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Nil Kamal Hazra

Publications and source records attributed to Nil Kamal Hazra.

At least 19 recordsLinked to original sources
arXiv2026

Extreme Population Selection under Multistage Sampling design With Applications

We study the problem of selecting the extreme (best or worst) population from among $K(\geq 2)$ populations, under the assumption that the extreme population is sufficiently separated from the nearest population. The selection is based on an appropriate measure, which may vary across different application domains. Since the actual value of the measure is unknown, we obtain its estimator using the generalized method of moments under a multistage sampling design. Using this estimator, we propose two sequential procedures, namely online algorithm and multi armed bandit based algorithm. Under suitable regularity conditions and without imposing parametric assumptions on the underlying distributions, both algorithms correctly identify the extreme population with a desired level of confidence. We illustrate the proposed algorithms through applications in econometrics and genetics. In the econometric application, the extreme population is selected using the Gini index as a measure of inequality and the performance of the proposed procedures is assessed through extensive Monte Carlo simulation studies conducted under various distributional settings. In the genetics application, the worst population is identified using a measure derived from the tumor mutation burden (TMB) score and the practical applicability of the proposed algorithms is demonstrated using the Memorial Sloan Kettering-IMPACT 50000 clinical sequencing cohort. Further, we use the proposed framework to identify an anomalous population, provided such a population exists.

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arXiv2026

Asymptotically Optimal Sequential Confidence Interval for the Gini Index Under Complex Household Survey Design with Sub-Stratification

We examine the optimality properties of the Gini index estimator under complex survey design involving stratification, clustering, and sub-stratification. While Darku et al. (Econometrics, 26, 2020) considered only stratification and clustering and did not provide theoretical guarantees, this study addresses these limitations by proposing two procedures - a purely sequential method and a two-stage method. Under suitable regularity conditions, we establish uniform continuity in probability for the proposed estimator, thereby contributing to the development of random central limit theorems under sequential sampling frameworks. Furthermore, we show that the resulting procedures satisfy both asymptotic first-order efficiency and asymptotic consistency. Simulation results demonstrate that the proposed procedures achieve the desired optimality properties across diverse settings. The practical utility of the methodology is further illustrated through an empirical application using data collected by the National Sample Survey agency of India

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arXiv2025

A Generalized Notion of Completeness and Its Application

From the perspective of data reduction, the notions of minimal sufficient and complete statistics together play an important role in determining optimal statistics (estimators). The classical notion of sufficiency and completeness are not adequate in many robust estimations that are based on different divergences. Recently, the notion of generalized sufficiency based on a generalized likelihood function was introduced in the literature. It is important to note that the concept of sufficiency alone does not necessarily produce optimal statistics (estimators). Thus, in line with the generalized sufficiency, we introduce a generalized notion of completeness with respect to a generalized likelihood function. We then characterize the family of probability distributions that possesses completeness with respect to the generalized likelihood function associated with the density power divergence (DPD). Moreover, we show that the family of distributions associated with the logarithmic density power divergence (LDPD) is not complete. Further, we extend the Lehmann-Scheffé theorem and the Basu's theorem for the generalized likelihood estimation. Subsequently, we obtain the generalized uniformly minimum variance unbiased estimator (UMVUE) for the $\mathcal{B^{(α)}}$-family. Further, we derive an formula of the asymptotic expected deficiency (AED) that is used to compare the performance between the minimum density power divergence estimator (MDPDE) and the generalized UMVUE for $\mathcal{B^{(α)}}$-family. Finally, we provide an application of the developed results in stress-strength reliability model.

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arXiv2025

On Generalized Likelihood Estimation Based on the Logarithmic Norm Relative Entropy

Traditional likelihood based methods for parameter estimation get highly affected when the given data is contaminated by outliers even in a small proportion. In this paper, we consider a robust parameter estimation method, namely the minimum logarithmic norm relative entropy (LNRE) estimation procedure, and study different (generalized) sufficiency principles associated with it. We introduce a new two-parameter power-law family of distributions (namely, $\mathcal{M}^{(α,β)}$-family), which is shown to have a fixed number of sufficient statistics, independent of the sample size, with respect to the generalized likelihood function associated with the LNRE. Then, we obtain the generalized minimal sufficient statistic for this family and derive the generalized Rao-Blackwell theorem and the generalized Cramér-Rao lower bound for the minimum LNRE estimation. We also study the minimum LNRE estimators (MLNREEs) for the family of Student's distributions particularly in detail. Our general results reduces to the classical likelihood based results under the exponential family of distributions at specific choices of the tuning parameter $α$ and $β$. Finally, we present simulation studies followed by a real data analysis, which highlight the practical utility of the MLNREEs for data contaminated by possible outliers. Along the way we also correct a mistake found in a recent paper on related theory of generalized likelihoods.

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arXiv2023

On multivariate orderings of some general ordered random vectors

Ordered random vectors are frequently encountered in many problems. The generalized order statistics (GOS) and sequential order statistics (SOS) are two general models for ordered random vectors. However, these two models do not capture the dependency structures that are present in the underlying random variables. In this paper, we study the developed sequential order statistics (DSOS) and developed generalized order statistics (DGOS) models that describe the dependency structures of ordered random vectors. We then study various univariate and multivariate ordering properties of DSOS and DGOS models under Archimedean copula. We consider both one-sample and two-sample scenarios and develop corresponding results.

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arXiv2020

Reliability study of series and parallel systems of heterogeneous component lifetimes under proportional odds model

In this paper, we investigate various stochastic orderings for series and parallel systems with independent and heterogeneous components having lifetimes following the proportional odds model. We also investigate comparisons between system with heterogeneous components and that with homogeneous components. This paper also studies relative ageing orders for two systems in the framework of components having lifetimes following the proportional odds model.

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arXiv2019

On Relative Ageing of Coherent Systems with Dependent Identically Distributed Components

Relative ageing describes how a system ages with respect to another one. The ageing faster orders are the ones which compare the relative ageings of two systems. Here, we study ageing faster orders in the hazard and the reversed hazard rates. We provide some sufficient conditions for proving that one coherent system dominates another system with respect to ageing faster orders. Further, we investigate whether the active redundancy at the component level is more effective than that at the system level with respect to ageing faster orders, for a coherent system. Furthermore, a used coherent system and a coherent system made out of used components are compared with respect to ageing faster orders.

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arXiv2019

Coherent systems with dependent and identically distributed components: A study of relative ageing based on cumulative hazard and cumulative reversed hazard rate functions

The relative ageing is an important notion which is useful to measure how a system ages relative to another one. Among all existing stochastic orders, there are two important orders describing the relative ageing of two systems, namely, ageing faster orders in the cumulative hazard and the cumulative reversed hazard rate functions. In this paper, we give some sufficient conditions under which one coherent system ages faster than another one with respect to the aforementioned stochastic orders. Further, we show that the proposed sufficient conditions are satisfied for $k$-out-of-$n$ systems. Moreover, some numerical examples are given to illustrate the developed results.

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arXiv2018

Generalization of the pairwise stochastic precedence order to the sequence of random variables

We discuss a new stochastic ordering for the sequence of independent random variables. It generalizes the stochastic precedence order that is defined for two random variables to the case $n>2$. All conventional stochastic orders are transitive, whereas the stochastic precedence order is not. Therefore, a new approach to compare the sequence of random variables had to be developed that resulted in the notion of the sequential precedence order. A sufficient condition for this order is derived and some examples are considered.

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arXiv2018

Comparing Lifetimes of Coherent Systems with Dependent Components Operating in Random Environments

We study an impact of a random environment on lifetimes of coherent systems with dependent components. There are two combined sources of this dependence. One results from the dependence of the components of the coherent system operating in a deterministic environment and the other is due to dependence of components of the system sharing the same random environment. We provide different sets of sufficient conditions for the corresponding stochastic comparisons and consider various scenarios, namely, (i) two different coherent systems operate under the same random environment; (ii) two coherent systems operate under two different random environments; (iii) one of the coherent systems operates under a random environment, whereas the other under a deterministic one. Some examples are given to illustrate the proposed reasoning.

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arXiv2017

On Component Redundancy Versus System Redundancy for a $k$-out-of-$n$ System

Precedence order is a natural type of comparison for random variables in numerous engineering applications (e.g., for the stress-strength modeling). In this note, we show that, for a $k$-out-of-$n$ system, redundancy at the component level is superior to that at the system level with respect to the stochastic precedence order. Cases of active and cold redundancy are considered. Similar results for other stochastic orders were intensively discussed in the literature.

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arXiv2016

On Stochastic Comparisons for Load-Sharing Series and Parallel Systems

We study the allocation strategies for redundant components in the load-sharing series/parallel systems. We show that under the specified assumptions, the allocation of a redundant component to the stochastically weakest (strongest) component of a series (parallel) system is the best strategy to achieve its maximal reliability. The results have been studied under cumulative exposure model and for a general scenario as well. They have a clear intuitive meaning, however, the corresponding additional assumptions are not obvious, which can be seem from the proofs of our theorems

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arXiv2016

Some Reliability Properties of Transformed-Transformer Family of Distributions

The Transformed-Transformer family of distributions are the resulting family of distributions as transformed from a random variable $T$ through another transformer random variable $X$ using a weight function $ω$ of the cumulative distribution function of $X$. In this paper, we study different stochastic ageing properties, as well as different stochastic orderings of this family of distributions. We discuss the results with several well known distributions.

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arXiv2016

On Stochastic Comparison Between Two Coherent Systems

Stochastic orders are very useful tool to compare the lifetimes of two coherent systems. We show that, under certain conditions, a coherent system of used components performs better (worse) than a used coherent system with respect to different stochastic orders. Some results on stochastic comparison between a coherent system of inactive components and an inactive coherent system are also discussed.

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arXiv2016

A Note on Warm Standby System

In this note, we investigate, under what circumstances, a warm standby system (formed by n active components and m warm standby components) has more number of surviving warm standby components than another similar system at the time of $k$th failure of the active component of the respective system. The number of such components being random, the comparison has been done with respect to different stochastic orders, viz. usual stochastic order, hazard rate and reversed hazard rate orders, and likelihood ratio order.

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arXiv2016

Ordering Properties of Order Statistics from Heterogeneous Generalized Exponential and Gamma Populations

Let $X_1, X_2,\ldots, X_n$ (resp. $Y_1, Y_2,\ldots, Y_n$) be independent random variables such that $X_i$ (resp. $Y_i$) follows generalized exponential distribution with shape parameter $θ_i$ and scale parameter $λ_i$ (resp. $δ_i$), $i=1,2,\ldots, n$. Here it is shown that if $\left(λ_1, λ_2,\ldots,λ_n\right)$ is $p$-larger than (resp. weakly supermajorizes) $\left(δ_1,δ_2,\ldots,δ_n\right)$, then $X_{n:n}$ will be greater than $Y_{n:n}$ in usual stochastic order (resp. reversed hazard rate order). That no relation exists between $X_{n:n}$ and $Y_{n:n}$, under same condition, in terms of likelihood ratio ordering has also been shown. It is also shown that, if $Y_i$ follows generalized exponential distribution with parameters $\left(\barλ,θ_i\right)$, where $\barλ$ is the mean of all $λ_i$'s, $i=1\ldots n$, then $X_{n:n}$ is greater than $Y_{n:n}$ in likelihood ratio ordering. Some new results on majorization have been developed which fill up some gap in the theory of majorization. Some results on multiple-outlier model are also discussed. In addition to this, we compare two series systems formed by gamma components with respect to different stochastic orders.

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arXiv2014

On Some Generalized Orderings: In the Spirit of Relative Ageing

We introduce some new generalized stochastic orderings (in the spirit of relative ageing) which compare probability distributions with the exponential distribution. These orderings are useful to understand the phenomenon of positive ageing classes and also helpful to guide the practitioners when there are crossing hazard rates and/or crossing mean residual lives. We study some characterizations of these orderings. Inter-relations among these orderings have also been discussed.

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