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Asok Nanda

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C. Michael Hall

University of Canterbury/Te Whare Wānanga o Waitaha

Plymouth University

Ioannis Ntzoufras

Athens University of Economics and Business

Kansas State University

Nader Ale Ebrahim نادر آل ابراهیم

Alzahra University

Gazi University

Erasmus University Rotterdam

Università di Verona

Alejandro Serna

The British Museum

Marcelo F Napoli

UFBA - Federal University of Bahia

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Papers by Asok Nanda

A Loglikelihood-Based Shape Measure of Past Lifetime Distribution

Calcutta Statistical Association Bulletin, 2009

Generalized past information measure based on Rényi entropy for a continuous random variable usef... more Generalized past information measure based on Rényi entropy for a continuous random variable useful in life testing is considered and its connection to the loglikelihood is established. From this relation, a loglikelihood-based past life distribution measure [Formula: see text] has been proposed. From the gradient of the spectrum of generalized past information, the [Formula: see text] measure has been calculated for some important parametric life distributions. Serving as a measure of the shape of a past life (inactivity time) distribution, [Formula: see text] can be used to compare the tails and shapes of various frequently used densities, where the traditional kurtosis measure is not applicable.

Conditional precedence orders for stochastic comparison of random variables

Statistics & Probability Letters

Most of the stochastic orders for comparing random variables, considered in the literature, are a... more Most of the stochastic orders for comparing random variables, considered in the literature, are afflicted with two main drawbacks: piq lack of connex property and piiq lack of consideration of any dependence structure between the random variables. Both these drawbacks can be overcome at the cost of transitivity with the stochastic precedence order, which may seem to be a good choice in particular when only two random variables are under consideration, a situation where the question of transitivity does not arise. In this paper, we show that even under such favorable conditions, stochastic precedence order may direct to misleading conclusion in certain situations and develop variations of the order to address the phenomenon.

Some Ageing Properties of Dynamic Additive Mean Residual Life Model

arXiv: Statistics Theory, 2017

Although proportional hazard rate model is a very popular model to analyze failure time data, som... more Although proportional hazard rate model is a very popular model to analyze failure time data, sometimes it becomes important to study the additive hazard rate model. Again, sometimes the concept of the hazard rate function is abstract, in comparison to the concept of mean residual life function. A new model called `dynamic additive mean residual life model' where the covariates are time-dependent has been defined in the literature. Here we study the closure properties of the model for different positive and negative ageing classes under certain condition(s). Quite a few examples are presented to illustrate different properties of the model.

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

Statistics, 2020

In this paper, we investigate various stochastic orderings for series and parallel systems with i... more 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.

Some Reliability Properties of Transformed-Transformer Family of Distributions

American Journal of Mathematical and Management Sciences, 2018

The Transformed-Transformer family of distributions are the resulting family of distributions as ... more 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.

The Hazard Rate and the Reversed Hazard Rate Orders, with Applications to Order Statistics

In this paper we first point out a simple observation that can be used successfully in order to t... more In this paper we first point out a simple observation that can be used successfully in order to translate results about the hazard rate order into results about the reversed hazard rate order. Using it, we derive some interesting new results which compare order statistics in the hazard and in the reversed hazard rate orders; as well as in the

Quantile based entropy function in past lifetime

Statistics & Probability Letters, 2013

Di Crescenzo and Longobardi (2002) introduced a measure of uncertainty in past lifetime distribut... more Di Crescenzo and Longobardi (2002) introduced a measure of uncertainty in past lifetime distributions and studied its relationship with residual entropy function. In the present paper, we introduce a quantile version of the entropy function in past lifetime and study its properties. Unlike the measure of uncertainty given in Di Crescenzo and Longobardi (2002) the proposed measure uniquely determines the underlying probability distribution. The measure is used to study two nonparametric classes of distributions. We prove characterizations theorems for some well known quantile lifetime distributions.

Reliability study of proportional odds family of discrete distributions

Communications in Statistics - Theory and Methods, 2017

The proportional odds model gives a method of generating new family of distributions by adding a ... more The proportional odds model gives a method of generating new family of distributions by adding a parameter, called tilt parameter, to expand an existing family of distributions. The new family of distributions so obtained is known as Marshall-Olkin family of distributions or Marshall-Olkin extended distributions. In this paper, we consider Marshall-Olkin family of distributions in discrete case with fixed tilt parameter. We study different ageing properties, as well as different stochastic orderings of this family of distributions. All the results of this paper are supported by several examples.

Some distributional results through past entropy

Journal of Statistical Planning and Inference, 2010

... View the MathML source. It is to be noted here that φ(t)=(1−m ′ (t))/m(t), where m ′ (t)=(d/d... more

On some generalized ageing orderings

Some partial orderings which compare probability distributions with the expo- nential distributio... more Some partial orderings which compare probability distributions with the expo- nential distribution, are found to be very useful to understand the phenomenon of ageing. Here, we introduce some new generalized partial orderings which de- scribe the same kind of characterization of some generalized ageing classes. We give some equivalent conditions for each of the orderings. Inter-relations among the generalized orderings have also been discussed.

On closure of some partial orderings under mixtures

Journal of applied probability, 1996

Recently new partial orderings s-SFR, s-FR, s-ST, s-CV and s-CX have been defined in the literatu... more Recently new partial orderings s-SFR, s-FR, s-ST, s-CV and s-CX have been defined in the literature. These are more general in the sense that most of the earlier known partial orderings reduce to particular cases of these orderings. Moreover, these orderings have helped in defining a new and useful ageing criterion. In this paper, we discuss preservation properties of these orderings under mixtures. Results are expected to be useful in engineering and reliability, where mixtures of distributions arise naturally in numerous situations. Moreover, existing results, specifically for LR, FR and ST orderings under the operation of mixtures, reduce to special cases.

Shannon's entropy and its Generalizations towards Statistics, Reliability and Information Science during 1948-2018

arXiv: Other Statistics, 2019

Starting from the pioneering works of Shannon and Weiner in 1948, a plethora of works have been r... more Starting from the pioneering works of Shannon and Weiner in 1948, a plethora of works have been reported on entropy in different directions. Entropy-related review work in the direction of statistics, reliability and information science, to the best of our knowledge, has not been reported so far. Here we have tried to collect all possible works in this direction during the period 1948-2018 so that people interested in entropy, specially the new researchers, get benefited.

Discrete life distributions with decreasing reversed hazard

Sankhyā: The Indian Journal of Statistics (2003-), 2005

This paper considers the class of discrete distributions for which the distribution function is a... more This paper considers the class of discrete distributions for which the distribution function is a log-concave sequence. It is shown that such distributions arise from a wide variety of circumstances in Reliability, and that these have a decreasing reversed hazard rate. After examining the closures of the class under certain key operations, sharp upper and lower bounds on the reliability function for the member distributions are given. Some useful inequalities for maintained systems are provided. Some results for the related class of discrete concave distributions are also given.

Errors due to departure from independence in exponential series system

Communications in Statistics - Theory and Methods

Optimal Replacement in a Discrete time Shock Model

OPSEARCH

ABSTRACT

Parameter estimates of general failure rate model: A Bayesian approach

Journal of Computational and Applied Mathematics

The failure rate function plays an important role in studying the lifetime distributions in relia... more The failure rate function plays an important role in studying the lifetime distributions in reliability theory and life testing models. A study of the general failure rate model r(t) = a + bt θ−1 , under squared error loss function taking a and b independent exponential random variables has been analyzed in the literature. In this article, we consider a and b not necessarily independent. The estimates of the parameters a and b under squared error loss, linex loss and entropy loss functions are obtained here.

Tests for Reversed Hazard Rate Function

Calcutta Statistical Association Bulletin

ABSTRACT The reversed hazard rate function of a random variable, defined as the ratio of the dens... more ABSTRACT The reversed hazard rate function of a random variable, defined as the ratio of the density to its distribution function, plays an important role in analyzing left-censored data. We develop tests based on the empirical distribution function for an assigned reversed hazard rate function against a one-sided alternative and two different tests for equality of two reversed hazard rate functions. The asymptotic normality of the proposed test statistics has been established.

On Improvement and Deterioration of A Repairable System Under Generalized Stochastic Orders

Calcutta Statistical Association Bulletin

ABSTRACT

Stochastic Orders in Terms of Laplace Transforms

Calcutta Statistical Association Bulletin

Recently s-FR and s-ST orderings have been defined in the literature. They are more general in th... more Recently s-FR and s-ST orderings have been defined in the literature. They are more general in the sense that most of the earlier known partial orderings reduce as particular cases of these orderings. Moreover, these orderings have helped in defining new and useful ageing criterion. In this paper, using Laplace transform, we characterize, by means of necessary and sufficient conditions. the property that two life distributions are ordered in the s-FR and s-ST sense. The characterization of LR, FR, MR, VR, STand HAMR orderings follow as particular cases.

Characterization of distributions through failure rate and mean residual life functions

Statistics Probability Letters, May 1, 2010

Makino [Makino, T., 1984. Mean hazard rate and its applications to the normal approximation of th... more Makino [Makino, T., 1984. Mean hazard rate and its applications to the normal approximation of the Weibull distribution. Naval Research Logistics Quarterly 31, 1-8] proves that, for any random variable X with finite mean µ, E(1/r(X)] 1/µ, where r(•) is the failure rate function of X , with equality if and only if X is exponentially distributed. Here we characterize exponential distribution and Rayleigh distribution through the expected values of r(X) and e(X), where e(•) is the mean residual life function of X .

A Loglikelihood-Based Shape Measure of Past Lifetime Distribution

Calcutta Statistical Association Bulletin, 2009

Generalized past information measure based on Rényi entropy for a continuous random variable usef... more Generalized past information measure based on Rényi entropy for a continuous random variable useful in life testing is considered and its connection to the loglikelihood is established. From this relation, a loglikelihood-based past life distribution measure [Formula: see text] has been proposed. From the gradient of the spectrum of generalized past information, the [Formula: see text] measure has been calculated for some important parametric life distributions. Serving as a measure of the shape of a past life (inactivity time) distribution, [Formula: see text] can be used to compare the tails and shapes of various frequently used densities, where the traditional kurtosis measure is not applicable.

Conditional precedence orders for stochastic comparison of random variables

Statistics & Probability Letters

Most of the stochastic orders for comparing random variables, considered in the literature, are a... more Most of the stochastic orders for comparing random variables, considered in the literature, are afflicted with two main drawbacks: piq lack of connex property and piiq lack of consideration of any dependence structure between the random variables. Both these drawbacks can be overcome at the cost of transitivity with the stochastic precedence order, which may seem to be a good choice in particular when only two random variables are under consideration, a situation where the question of transitivity does not arise. In this paper, we show that even under such favorable conditions, stochastic precedence order may direct to misleading conclusion in certain situations and develop variations of the order to address the phenomenon.

Some Ageing Properties of Dynamic Additive Mean Residual Life Model

arXiv: Statistics Theory, 2017

Although proportional hazard rate model is a very popular model to analyze failure time data, som... more Although proportional hazard rate model is a very popular model to analyze failure time data, sometimes it becomes important to study the additive hazard rate model. Again, sometimes the concept of the hazard rate function is abstract, in comparison to the concept of mean residual life function. A new model called `dynamic additive mean residual life model' where the covariates are time-dependent has been defined in the literature. Here we study the closure properties of the model for different positive and negative ageing classes under certain condition(s). Quite a few examples are presented to illustrate different properties of the model.

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

Statistics, 2020

In this paper, we investigate various stochastic orderings for series and parallel systems with i... more 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.

Some Reliability Properties of Transformed-Transformer Family of Distributions

American Journal of Mathematical and Management Sciences, 2018

The Transformed-Transformer family of distributions are the resulting family of distributions as ... more 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.

The Hazard Rate and the Reversed Hazard Rate Orders, with Applications to Order Statistics

In this paper we first point out a simple observation that can be used successfully in order to t... more In this paper we first point out a simple observation that can be used successfully in order to translate results about the hazard rate order into results about the reversed hazard rate order. Using it, we derive some interesting new results which compare order statistics in the hazard and in the reversed hazard rate orders; as well as in the

Quantile based entropy function in past lifetime

Statistics & Probability Letters, 2013

Di Crescenzo and Longobardi (2002) introduced a measure of uncertainty in past lifetime distribut... more Di Crescenzo and Longobardi (2002) introduced a measure of uncertainty in past lifetime distributions and studied its relationship with residual entropy function. In the present paper, we introduce a quantile version of the entropy function in past lifetime and study its properties. Unlike the measure of uncertainty given in Di Crescenzo and Longobardi (2002) the proposed measure uniquely determines the underlying probability distribution. The measure is used to study two nonparametric classes of distributions. We prove characterizations theorems for some well known quantile lifetime distributions.

Reliability study of proportional odds family of discrete distributions

Communications in Statistics - Theory and Methods, 2017

The proportional odds model gives a method of generating new family of distributions by adding a ... more The proportional odds model gives a method of generating new family of distributions by adding a parameter, called tilt parameter, to expand an existing family of distributions. The new family of distributions so obtained is known as Marshall-Olkin family of distributions or Marshall-Olkin extended distributions. In this paper, we consider Marshall-Olkin family of distributions in discrete case with fixed tilt parameter. We study different ageing properties, as well as different stochastic orderings of this family of distributions. All the results of this paper are supported by several examples.

Some distributional results through past entropy

Journal of Statistical Planning and Inference, 2010

... View the MathML source. It is to be noted here that φ(t)=(1−m ′ (t))/m(t), where m ′ (t)=(d/d... more

On some generalized ageing orderings

Some partial orderings which compare probability distributions with the expo- nential distributio... more Some partial orderings which compare probability distributions with the expo- nential distribution, are found to be very useful to understand the phenomenon of ageing. Here, we introduce some new generalized partial orderings which de- scribe the same kind of characterization of some generalized ageing classes. We give some equivalent conditions for each of the orderings. Inter-relations among the generalized orderings have also been discussed.

On closure of some partial orderings under mixtures

Journal of applied probability, 1996

Recently new partial orderings s-SFR, s-FR, s-ST, s-CV and s-CX have been defined in the literatu... more Recently new partial orderings s-SFR, s-FR, s-ST, s-CV and s-CX have been defined in the literature. These are more general in the sense that most of the earlier known partial orderings reduce to particular cases of these orderings. Moreover, these orderings have helped in defining a new and useful ageing criterion. In this paper, we discuss preservation properties of these orderings under mixtures. Results are expected to be useful in engineering and reliability, where mixtures of distributions arise naturally in numerous situations. Moreover, existing results, specifically for LR, FR and ST orderings under the operation of mixtures, reduce to special cases.

Shannon's entropy and its Generalizations towards Statistics, Reliability and Information Science during 1948-2018

arXiv: Other Statistics, 2019

Starting from the pioneering works of Shannon and Weiner in 1948, a plethora of works have been r... more Starting from the pioneering works of Shannon and Weiner in 1948, a plethora of works have been reported on entropy in different directions. Entropy-related review work in the direction of statistics, reliability and information science, to the best of our knowledge, has not been reported so far. Here we have tried to collect all possible works in this direction during the period 1948-2018 so that people interested in entropy, specially the new researchers, get benefited.

Discrete life distributions with decreasing reversed hazard

Sankhyā: The Indian Journal of Statistics (2003-), 2005

This paper considers the class of discrete distributions for which the distribution function is a... more This paper considers the class of discrete distributions for which the distribution function is a log-concave sequence. It is shown that such distributions arise from a wide variety of circumstances in Reliability, and that these have a decreasing reversed hazard rate. After examining the closures of the class under certain key operations, sharp upper and lower bounds on the reliability function for the member distributions are given. Some useful inequalities for maintained systems are provided. Some results for the related class of discrete concave distributions are also given.

Errors due to departure from independence in exponential series system

Communications in Statistics - Theory and Methods

Optimal Replacement in a Discrete time Shock Model

OPSEARCH

ABSTRACT

Parameter estimates of general failure rate model: A Bayesian approach

Journal of Computational and Applied Mathematics

The failure rate function plays an important role in studying the lifetime distributions in relia... more The failure rate function plays an important role in studying the lifetime distributions in reliability theory and life testing models. A study of the general failure rate model r(t) = a + bt θ−1 , under squared error loss function taking a and b independent exponential random variables has been analyzed in the literature. In this article, we consider a and b not necessarily independent. The estimates of the parameters a and b under squared error loss, linex loss and entropy loss functions are obtained here.

Tests for Reversed Hazard Rate Function

Calcutta Statistical Association Bulletin

ABSTRACT The reversed hazard rate function of a random variable, defined as the ratio of the dens... more ABSTRACT The reversed hazard rate function of a random variable, defined as the ratio of the density to its distribution function, plays an important role in analyzing left-censored data. We develop tests based on the empirical distribution function for an assigned reversed hazard rate function against a one-sided alternative and two different tests for equality of two reversed hazard rate functions. The asymptotic normality of the proposed test statistics has been established.

On Improvement and Deterioration of A Repairable System Under Generalized Stochastic Orders

Calcutta Statistical Association Bulletin

ABSTRACT

Stochastic Orders in Terms of Laplace Transforms

Calcutta Statistical Association Bulletin

Recently s-FR and s-ST orderings have been defined in the literature. They are more general in th... more Recently s-FR and s-ST orderings have been defined in the literature. They are more general in the sense that most of the earlier known partial orderings reduce as particular cases of these orderings. Moreover, these orderings have helped in defining new and useful ageing criterion. In this paper, using Laplace transform, we characterize, by means of necessary and sufficient conditions. the property that two life distributions are ordered in the s-FR and s-ST sense. The characterization of LR, FR, MR, VR, STand HAMR orderings follow as particular cases.

Characterization of distributions through failure rate and mean residual life functions

Statistics Probability Letters, May 1, 2010

Makino [Makino, T., 1984. Mean hazard rate and its applications to the normal approximation of th... more Makino [Makino, T., 1984. Mean hazard rate and its applications to the normal approximation of the Weibull distribution. Naval Research Logistics Quarterly 31, 1-8] proves that, for any random variable X with finite mean µ, E(1/r(X)] 1/µ, where r(•) is the failure rate function of X , with equality if and only if X is exponentially distributed. Here we characterize exponential distribution and Rayleigh distribution through the expected values of r(X) and e(X), where e(•) is the mean residual life function of X .

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

by Pradip Kundu and Asok Nanda

Statistics, 2020

In this paper, we investigate various stochastic orderings for series and parallel systems with i... more 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.