y e dy is the upper incomplete gamma function.
θx
Proof : We begin the proof with integrating the PDF of the proposed GW CJ (x, θ, λ, β) given in equation (5). Z x Z x Z x θ β−1 −θx β+1 −θx F(x, θ, λ, β) = f(t, θ, λ, β)dt = λ x e dx + θ x e (9) 0 [θλ + β(β + 1)] 0 0
dy
define y = θx then dx = , hence ( Z Z )
θx θx
1 β−1 −y β+1 −y θλγ(β, θx) + γ(β + 2, θx) F(x, θ, λ, β) = θλ y e dy + y e dy = Γ (β) (θλ + β(β + 1)) 0 0 Γ (β) (θλ + β(β + 1)) Note: The following special derivations from the gamma function were employed;
β −x β −θx β+1 −θx
Γ(β + 1, x) = βΓ(β, x)+x e ; Γ(β) = γ(β, x)+Γ(β, x); Γ(β + 2, θx) = β(β+1)Γ(β, θx)+(β+1)(θx) e +(θx) e Making appropriate substitutions from the special derivations into equation (10), we complete the proof. □
It is easy to define from equation (8) the survival function of GWCJ distribution Let X ∼ GW CJ(θ, λ, β), the survival function is given as Γ(β, θx) 1 β β −θx S(x, θ, λ, β) = + (1 + β + θx) θ x e (11) Γ(β) θλ + β (β + 1)
We use the survival function as a measure of system reliability. The hazard rate function which measures the likelihood that an item will survive to a certain point in time based on its survival to an earlier time t. Let X ∼ GW CJ(θ, λ, β), the hazard rate function is given as
β+1 β−1 2 −θx
θ x λ + θx e hrf(x, θ, λ, β) = (12)
β β −θx
Γ(β, θx) (θλ + β(β + 1)) + Γ(β) (1 + β + θx) θ x e Another very important measure is the reversed hazard rate function. It is the conditional probability of failures of an item in the next dt units of time given that it did not fail before t. It is essential in analyzing censored data and is commonly used in Forensic sciences (see Kayid [2]). Let X ∼ GW CJ(θ, λ, β), the reversed hazard rate function is given as
β+1 β−1
θ x 2 −θx
λ + θx e
θλ+β(β+1) Γ(β)
rhrf(x, θ, λ, β) = h i (13)
Γ(β,θx) 1 β β −θx
1 − − (1 + β + θx) θ x e
Γ(β) θλ+β(β+1)
The cumulative hazard rate function provides the total accumulated risk of experiencing the event of interest that has been gained by progressing to time t. While the instantaneous hazard rate hrf(t) can increase or decrease with time, the cumulative hazard rate can only increase or remain the same. Let X ∼ GW CJ(θ, λ, β), the cumulative hazard rate function is given as Γ(β, θx) 1 β β −θx chrf(x, θ, λ, β) = − ln S(x, θ, λ, β) = − ln + (1 + β + θx) θ x e (14) Γ(β) θλ + β (β + 1)
Mathematical Properties
In this section, we derive some important properties of the GWCJ distribution
th
Moment. Let X ∼ GW CJ(θ, λ, β), the r non-central moment is given as
survival and hazard rate function of the GWCJ distribution
[Variance of the GWCJ distribution] The variance of a random variance is a measure of spread which is generally denoted by σ = µ2 − (µ1) . From a random variable X that is GWCJ distributed, the variance is given as
θ λ + θ β (β + 1) θ λ + θβ (β + 1) [Kurtosis] Kurtosis is another measure that explains the peakness of a distribution. Suppose X ∼ GWCJ (θ, λ, β), the kurtosis can be measured as follows;
[Skewness] Skewness is an important measure that indicates where the weight of the distribution is. Given X ∼ GWCJ(θ, λ, β), the measure of skewness is expressed as;
An odd function is a reliability tool for modeling a data set that shows a non-monotone hazard rate. It is defined to be the ratio of the CDF to the Survival function FGW CJ (x; θ, λ, β) OGW CJ (x; θ, λ, β) = (26) SGW CJ (x; θ, λ, β)
Γ(β,θx) 1 β β −θx
1 − − (1 + β + θx) θ x e Γ(β) θλ+β(β+1) Γ(β) θλ + β(β + 1)
−β −β θx
OGW CJ (x; θ, λ, β) = = + θ x e −1 Γ(β,θx) 1 β β −θx Γ(β, θx) 1 + β + θx + (1 + β + θx) θ x e
Γ(β) θλ+β(β+1)
Index of Dispersion
Index of dispersion is a normalized measure of the dispersion of a probability distribution which is used to quantify whether a set of observed occurrences are clustered or dispersed compared to a standard statistical model.
Given an ordered sample X(1), X(2), · · · , X(n) from a GWCJ distribution. The distribution of the statistics is n! r−1 n−r fr:n(x; θ, λ, β) = fGW CJ (x; θ, λ, β) FGW CJ (x; θ, λ, β) 1 − FGW CJ (x; θ, λ, β) (29) (r − 1)!(n − r)!
Mean of GWCJ distribu- Figure 4. Variance of GWCJ distri-tion bution
Skewness of GWCJ distri- Figure 6. Kurtosis of GWCJ distri-bution bution
where fGW CJ (x; θ, λ, β) and FGW CJ (x; θ, λ, β) are the pdf and cdf of GWCJ distribution respectively. Hence, we have fr:n(x; θ, λ, β) =
The PDF of the largest order statistics is gotten by setting r = n ( )n−1
β+1 β−1
nθ x 2 −θx Γ(β, θx) 1 β β −θx λ + θx e 1 − − (1 + β + θx) θ x e (31) θλ + β (β + 1) Γ(β) Γ(β) θλ + β (β + 1)
The PDF of the smallest order statistics is gotten by setting r = 1 ( )n−1
β+1 β−1
nθ x 2 −θx Γ(β, θx) 1 β β −θx λ + θx e − (1 + β + θx) θ x e (32) θλ + β (β + 1) Γ(β) Γ(β) θλ + β (β + 1)
Generating Functions
An endless series of numbers can be encoded using a generating function in mathematics by treating them as the coefficients of a formal power series.
Moment Generating Function
The moment-generating function of a real-valued random variable is an alternative specification of its probability distribution in probability theory and statistics. Instead of using probability density function or cumulative distribution functions directly, it provides the foundation for an alternative path to analytical solutions.
The moment generating function of a X ∼ GWCJ (θ, λ, β) is given by
tx tx
MX(t) = E(e ) = e f(x)dx
θ tx β−1 2 −θx = e x (λ + θx )e dx θλ + β(β + 1)Γ(β) 0 θ β−1 2 −(θ−t)x = x (λ + θx )e dx θλ + β(β + 1)Γ(β) (33) " Z Z #
β+1 ∞ ∞
θ β−1 −(θ−t)x β+1 −(θ−t)x = λ x e dx + θ x e dx θλ + β(β + 1)Γ(β) 0 0
θ λΓ(β) θΓ(β + 2)
β β+2
θλ + β(β + 1)Γ(β) (θ − t) (θ − t)
Characteristic Generating Function
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution. Unlike the moment-generating function, the characteristic function always exists when treated as a function of a real-valued argument.
The characteristic function of a X ∼ GWCJ(θ, λ, β) is given by
θ λΓ(β) θΓ(β + 2) ϕX(it) = + (34)
β β+2
θλ + β(β + 1)Γ(β) (θ − it) (θ − it)
Maximum Likelihood Estimation of the Parameters
Let x1, x2, ..., xn be a random sample of size n drawn from GWCJ distribution, then the likelihood function is given as
n β+1 β−1 n(β+1) −θ x n
Y i Y θ xi 2 −θx θ e β−1 2 L fGW CJ (x; θ, λ, β) = (λ + θxi )e = xi (λ + θxi ) θλ + β(β + 1) Γ(β) (θλ + β(β + 1)) Γ(β)
i=1 i=1
Taking the natural log of L(.) and set log Lϕ, we have
X X X
β−1 2
ϕ = n(β + 1) log θ − θ xi − n log Γ(β) − n log(θλ + β(β + 1)) + log xi + log(λ + θxi ) (36)
Differentiating partially with respect to θ, λ and β yields the following;
n n 2
∂ϕ n(β + 1) X nλ X x = − xi − + ∂θ θ θλ + β(β + 1) λ + θxi
i=1 i=1
∂ϕ nθ X 1 = − + (37) ∂λ θλ + β(β + 1) λ + θxi
∂ϕ nΓ (β) n(2β + 1) X = n log θ − − + log xi ∂β Γ(β) θλ + β(β + 1)
The system of non-linear equations in (37) do not possess analytical solution hence optim() function in R will be used to obtain the solution, see [42].
Numerical Analysis
This section is used to illustrate the applicability of the proposed distribution to life time data. Data-I is the remission times (in months) of a sample of 128 bladder cancer patients documented by Lee and Wang [5], see table 1. Data-II and III consist of the mortality rate of children in Japan and Ireland under five years of age from 1969 to 2021 respectively. The data are presented in table 2 and 3 which were obtained https://data.worldbank.org/indicator/SP.DYN.IMRT.INhttps://data.worldbank.org/indicator/SP.DYN.IMRT.IN (accessed on 21 July 2024). We provide the model comparison using the following measures; log-likelihood (LL) statistic, Akaike Information Criterion (AIC), Corrected Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC) and the Hannan Quinn Information Criterion (HQIC). Further, we provide the fitness of the model to the data sets using the Cramér von Mises (W) statistics, Anderson-Darling (A) statistics, Kolmogorov-Smirnov (KS) statistics with its p-value. From the results in table 4, the GWCJ distribution fits the data better than the rest of the distributions compared and also performed adequately in estimating the parameters. However, from the bloated value of λ ˆ , there is indication of poor estimation. In table 5, the
MLE
GWCJ competes favourably while it is better than the competitors in table 6. Figures 7, 8 and 9 represent the histogram, CDF, survival function, TTT plot, PP and QQ plots of the three data sets respectively. They visually tell how well the distribution fits the associated data sets.
Conclusion
Generalization of distributions has inadvertently proved useful in the statistical literature especially in shaping the decision on the choice of the number of parameters and weighting a random variable. In this article, we generalized the length-biased Chris-Jerry distribution proposed by Subramanian and Subhashree [6] and the weighted Chris-Jerry distribution by Praseeja et al. [7]. The functional form of the distribution including the Dist LL AIC CAIC BIC HQIC W A KS p-value θ ˆ λ ˆ β ˆ
Data-I The remission times (in months) of a sample of 128 bladder cancer patients
This table is shown in the PDF version of the article.
Data-II The mortality rate of children in Japan under five years of age from 1969 to
This table is shown in the PDF version of the article.
Data-III The mortality rate of children in Ireland under five years of age from 1969 to 2021
This table is shown in the PDF version of the article.
Measures of Model Adequacy using Data-I
This table is shown in the PDF version of the article.
PDF, CDF, Survival function and hazard were presented. The graphical illustration of these functions were also provided. The characteristics of the distribution were studied. The parameters of the model were estimated using the maximum likelihood estimation procedure. To illustrate the importance of the model, three lifetime data sets were used and the model proved better than some classical models.
Measures of Model Adequacy using Data-III
This table is shown in the PDF version of the article.
Dist LL AIC CAIC BIC HQIC W A KS p-value θ ˆ λ ˆ β ˆ
The authors would like to express their sincere gratitude to the editors and anonymous reviewers for their invaluable comments and constructive feedback, which significantly contributed to the enhancement of this paper.
Author Contribution
Ibeakuzie, Igbokwe & Ekemezie: methodology, software, and editing. Anabike: conceptualization. Nwankwo & Chinedu: writing and editing. All authors have read and agreed to the published version of the manuscript.
Funding
The authors declare that no external funding or support was received for the research presented in this paper, including administrative, technical, or in-kind contributions. Data Availability All data supporting the reported findings in this research paper are provided within the manuscript.
Conflicts of Interest
The authors declare that there is no conflict of interest concerning the reported research findings. Funders played no role in the study’s design, in the collection, analysis, or interpretation of the data, in the writing of the manuscript, or in the decision to publish the results.
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