Weighted distribution method for constructing flexible probability models with applications

Introduction

Statistical distributions play a fundamental role in data science by offering a structured framework for understanding the behavior of random variables. A probability distribution characterizes the possible values a variable may assume and facilitates the computation of probabilities associated with different events. As a result, statistical modeling and real-world data analysis rely extensively on probability distributions. Nevertheless, data arising from diverse domains such as environmental studies, finance, biomedical sciences, and other applied disciplines often deviate from classical distributional assumptions. This inadequacy has led to the development of new and more flexible distributional models capable of capturing varying degrees of skewness and kurtosis, thereby enhancing goodness-of-fit and improving the accuracy of empirical data modeling. Many widely used classical probability distributions are often inadequate for accurately modeling and predicting data arising in applied fields such as biology, environmental sciences, engineering, medical sciences, finance, and economics. Consequently, numerous generalized families of distributions have been developed as extensions of traditional models. These newly proposed families have been rigorously investigated and demonstrated to provide greater flexibility and improved adaptability in practical applications. Such extended distributional frameworks have been successfully utilized to model diverse types of data across various scientific disciplines. A common and effective strategy in distribution theory involves augmenting existing families with additional parameters. The introduction of extra parameters typically enhances model flexibility, enabling better accommodation of complex data structures and improving overall model performance. The concept of weighted distributions (WD) provides an effective framework for addressing challenges in data interpretation and model construction. The idea was initially introduced by Fisher [1] to account for ascertainment bias in observed data, and later systematically formalized by Rao [2] , who developed a structured approach for modeling situations where conventional probability distributions fail to adequately represent observations occurring with unequal selection probabilities. Weighted distributions have since found wide applications in diverse areas such as ecology, biomedicine, branching processes, and reliability analysis. This methodology provides an alternative mechanism for generating modified distributions by incorporating a weight function into the baseline model. When samples may arise from both the original and the weighted versions of a distribution, the weighted distribution framework facilitates model fitting even when the weight function is unknown. A notable special case occurs when the weight function is proportional to the unit length, leading to the well-known length-biased distribution. A brief review, perspective, and characterization of Weighted distributions studied by Saghir. et al., [3] . Area-biased Poisson exponential distribution with applications explained by Fazal [4] . Mathew [5] detailed some length-biased distribution and its characteristics. Over time, numerous researchers have explored various forms of weighted probability models and demonstrated their usefulness across different applied contexts. The Sauleh distribution is a composite model derived from a combination of the Gamma and Exponential distributions. It is a newly proposed one-parameter lifetime distribution introduced by Aijaz Ahmad et al. [6] , and is designed for modeling lifetime data with enhanced flexibility compared to classical single-parameter models.

Weighted Distributions (WD)

Let X be a non-negative random variable with pdf ) (x f , then the pdf of a weighted random variable w X is presented as, For various selections of weight function w (x) weighted models are of many forms when w (x) = x c , the resulting distribution is termed as WD. The nomenclature of a newly derived distribution need not necessarily reflect its parent distribution, particularly when the resulting model does not preserve the fundamental characteristics of the original one. In many instances, the newly constructed distribution exhibits structural and statistical properties that are substantially different from those of its parent model. This transformation may be likened to the formation of water from oxygen and hydrogen, where the resulting compound possesses properties entirely distinct from those of its constituent elements.

General case: KrishSupra-P distribution

Consider the initial distribution as Sauleh distribution with probability density function (pdf)

and the cumulative distribution function (cdf) 0 , 0 ; 6 2

The weighted version of the Sauleh distribution by letting w (x) = x c and the new distribution is termed as the KrishSupra-P Distribution (KSPD). The pdf of the WD is provided by

Substituting the eqs. ( 1) and ( 5) in eq. ( 4), the KSPD's pdf as

And the cdf of KSPD is attained as

Here γ(c+1,θx), γ(c+3,θx) & γ(c+4,θx) represents the Incomplete Gamma function.

The survival function of KSPD is, To compare the KSPD with Sauleh distributions, Exponential distributions, as well as LDs, we consider the parameters AICC ("Akaike Information Criterion Corrected"), AIC ("Akaike Information Criterion"), BIC ("Bayesian Information Criterion"), and -2logL. The distribution with lower values of -2logL, AICC, BIC, &AIC is the better one. For determining the criteria AIC, AICC, BIC, and -2logL formulas mentioned below are applied.

Analysis of data

Here, n represents the size of the sample, k indicates the number of parameters inside the statistical model, and -2logLpresents "the maximized value of the log-likelihood function in the model. Table 2 implies the KSPD has the lesser values of -2logL, BIC, AICC, and AIC as compared to exponential, Sauleh, and Lindley distributions. Hence, it shows the KSPD leads to a better fit than Sauleh, exponential as well and Lindley distributions.

Conclusion

This study introduces a novel probability distribution, KSPD, and examines its key statistical properties. Application to a real data set demonstrates that KSPD outperforms the Exponential, Sauleh, and Lindley distributions in terms of model fit.

Area biased distribution: PrasNikh-N distribution

The '2-parameter' Akash distribution (TAD) is a freshly proposed life-time distribution introduced by Shanker &Shukla [7] of which the Akash distribution with one parameter is a specific case. Shanker [8] explained Akash distribution and applications. the pdf (probability density function) of TAD is, (10) The cdf (Cumulative distribution function) of TAD is,

The weighted random variable w X is with pdf,

Since the weight function w (x) = x 2 , the new distribution is called Area Biased Distribution with pdf, ) (

In equation ( 12), apply the equations ( 10) and (13) we get the pdf of new distribution which is termed as PrasNikh-N Distribution (P2ND) as,

Application

Table 3 presents real medical lifetime data comprising the weight loss (kg) observed after the first cycle of chemotherapy for 60 randomly selected patients from a hospital in Thrissur district, Kerala. All patients were diagnosed with gastrointestinal (GI) cancer, which includes malignancies of the digestive tract such as the small and large intestine, stomach, colon, pancreas, anus, rectum, biliary system, and liver.

Conclusion

A generalized form of the TAD distribution is proposed and termed P2ND. The properties of P2ND are thoroughly examined, and its practical relevance is illustrated using medical data. Studying the characteristics of such biomedical data is essential for reliable modeling and inference. The results indicate that P2ND provides a better fit compared to the TAD, Akash, Exponential, and Lindley distributions.

Length-biased distribution: ChandBhas-P Distribution (CBPD)

The two-parameter Sujatha distribution is a recently proposed lifetime model introduced by Tesfay and Shanker [9] , wherein the one-parameter Akash and Sujatha distributions emerge as special cases. In a subsequent work [10] , Tesfay and Shanker further developed a two-parameter Sujatha model that incorporates both the size-biased Lindley and the Sujatha distributions.

The probability density function (pdf) of the new two-parameter Sujatha distribution is,  

and the cumulative density function (cdf) of the new two-parameter Sujatha distribution is presented by

,the pdf of a non-negative random Variable X, then the pdf of weighted w X is,

Where w (x) be the non-negative weight function The distribution that results from our consideration of the "weight function" at w (x) = x is termed the "length-biased distribution". The New Two -Parameter Sujatha distribution is termed as ChandBhas-P Distribution (CBPD) which is a length-biased transformation with pdf, ) ((

Thereby the pdf of the CBPD by putting eqs. ( 17) & (20) into eq. ( 19).

And the cdf of CBP Distribution is,

By simplifying the eq. ( 6), the cdf of CBPD is, The Survival function is,

The Hazard Rate of the distribution is, 10. The nature of hazard rate is noticed from Figure 11. 5 represents the birth weight (Kg) of new born babies, randomly selected from a hospital at Thrissur, Kerala. Data set 5.2: Table 6 shows the weight (Kg) of 100 randomly selected children-of age range between 3 months to 4 years-from a few babysitting centres and play schools across Thrissur. The distribution with the smallest values of AIC, AICC, BIC, and -2logL is regarded as the best-fitting model. From Tables 7 and8, the CBP distribution exhibits lower AIC, AICC, BIC, and -2logL values compared to the New Two-Parameter Sujatha, Sujatha, Two-Parameter Sujatha, Lindley, and Exponential distributions. This confirms that the CBP distribution provides a superior fit to the given biomedical data. Hence, the effectiveness and significance of the proposed distribution are established.

Conclusion

In this study, a novel distribution, termed the ChandBhas-P (CBP) distribution, is introduced using the length-biased approach as a modification of the conventional distribution. The superiority of the proposed model is evaluated using two biomedical data sets.

The results demonstrate that the CBP distribution provides a substantially better fit than the New Two-Parameter Sujatha, Sujatha, Two-Parameter Sujatha, Exponential, and Lindley distributions for such biological data.

References

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