On generalized conditional cumulative residual inaccuracy measure

Abstract

Recently, the notion of cumulative residual Rényi’s entropy has been proposed in the literature as a measure of information that parallels Rényi’s entropy. Motivated by this, here we introduce a generalized measure of it, namely cumulative residual inaccuracy of order α. We study the proposed measure for conditionally specified models of two components having possibly different ages called generalized conditional cumulative residual inaccuracy measure. Several properties of generalized conditional cumulative residual inaccuracy measure including the effect of monotone transformation are investigated. Further, we provide some bounds on using the usual stochastic order and characterize some bivariate distributions using the concept of conditional proportional hazard rate model.     

On weighted measure of inaccuracy for doubly truncated random variables

Recently, authors have studied the weighted version of Kerridgeinaccuracy measure for left/right truncated distributions. In the present communication we introduce the notion of weighted interval inaccuracy measure and study it in the context of two-sided truncated random variables. In reliability theory and survival analysis, this measure may help to study the various characteristics of a system/component when it fails between two time points. We study various properties of this measure, including the effect of monotone transformations, and obtain its upper and lower bounds. It is shown that the proposed measure can uniquely determine the distribution function and characterizations of some important life distributions have been provided. This new measure is a generalization of recent weighted residual (past) inaccuracy measure.     

Estimation of the process incapability index

Greenwich and Jahr-Schaffrath (1995) introduced a new index C _pp a simple transformation of the index C _pm , which provides an uncontaminated separation between information concerning process accuracy and process precision. Under the assumption of normality, we first show that the estimators of C _pp proposed by Greenwich and Jahr-Schaffrath (1995) are UMVU estimators. We also show that for the inaccuracy index, the variance of the unbiased estimator is smaller than the mean squared error (MSE) of the natural (biased) estimator for n > 3. In addition, we obtain the r-th moment and the probability density function of these estimators.     

Characterization of the quantitative-qualitative measure of inaccuracy for discrete generalized probability distributions

A characterization of the quantitative-qualitative measure of inaccuracy depending upon the additivity postulate and the mean value property for discrete generalized probability distribution has been provided.     

On some dynamic generalized information measures for bivariate lifetimes

Several generalizations to the concept of Kullback-Leibler divergence measure and Kerridge inaccuracy measure are available in the literature. In a recent paper Kundu (Metrika, 78:415–435, 2015 Kundu, C. 2015. Generalized measures of information for truncated random variables. Metrika 78:415–35.[Crossref], [Web of Science ®] , [Google Scholar]) considered a generalized K-L divergence measure of order (α, β). Nath (Metrika, 13:123–135, 1968 Nath, P. 1968. Inaccuracy and coding theory. Metrika 13:123–35.[Crossref] , [Google Scholar]) has also proposed generalized inaccuracy measure of order α. Here we address the question of extending these measures to higher dimensions with reference to residual lifetimes. In the present work, the generalized divergence and inaccuracy measures are extended for conditional lifetimes of two components having possibly different ages. Several properties, including monotonicity, and bounds of these measures are obtained for conditional random variables. Moreover, we study the effect of (increasing) monotone transformation on these generalized measures.     

Generalized Kerridge’s inaccuracy measure for conditionally specified models

Recently, conditional Renyi’s divergence of order α and Kerridge’s inaccuracy measures are studied by Navarro et al. (2014 Navarro, J., Sunoj, S.M., Linu, M.N. (2014). Characterizations of bivariate models using some dynamic conditional information divergence measures. Commun. Stat. Theory Methods 43:1939–1948.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). In the present article, a generalized dynamic conditional Kerridge’s inaccuracy measure is introduced, which can be represented as the sum of conditional Renyi’s divergence and Renyi’s entropy. Some useful bounds are obtained using the concept of likelihood ratio order. The results are extended to weighted distributions. Sufficient conditions are obtained for the monotonicity of the proposed measure. Characterizations for bivariate exponential conditional distribution are presented based on the proposed measure.     

Estimation for a scale parameter with known coefficient of variation

A loss function proposed by Wasan (1970) is well-fitted for a measure of inaccuracy for an estimator of a scale parameter of a distribution defined onR ⁺=(0, ∞). We refer to this loss function as the K-loss function. A relationship between the K-loss and squared error loss functions is discussed. And an optimal estimator for a scale parameter with known coefficient of variation under the K-loss function is presented.     

Robust methods of estimation of regression coefficients 1

Recently, cumulative residual entropy (CRE) has been found to be a new measure of information that parallels Shannon's entropy (see Rao et al. [Cumulative residual entropy: A new measure of information, IEEE Trans. Inform. Theory. 50(6) (2004), pp. 1220–1228] and Asadi and Zohrevand [On the dynamic cumulative residual entropy, J. Stat. Plann. Inference 137 (2007), pp. 1931–1941]). Motivated by this finding, in this paper, we introduce a generalized measure of it, namely cumulative residual Renyi's entropy, and study its properties. We also examine it in relation to some applied problems such as weighted and equilibrium models. Finally, we extend this measure into the bivariate set-up and prove certain characterizing relationships to identify different bivariate lifetime models.     

Large Deviations Limit Theorems for the Kernel Density Estimator

We establish pointwise and uniform large deviations limit theorems of Chernoff-type for the non-parametric kernel density estimator based on a sequence of independent and identically distributed random variables. The limits are well-identified and depend upon the underlying kernel and density function. We derive then some implications of our results in the study of asymptotic efficiency of the goodness-of-fit test based on the maximal deviation of the kernel density estimator as well as the inaccuracy rate of this estimate     

On the Percentile Residual Lifetime of Parallel Systems

In this paper, we consider a parallel system consisting of n components. Then, the percentile residual lifetime of the system given survival of at least n ? r + 1, r = 1, 2, …, n component(s) has been introduced, and some properties of this measure have been investigated. We show that the system accommodates decreasing percentile residual lifetime function, provided the components have increasing hazard rate functions. Different parallel systems have been compared with each other in terms of the introduced measure. Furthermore, behavior of the percentile residual lifetime of the system and the components have been compared in terms of some reliability notions. Also, a characterization result has been presented.     

Comparison of two sampling schemes for generating record-breaking data from the proportional hazard rate models

ABSTRACT

In this article, we consider a sampling scheme in record-breaking data set-up, as record ranked set sampling. We compare the proposed sampling with the well-known sampling scheme in record values known as inverse sampling scheme when the underlying distribution follows the proportional hazard rate model. Various point estimators are obtained in each sampling schemes and compared with respect to mean squared error and Pitman measure of closeness criteria. It is observed in most of the situations that the new sampling scheme provides more efficient estimators than their counterparts. Finally, one data set has been analyzed for illustrative purposes.     

An operative extension of the likelihood ratio test from fuzzy data

In the present paper we are going to extend the likelihood ratio test to the case in which the available experimental information involves fuzzy imprecision (more precisely, the observable events associated with the random experiment concerning the test may be characterized as fuzzy subsets of the sample space, as intended by Zadeh, 1965). In addition, we will approximate the immediate intractable extension, which is based on Zadeh’s probabilistic definition, by using the minimum inaccuracy principle of estimation from fuzzy data, that has been introduced in previous papers as an operative extension of the maximum likelihood method.     

Characterizations based on generalized cumulative residual entropy functions

The Shannon entropy and the cumulative residual entropy (CRE) of a random variable are useful tools in probability theory. Recently, a new concept called generalized cumulative residual entropy (GCRE) of order n was introduced and studied. It is related with the record values of a sequence of i.i.d. random variables and with the relevation transform. In this paper, we show that, under some assumptions, the GCRE function of a fixed order n uniquely determines the distribution function. Some characterizations of particular probability models are obtained from this general result.     

On cumulative entropies

In analogy with the cumulative residual entropy recently proposed by Wang et al. [2003a. A new and robust information theoretic measure and its application to image alignment. In: Information Processing in Medical Imaging. Lecture Notes in Computer Science, vol. 2732, Springer, Heidelberg, pp. 388–400; 2003b. Cumulative residual entropy, a new measure of information and its application to image alignment. In: Proceedings on the Ninth IEEE International Conference on Computer Vision (ICCV’03), vol. 1, IEEE Computer Society Press, Silver Spring, MD, pp. 548–553], we introduce and study the cumulative entropy, which is a new measure of information alternative to the classical differential entropy. We show that the cumulative entropy of a random lifetime X can be expressed as the expectation of its mean inactivity time evaluated at X. Hence, our measure is particularly suitable to describe the information in problems related to ageing properties of reliability theory based on the past and on the inactivity times. Our results include various bounds to the cumulative entropy, its connection to the proportional reversed hazards model, and the study of its dynamic version that is shown to be increasing if the mean inactivity time is increasing. The empirical cumulative entropy is finally proposed to estimate the new information measure.     

Characterizations of Bivariate Models Using Some Dynamic Conditional Information Divergence Measures

In this article, we study some relevant information divergence measures viz. Renyi divergence and Kerridge’s inaccuracy measures. These measures are extended to conditionally specified models and they are used to characterize some bivariate distributions using the concepts of weighted and proportional hazard rate models. Moreover, some bounds are obtained for these measures using the likelihood ratio order.     

Comparing the Fisher information in record data and random observations

We compare the Fisher information (FI) contained in the firstn record values and record times with the FI inn i. i. d. observations. General results are established for exponential family and Weibull type setups, and a summary table is provided listing several common distributions. We show that the FI in record data improves notably once the record times are included, often changing from being less to being equal or greater than the FI in a random sample of the same size. The behavior in the Weibull case is surprising. There it depends onn, whether the record or the i.i. d. observations have more FI. We propose new estimators based on record data. The results may be of interest in some life testing situations. Supported in part by Fondo Nacional de Desarrollo Cientifico y Tecnologico (FONDECYT) grant # 1010222 of Chile.     

Characterizations of discrete distributions based on conditional expectations of record values

r -th record values subject to (r + 1)-th record values, record mean function, from a distribution of discrete type. We give some properties of the record mean function and an explicit expression for the distribution function based on its record mean function, which allows us to characterize particular discrete distributions using the record mean functions. Received: January 4, 1999; revised version: September 27, 1999     

Cumulative ratio information based on general cumulative entropy

In this paper, we suggest an extension of the cumulative residual entropy (CRE) and call it generalized cumulative entropy. The proposed entropy not only retains attributes of the existing uncertainty measures but also possesses the absolute homogeneous property with unbounded support, which the CRE does not have. We demonstrate its mathematical properties including the entropy of order statistics and the principle of maximum general cumulative entropy. We also introduce the cumulative ratio information as a measure of discrepancy between two distributions and examine its application to a goodness-of-fit test of the logistic distribution. Simulation study shows that the test statistics based on the cumulative ratio information have comparable statistical power with competing test statistics.     

Upper and lower bounds on the variances of linear combinations of the kth records

We describe a method of determining upper bounds on the variances of linear combinations of the kth records values from i.i.d. sequences, expressed in terms of variances of parent distributions. We also present conditions for which the bounds are sharp, and those for which the respective lower ones are equal to zero. A special attention is paid to the case of the kth record spacings, i.e. the differences of consecutive kth record values.