American Journal of Mathematical and Management Sciences, 2020
Jiang et al. have introduced a quantitative measure known as the ageing intensity function for ev... more Jiang et al. have introduced a quantitative measure known as the ageing intensity function for evaluating the ageing properties of a component/system. The present study extends this ageing intensity function to the conditionally specified and conditional survival models. In a two component system, these two conditional ageing intensity functions provide the ageing patterns of one component when the other component has either failed or survived a specified period of time. The proposed conditionally ageing intensity function models provide simple expressions for some commonly used bivariate lifetime models. Characterization relationships are established for bivariate exponential, bivariate Weibull, conditional proportional hazards and bivariate weighted models.
In reliability and life testing when the exponentially distributed components are put in series, ... more In reliability and life testing when the exponentially distributed components are put in series, it is generally assumed that the lifetimes of the components are independently distributed, which leads to some errors if they are not actually independent. In this paper, we study the relative errors incurred in different reliability measures due to such assumptions when actually they follow some bivariate exponential distributions.
We introduce some new generalized stochastic orderings (in the spirit of relative ageing) which c... more 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. Interrelations among these orderings have also been discussed.
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.
International Journal of Mathematics and Statistics, Jan 26, 2013
Marshall and Olkin (1997) [Biometrika, 84, pp. 641-652] have discussed a general method of introd... more Marshall and Olkin (1997) [Biometrika, 84, pp. 641-652] have discussed a general method of introducing a parameter, called tilt parameter, to give more flexibility in modeling. In this paper, we take the tilt parameter as a random variable and study its properties for different ageing classes. It is noted that although some ageing classes are closed under this transformation for a fixed tilt parameter family, they are not closed when the tilt parameter is random. We also give upper and lower bounds to the residual survival function as well as of the failure rate function of the transformed model.
International Journal of Quality & Reliability Management, Feb 5, 2018
Purpose The purpose of this paper is to introduce a new probability density function having both ... more Purpose The purpose of this paper is to introduce a new probability density function having both unbounded and bounded support with a wider applicability. While the distribution with bounded support on [0, 1] has applications in insurance and inventory management with ability to fit risk management data on proportions better than existing bounded distributions, the same with unbounded support is used as a lifetime model and is considered as an attractive alternative to some existing models in the reliability literature. Design/methodology/approach The new density function, called modified exponential-geometric distribution is derived from the exponential-geometric distribution introduced by Adamidis and Loukas (1998). The support of the density function is shown to be both unbounded and bounded depending on the values of one of the shape parameters. Various properties of the density function are studied in detail and the parameters are estimated through maximum likelihood method of estimation. A number of applications related to reliability, insurance and inventory management are exhibited along with some useful data analysis. Findings A single probability distribution with both unbounded and bounded support, which does not seem to exist in the reliability literature, is introduced in this paper. The proposed density function exhibits varying shapes including U-shape, and the failure rate also shows increasing, decreasing and bathtub shapes. The Monte Carlo simulation shows that the estimates of the parameters are quite stable with low standard errors. The distribution with unbounded support is shown to have competitive features for lifetime modeling through analysis of two data sets. The distribution with bounded support on [0, 1] is shown to have application in insurance and inventory management and is found to t data on proportions related to risk management better than some existing bounded distributions. Originality/value The authors introduce an innovative probability distribution which contributes significantly in insurance and inventory management besides its remarkable statistical and reliability properties.
We consider stochastic comparisons of minimum order statistics from the location-scale family of ... more We consider stochastic comparisons of minimum order statistics from the location-scale family of distributions that contain most of the popular lifetime distributions. Under certain assumptions, we show that the minimum order statistic of one set of random variables dominates that of another set of random variables with respect to different stochastic orders. Furthermore, we illustrate our results using some well-known specific distributions.
Quality Technology and Quantitative Management, 2010
A generalized process capability index, defined as the ratio of proportion of specification confo... more A generalized process capability index, defined as the ratio of proportion of specification conformance (or, process yield) to proportion of desired (or, natural) conformance, has been developed. Almost all the process capabilities defined in the literature are directly or indirectly associated with this generalized index. Normal as well as non-normal and continuous as well as discrete random variables could be covered by this new index. It can also be assessed under either unilateral or bilateral specifications. We deal with the proposed index in case of normal, exponential and Poisson processes. Under each distributional assumption, point estimators for the proposed index are suggested and compared through simulation study. Two real-world applications have been discussed using the proposed index.
We consider the location-scale family of distributions, which contains many standard lifetime dis... more We consider the location-scale family of distributions, which contains many standard lifetime distributions. We give conditions under which the largest order statistic of a set of random variables with different/the same location as well as different/the same scale parameters dominates that of another set of random variables with respect to various stochastic orders. Along with general results, we consider important special cases, namely, the Feller-Pareto, generalized Pareto, Burr, exponentiated Weibull, Power generalized Weibull, generalized gamma, Half-normal and Fréchet distributions.
In the literature, stochastic orders are used to compare the lifetimes of two coherent systems. W... more In the literature, stochastic orders are used to compare the lifetimes of two coherent systems. We show that, under certain conditions, a coherent system made out of independent components, which have been used for some units of time, performs better (worse) than a used coherent system (i.e., a system is formed by some new independent components and then the system has been used for sometime) with respect to different stochastic orders. Some results on stochastic comparisons between a coherent system of inactive components and an inactive coherent system are also discussed.
Stochastic orders are very useful tools to compare the lifetimes of two systems. Optimum lifetime... more Stochastic orders are very useful tools to compare the lifetimes of two systems. Optimum lifetime of a series (resp. parallel) system with general standby component(s) depends on allocation strategy of standby component(s) into the system. Here, we discuss three different models of one or more standby components. In each model, we compare different series (resp. parallel) systems (which are formed through different allocation strategies of standby component(s)) with respect to the usual stochastic order and the stochastic precedence order. The results related to the cold as well as the hot standby models are obtained as particular cases of the results discussed in this paper because the model considered here is a general one.
Calcutta Statistical Association Bulletin, Sep 1, 2006
In this paper a stochastic order and an aging property have been defined based on R-norm residual... more In this paper a stochastic order and an aging property have been defined based on R-norm residual entropy, and their properties have been studied. A few useful parametric distributions have been characterized with the help of this entropy. A couple of results have been developed for discrete random variables.
Calcutta Statistical Association Bulletin, Mar 1, 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.
Let X 1 , X 2 ,. .. , X n (resp. Y 1 , Y 2 ,. .. , Y n) be independent random variables such that... more Let X 1 , X 2 ,. .. , X n (resp. Y 1 , Y 2 ,. .. , 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,. .. , n. Here it is shown that if λ = (λ 1 , λ 2 ,. .. , λ n) majorizes δ = (δ 1 , δ 2 ,. .. , δ n) then X n:n will be greater than Y n:n in reversed hazard rate ordering. 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 λ, θ i , where λ is the mean of all λ i 's, i = 1. .. n, then X n:n is greater than Y n:n in likelihood ratio ordering. In this context, an error in Marshall, Olkin and Arnold [Inequalities: Theory of Majorization and Its applications (2011)] has been corrected, and some new results on majorization have been developed.
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.
There are many measures for qualitative and quantitative analysis of ageing phenomena of a system... more There are many measures for qualitative and quantitative analysis of ageing phenomena of a system with a well defined statistical distribution, namely, hazard rate, ageing intensity (AI) function e...
SummaryStarting from the pioneering works of Shannon and Weiner in 1948, a plethora of works have... more SummaryStarting 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 statistical inference, 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 last seven decades so that people interested in entropy, specially the new researchers, get benefited.
In the study of probabilistic orders iterated integrals of survival functions have been used by m... more In the study of probabilistic orders iterated integrals of survival functions have been used by many authors. Recently, dispersion-typestochastic orders have been defined by Belzunce, Hu and Khaledi (2003) [ Probability in the Engineering and Informational Sciences, Vol. 17, pp. 305–334]. The purpose of this paper is to generalize the dispersion-type stochastic orders using generalized iterated survival functions. The relationships between generalized stochastic orders and the stochastic orders of dispersion-type have been studied. It is shown that under certain condition, generalized down stochastic orders imply generalized stochastic orders of dispersion-type which, in turn, imply generalized stochastic orders. Generalized stochastic orders of dispersion-type of residual random variables are discussed. Some characterizations of generalized stochastic orders of dispersion-type have also been studied. Some results regarding generalized orders are also discussed when the residual tim...
Journal of Computational and Applied Mathematics, 2016
Majorization is a key concept in studying the Schur-convex property of a function, which is very ... more Majorization is a key concept in studying the Schur-convex property of a function, which is very useful in the study of stochastic orders. In this paper, some results on Schur-convexity have been developed. We have studied the conditions under which a function ? defined by ? ( x ) = ? i = 1 n u i g ( x i ) will be Schur-convex. This fills some gap in the theory of majorization. The results so developed have been used in the case of generalized exponential and gamma distributions. During this, we have also developed some stochastic properties of order statistics. Some useful results on majorization are developed.This enriches the theory of majorization.As applications, some distributions have been studied.
American Journal of Mathematical and Management Sciences, 2020
Jiang et al. have introduced a quantitative measure known as the ageing intensity function for ev... more Jiang et al. have introduced a quantitative measure known as the ageing intensity function for evaluating the ageing properties of a component/system. The present study extends this ageing intensity function to the conditionally specified and conditional survival models. In a two component system, these two conditional ageing intensity functions provide the ageing patterns of one component when the other component has either failed or survived a specified period of time. The proposed conditionally ageing intensity function models provide simple expressions for some commonly used bivariate lifetime models. Characterization relationships are established for bivariate exponential, bivariate Weibull, conditional proportional hazards and bivariate weighted models.
In reliability and life testing when the exponentially distributed components are put in series, ... more In reliability and life testing when the exponentially distributed components are put in series, it is generally assumed that the lifetimes of the components are independently distributed, which leads to some errors if they are not actually independent. In this paper, we study the relative errors incurred in different reliability measures due to such assumptions when actually they follow some bivariate exponential distributions.
We introduce some new generalized stochastic orderings (in the spirit of relative ageing) which c... more 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. Interrelations among these orderings have also been discussed.
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.
International Journal of Mathematics and Statistics, Jan 26, 2013
Marshall and Olkin (1997) [Biometrika, 84, pp. 641-652] have discussed a general method of introd... more Marshall and Olkin (1997) [Biometrika, 84, pp. 641-652] have discussed a general method of introducing a parameter, called tilt parameter, to give more flexibility in modeling. In this paper, we take the tilt parameter as a random variable and study its properties for different ageing classes. It is noted that although some ageing classes are closed under this transformation for a fixed tilt parameter family, they are not closed when the tilt parameter is random. We also give upper and lower bounds to the residual survival function as well as of the failure rate function of the transformed model.
International Journal of Quality & Reliability Management, Feb 5, 2018
Purpose The purpose of this paper is to introduce a new probability density function having both ... more Purpose The purpose of this paper is to introduce a new probability density function having both unbounded and bounded support with a wider applicability. While the distribution with bounded support on [0, 1] has applications in insurance and inventory management with ability to fit risk management data on proportions better than existing bounded distributions, the same with unbounded support is used as a lifetime model and is considered as an attractive alternative to some existing models in the reliability literature. Design/methodology/approach The new density function, called modified exponential-geometric distribution is derived from the exponential-geometric distribution introduced by Adamidis and Loukas (1998). The support of the density function is shown to be both unbounded and bounded depending on the values of one of the shape parameters. Various properties of the density function are studied in detail and the parameters are estimated through maximum likelihood method of estimation. A number of applications related to reliability, insurance and inventory management are exhibited along with some useful data analysis. Findings A single probability distribution with both unbounded and bounded support, which does not seem to exist in the reliability literature, is introduced in this paper. The proposed density function exhibits varying shapes including U-shape, and the failure rate also shows increasing, decreasing and bathtub shapes. The Monte Carlo simulation shows that the estimates of the parameters are quite stable with low standard errors. The distribution with unbounded support is shown to have competitive features for lifetime modeling through analysis of two data sets. The distribution with bounded support on [0, 1] is shown to have application in insurance and inventory management and is found to t data on proportions related to risk management better than some existing bounded distributions. Originality/value The authors introduce an innovative probability distribution which contributes significantly in insurance and inventory management besides its remarkable statistical and reliability properties.
We consider stochastic comparisons of minimum order statistics from the location-scale family of ... more We consider stochastic comparisons of minimum order statistics from the location-scale family of distributions that contain most of the popular lifetime distributions. Under certain assumptions, we show that the minimum order statistic of one set of random variables dominates that of another set of random variables with respect to different stochastic orders. Furthermore, we illustrate our results using some well-known specific distributions.
Quality Technology and Quantitative Management, 2010
A generalized process capability index, defined as the ratio of proportion of specification confo... more A generalized process capability index, defined as the ratio of proportion of specification conformance (or, process yield) to proportion of desired (or, natural) conformance, has been developed. Almost all the process capabilities defined in the literature are directly or indirectly associated with this generalized index. Normal as well as non-normal and continuous as well as discrete random variables could be covered by this new index. It can also be assessed under either unilateral or bilateral specifications. We deal with the proposed index in case of normal, exponential and Poisson processes. Under each distributional assumption, point estimators for the proposed index are suggested and compared through simulation study. Two real-world applications have been discussed using the proposed index.
We consider the location-scale family of distributions, which contains many standard lifetime dis... more We consider the location-scale family of distributions, which contains many standard lifetime distributions. We give conditions under which the largest order statistic of a set of random variables with different/the same location as well as different/the same scale parameters dominates that of another set of random variables with respect to various stochastic orders. Along with general results, we consider important special cases, namely, the Feller-Pareto, generalized Pareto, Burr, exponentiated Weibull, Power generalized Weibull, generalized gamma, Half-normal and Fréchet distributions.
In the literature, stochastic orders are used to compare the lifetimes of two coherent systems. W... more In the literature, stochastic orders are used to compare the lifetimes of two coherent systems. We show that, under certain conditions, a coherent system made out of independent components, which have been used for some units of time, performs better (worse) than a used coherent system (i.e., a system is formed by some new independent components and then the system has been used for sometime) with respect to different stochastic orders. Some results on stochastic comparisons between a coherent system of inactive components and an inactive coherent system are also discussed.
Stochastic orders are very useful tools to compare the lifetimes of two systems. Optimum lifetime... more Stochastic orders are very useful tools to compare the lifetimes of two systems. Optimum lifetime of a series (resp. parallel) system with general standby component(s) depends on allocation strategy of standby component(s) into the system. Here, we discuss three different models of one or more standby components. In each model, we compare different series (resp. parallel) systems (which are formed through different allocation strategies of standby component(s)) with respect to the usual stochastic order and the stochastic precedence order. The results related to the cold as well as the hot standby models are obtained as particular cases of the results discussed in this paper because the model considered here is a general one.
Calcutta Statistical Association Bulletin, Sep 1, 2006
In this paper a stochastic order and an aging property have been defined based on R-norm residual... more In this paper a stochastic order and an aging property have been defined based on R-norm residual entropy, and their properties have been studied. A few useful parametric distributions have been characterized with the help of this entropy. A couple of results have been developed for discrete random variables.
Calcutta Statistical Association Bulletin, Mar 1, 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.
Let X 1 , X 2 ,. .. , X n (resp. Y 1 , Y 2 ,. .. , Y n) be independent random variables such that... more Let X 1 , X 2 ,. .. , X n (resp. Y 1 , Y 2 ,. .. , 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,. .. , n. Here it is shown that if λ = (λ 1 , λ 2 ,. .. , λ n) majorizes δ = (δ 1 , δ 2 ,. .. , δ n) then X n:n will be greater than Y n:n in reversed hazard rate ordering. 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 λ, θ i , where λ is the mean of all λ i 's, i = 1. .. n, then X n:n is greater than Y n:n in likelihood ratio ordering. In this context, an error in Marshall, Olkin and Arnold [Inequalities: Theory of Majorization and Its applications (2011)] has been corrected, and some new results on majorization have been developed.
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.
There are many measures for qualitative and quantitative analysis of ageing phenomena of a system... more There are many measures for qualitative and quantitative analysis of ageing phenomena of a system with a well defined statistical distribution, namely, hazard rate, ageing intensity (AI) function e...
SummaryStarting from the pioneering works of Shannon and Weiner in 1948, a plethora of works have... more SummaryStarting 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 statistical inference, 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 last seven decades so that people interested in entropy, specially the new researchers, get benefited.
In the study of probabilistic orders iterated integrals of survival functions have been used by m... more In the study of probabilistic orders iterated integrals of survival functions have been used by many authors. Recently, dispersion-typestochastic orders have been defined by Belzunce, Hu and Khaledi (2003) [ Probability in the Engineering and Informational Sciences, Vol. 17, pp. 305–334]. The purpose of this paper is to generalize the dispersion-type stochastic orders using generalized iterated survival functions. The relationships between generalized stochastic orders and the stochastic orders of dispersion-type have been studied. It is shown that under certain condition, generalized down stochastic orders imply generalized stochastic orders of dispersion-type which, in turn, imply generalized stochastic orders. Generalized stochastic orders of dispersion-type of residual random variables are discussed. Some characterizations of generalized stochastic orders of dispersion-type have also been studied. Some results regarding generalized orders are also discussed when the residual tim...
Journal of Computational and Applied Mathematics, 2016
Majorization is a key concept in studying the Schur-convex property of a function, which is very ... more Majorization is a key concept in studying the Schur-convex property of a function, which is very useful in the study of stochastic orders. In this paper, some results on Schur-convexity have been developed. We have studied the conditions under which a function ? defined by ? ( x ) = ? i = 1 n u i g ( x i ) will be Schur-convex. This fills some gap in the theory of majorization. The results so developed have been used in the case of generalized exponential and gamma distributions. During this, we have also developed some stochastic properties of order statistics. Some useful results on majorization are developed.This enriches the theory of majorization.As applications, some distributions have been studied.
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