A study on the weighted Ola distribution: Statistical properties and lifetime data applications

Helal et al. (2022) [13] proposed the weighted Shanker distribution; Mohiuddin et al. (2022) [14] introduced the weighted Amarendra distribution for medical data; and Chesneau et al. (2022) [15] suggested a modified weighted exponential model for skewed datasets. Ganaie and Rajagopalan (2023) [16] developed the weighted power quasi-Lindley distribution, and Hashempour and Alizadeh (2023) [17] presented a weighted half-logistic distribution with different estimation methods. Shanker R, Ray M & Prodhani HR (2023) [18] proposed Weighted Komal distribution with properties and applications to model failure-time data from engineering, while Ranade and Rather (2023) [19] introduced the weighted Sabur distribution for lifetime data modeling Abu Thaimer and Al-Omari (2025) [25] proposed the weighted Gamma-Lindley distribution, which they applied to COVID-19 data. Soujanya and Vijayasankara (2025) [22] introduced the weighted X-Rani distribution with a focus on reliability studies. These recent models demonstrate the continued interest in improving distributional models to better reflect real-world conditions and datasets. In this research, we propose a new two-parameter lifetime distribution, referred to as the Weighted Ola Distribution (WOD). This model is developed by applying a weight function to the Ola distribution, which was introduced by Al-Ta'ani and Gharaibeh (2023) [20] as a flexible one-parameter model suitable for lifetime data analysis. The proposed WOD enhances the flexibility of the original model and demonstrates superior performance when applied to real-life datasets, including cancer survival data, compared to several existing distributions.

Weighted Ola distribution (WOD)

A new one parameter life distribution name as Ola distribution. the probability density function (pdf) of the Ola distribution is given by 𝑓(𝑥; 𝛽) = 𝛽 8 ; 𝑥, 𝛽 > 0

A random variable x with a probability density function 𝑓(𝑥) has been examined. Let 𝑤(𝑥)be a weight function that is not negative. A new probability density function should be indicated. ; 𝑥 > 0, 𝛽 > 0, 𝛼 > 0

After simplification we get

We can obtain the pdf of the WUD by substituting the values of equations ( 1

Reliability analysis

In this part, we will look at the reliability function, hazard rate, reverse hazard function, odds rate and mills ratio for the proposed weighted Ola distribution.

Reliability function

The reliability or survival function 𝑆(𝑥) is the probability of an item that will survive after a time x. using the reliability function of the weighted Ola distribution is given by 𝛽 4 ((𝛼 + 7)! + 𝛽 4 (𝛼 + 3)! + 𝛽 7 (𝛼)!)

Harmonic Mean

The harmonic Mean of the weighted Ola distribution is defined as

Moment Generating function and Characteristic function

Let 𝑋 𝑤 follows weighted Ola distribution then the Moment Generating Function (MGF) of X is obtained as

Using Taylor's series expansion

Similarly, we can get the characteristic function of weighted Ola distribution can be obtained as

Order Statistics

Let 𝑋 (1) , 𝑋 (2) , 𝑋 (3) , … . , 𝑋 (𝑛) be the order statistics of a random sample 𝑋 1 , 𝑋 2 , 𝑋 3 , … . , 𝑋 𝑛 drawn from the Continuous Population with probability density function 𝑓 𝑋 (𝑥) and 𝐹 𝑋 (𝑥) then the pdf of 𝑟 𝑡ℎ order statistics 𝑋 (𝑟) is given by

Inserting equation ( 5) and ( 6) in equation ( 8), the probability density function of 𝑟 𝑡ℎ order statistic 𝑋 (𝑟) of the weighted Ola distribution is given by

Therefore, the pdf of higher order statistics 𝑿 (𝒏) of weighted Ola distribution can be obtained as

For r = 1, we will determine the probability density function of first order statistic 𝑿 (𝟏) of weighted Ola distribution as

Quantile function

The quantile function of a distribution with cumulative distribution function 𝐹 𝑤 (𝑥; 𝛽, 𝛼) is defined by 𝑞 = 𝐹 𝑞 (𝑥; 𝛽, 𝛼) whether 0<q<1. Thus, the quantile function of weighted Ola distribution is the real solution of the equation.

1 -𝑞 = 1 -𝛾(𝛼 + 8, 𝛽𝑥) + 𝛽 4 𝛾(𝛼 + 4, 𝛽𝑥) + 𝛽 7 𝛾(𝛼 + 1, 𝛽𝑥) (𝛼 + 7)! + 𝛽 4 (𝛼 + 3)! + 𝛽 7 (𝛼)!

Likelihood Ratio Test

This section uses the weighted Ola distribution to derive the likelihood ratio test.

Consider 𝑋 (1) , 𝑋 (2) , 𝑋 (3) , … . , 𝑋 (𝑛) as a random sample from the weighted Ola distribution.

The null and alternative hypotheses are used to test the hypothesis.

𝐻 0: 𝑓(𝑥) = 𝑓(𝑥; 𝛽) 𝑎𝑔𝑎𝑖𝑛𝑠𝑡 𝐻 1: 𝑓(𝑥) = 𝑓 𝑤 (𝑥; 𝛽, 𝛼)

The following test statistics are used to determine if a randomly selected sample of size n originates from the Ola distribution or the weighted Ola distribution. is the observed value of the statistics ∆ * .

∆=

Bonferroni and Lorenz Curves and Gini Index

In this section, we have derived the Bonferroni and Lorenz curves and Gini index from the weighted Ola distribution. The Bonferroni and Lorenz curve is a powerful tool in the analysis of distributions and has applications in many fields, such as economies, insurance, income, reliability, and medicine. The Bonferroni and Lorenz cures for a be the random variable of a unit and 𝑓(𝑥)be the probability density function of x. 𝑓(𝑥)𝑑𝑥 will be represented by the probability that a unit selected at random is defined as 𝑝(Γ(𝛼 + 9) + 𝛽 4 Γ(𝛼 + 5) + 𝛽 7 Γ(𝛼 + 2)) (

𝑝(Γ(𝛼 + 9) + 𝛽 4 Γ(𝛼 + 5) + 𝛽 ) )

Gini index

The information in the Lorenz Curve is often summarized in a single measure called the Gini index (proposed in a 1912 paper by the Italian statistician Corrado Gini. It is often used as a gauge of economic inequality, measuring income distribution. The Gini index is defined as Therefore, the Gini index is for weighted Ola distribution 𝐺 = 1 -2 ∫ 𝐿(𝑝)𝑑𝑝 ∂log Λ 𝜕𝑥 = (𝛼 + 7)𝑥 (𝛼+6) + (𝛼 + 3)𝑥 (𝛼+2) + 𝛼𝑥 (𝛼-1) 𝑥 (𝛼+7) + 𝑥 (𝛼+3) + 𝑥 𝛼 -(𝜆 + 7)𝑥 (𝜆+6) + (𝜆 + 3)𝑥 (𝜆+2) + 𝜆𝑥 (𝜆-1) 𝑥 (𝜆+7) + 𝑥 (𝜆+3) + 𝑥 𝜆 + (𝛽 -𝜃) = 0

Hence 𝜕𝑙𝑜𝑔 [𝛬] 𝜕𝑥 < 0 𝑖𝑓 β < θ and 𝛼 < λ

Entropies

In this part, we used the weighted Ola distribution to calculate Shannon, Renyi, and Tsallis entropies.

It is commonly known that entropy and information can be used to measure uncertainty or the randomness of probability distributions. It is used in a variety of disciplines, including engineering, finance, information theory, and biomedicine. The entropy functionals for the probability distribution were developed using a variational concept of uncertainty.

Shannon Entropy

The Shannon entropy of the random variable X that defines the weighted Ola distribution

; 𝜆 > 0, 𝜆 ≠ 1 𝑠 𝜆 = -∫ 𝑓(𝑥; 𝛽, 𝛼) log(𝑓(𝑥; 𝛽, 𝛼))𝑑𝑥 ∞ 0 𝑠 𝜆 = -∫ ( 𝑥 𝛼 𝛽 𝛼+8 (𝑥 7 + 𝑥 3 + 1)𝑒 -𝛽𝑥 (𝛼 + 7)! + 𝛽 4 (𝛼 + 3)! + 𝛽 7 (𝛼)!

) log ( 𝑥 𝛼 𝛽 𝛼+8 (𝑥 7 + 𝑥 3 + 1)𝑒 -𝛽𝑥 (𝛼 + 7)! + 𝛽 4 (𝛼 + 3)! + 𝛽 7 (𝛼)!

Renyi Entropy

Entropy is described as a random variable. X represents the variation of the uncertainty. Engineering, statistical mechanics, finance, information theory, biology, and economics are among the domains where it is applied. The entropy is the Renyi of order, which is defined as

Using Binomial expansion, we get

Tsallis Entropy

The Boltzmann-Gibbs (B-G) statistical properties initiated by Tsallis have received a great deal of attention. This generalization of (B-G) statistics was first proposed by introducing the mathematical expression of Tsallis entropy (Tsallis, (1988) for continuous random variables, which is defined as 4 (𝛼 + 3)! + 𝛽 7 The log likelihood function is given by log 𝐿 = log ( 𝛽 𝛼+8 (𝛼 + 7)! + 𝛽 4 (𝛼 + 3)! + 𝛽 7 The maximum likelihood estimates of the parameters of the distribution are obtained by solving this nonlinear system of equations. Therefore, we use R and wolfram mathematics for estimating the parameters of the newly proposed distribution. We apply asymptotic normality results to get the confidence interval. If 𝜆 ̂= (𝜃 ̂) represents the MLE of 𝜆 = (𝜃), we can state the following results:√𝑛(𝜆 ̂-𝜆) → 𝑁 2 (0, 𝐼 ]

Since 𝜆 being unknown, we estimate 𝐼 -1 (𝜆) by 𝐼 -1 (𝜆 ̂) and this can be used to obtain asymptotic confidence interval for 𝛽, 𝛼.

Applications Data set 1

This application is from [3] and it is about the remission times (in months) of 128 patients suffering from bladder cancer. This data has been analyzed recently in many papers, such as [10,12] . Where, k=number of parameters, n sample size and -2log L is the maximized value of loglikelihood function. From table 1, and 2 it can be clearly observed and seen from the results that the weighted Ola distribution have the lesser AIC, BIC, AICC, -2log𝐿, and values as compared to the Ola, exponential and Lindley distributions, which indicates that the weighted Ola distribution better fits than the Ola, exponential and Lindley distributions. Therefore, it can be concluded that the weighted Ola distribution provides a better fit than the other compared distributions.

Conclusion

In this study, we introduced the Weighted Ola distribution as an extended form of the standard Ola distribution to provide greater flexibility for modeling lifetime data. We derived several statistical properties of the proposed distribution, including moments, reliability functions, and order statistics. The parameters were estimated using the method of maximum likelihood. The practical application of the Weighted Ola distribution was demonstrated using real-life cancer survival data. The model was compared with the original Ola, Exponential, and Lindley distributions. Based on model selection criteria such as AIC, BIC, AICC, and -2LogL, the Weighted Ola distribution showed the best fit among all the models considered. Overall, the results confirm that the Weighted Ola distribution is a more accurate and flexible model for lifetime data analysis and can be effectively used in medical and reliability studies.

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