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Weighted Tail Random Variable: A Novel Framework with Stochastic Properties and Applications

Sarikul Islam, Nitin Gupta

TL;DR

The paper presents a novel framework to construct non-negative continuous distributions by combining a base variable's survival function with an increasing weight function, producing the weighted tail random variable $X_w$ with $f_{X_w}(x) = \frac{w'(x)\overline{F}_X(x)}{\mathbb{E}[w(X)]}$. It analyzes reliability and stochastic-order properties of $X_w$, including preservation of IFR/DFR and various orderings under conditions on $w$ and the base variable, and shows how equilibrium distributions arise as a special case when $w(x)=x$. The framework is applied to Kumaraswamy distributions to form the weighted Kumaraswamy (WK) family, deriving closed-form PDFs and moments, and demonstrated on monsoon rainfall data where WK provides superior goodness-of-fit over the base Kw. The results highlight the approach’s potential to yield flexible, tail-aware models with practical benefits for reliability, survival analysis, and environmental data modeling. Overall, the weighted tail framework offers a principled, extensible method to tailor distributions by leveraging survival structure and a user-defined weight function, with demonstrated gains in empirical fit and interpretability.

Abstract

This paper introduces a novel framework to construct the probability density function (PDF) of non-negative continuous random variables. The proposed framework uses two functions: one is the survival function (SF) of a non-negative continuous random variable, and the other is a weight function, which is an increasing and differentiable function satisfying some properties. The resulting random variable is referred to as the weighted tail random variable (WTRV) corresponding to the given random variable and the weight function. We investigate several reliability properties of the WTRV and establish various stochastic orderings between a random variable and its WTRV, as well as between two WTRVs. Using this framework, we construct a WTRV of the Kumaraswamy distribution. We conduct goodness-of-fit tests for two real-world datasets, applied to the Kumaraswamy distribution and its corresponding WTRV. The test results indicate that the WTRV offers a superior fit compared to the Kumaraswamy distribution, which demonstrates the utility of the proposed framework.

Weighted Tail Random Variable: A Novel Framework with Stochastic Properties and Applications

TL;DR

The paper presents a novel framework to construct non-negative continuous distributions by combining a base variable's survival function with an increasing weight function, producing the weighted tail random variable with . It analyzes reliability and stochastic-order properties of , including preservation of IFR/DFR and various orderings under conditions on and the base variable, and shows how equilibrium distributions arise as a special case when . The framework is applied to Kumaraswamy distributions to form the weighted Kumaraswamy (WK) family, deriving closed-form PDFs and moments, and demonstrated on monsoon rainfall data where WK provides superior goodness-of-fit over the base Kw. The results highlight the approach’s potential to yield flexible, tail-aware models with practical benefits for reliability, survival analysis, and environmental data modeling. Overall, the weighted tail framework offers a principled, extensible method to tailor distributions by leveraging survival structure and a user-defined weight function, with demonstrated gains in empirical fit and interpretability.

Abstract

This paper introduces a novel framework to construct the probability density function (PDF) of non-negative continuous random variables. The proposed framework uses two functions: one is the survival function (SF) of a non-negative continuous random variable, and the other is a weight function, which is an increasing and differentiable function satisfying some properties. The resulting random variable is referred to as the weighted tail random variable (WTRV) corresponding to the given random variable and the weight function. We investigate several reliability properties of the WTRV and establish various stochastic orderings between a random variable and its WTRV, as well as between two WTRVs. Using this framework, we construct a WTRV of the Kumaraswamy distribution. We conduct goodness-of-fit tests for two real-world datasets, applied to the Kumaraswamy distribution and its corresponding WTRV. The test results indicate that the WTRV offers a superior fit compared to the Kumaraswamy distribution, which demonstrates the utility of the proposed framework.

Paper Structure

This paper contains 11 sections, 23 theorems, 46 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Notations, Preliminary Concepts, and Lemmas
  3. Methodology of the Framework
  4. Derivation of the Framework
  5. Reliability Properties of a Random Variable and Its WTRV
  6. Stochastic Ordering Properties of Random Variables and Their WTRVs
  7. An Application of the Framework
  8. Model Description
  9. Estimation of Model Parameters
  10. Statistical Testing for the Goodness-of-Fit of the Datasets
  11. Conclusion

Key Result

Lemma 1

Consider the independent variables $X$ and $Y$ having their respective supports $S_X=(l_X,\,u_X)$ and $S_Y=(l_Y,\,u_Y)$. Then $X \leq_{st} Y$ if and only if $l_X \leq l_Y$, $u_X \leq u_Y$, and $\mathbb{E}[\phi_2(X,Y)] \geq \mathbb{E}[\phi_1(X,Y)]$ for all such functions $\phi_2(\cdot,\cdot)$, $\phi_ where $\overline{S}_X = [l_X, u_X]$ and $\overline{S}_Y = [l_Y, u_Y]$ denote the closure of $S_X$ a

Figures (3)

  • Figure 1: Plot of the function $f_{Y_{w_2}}(\cdot)/f_{X_{w_1}}(\cdot)$.
  • Figure 2: Density plot of weighted Kumaraswamy distribution with parameters $a,\,b,\,c$.
  • Figure 3: Plots of the histograms and corresponding fitted densities of the datasets.

Theorems & Definitions (41)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Proposition 1
  • Example 1
  • ...and 31 more