David D. Hanagal

Some inference results in bivariate exponential distributions rased on censored samples

In this paper, two bivariate exponential distributions based on time(right) censored samples are presented. We assume that the censoring time is independent of the life-times of the two components. This paper obtains comparison of different tests for testing zero and non-zero values of the parameter λ3 which measures the degree of dependence between the two components and also testing symmetry of the two components or λ1=λ2 in the bivariate exponential distribution (BVED) formulated by Marshall and Olkin (1967) based on the above censored sample. It is observed from simulated study that the test based on MLEs performs better in both tests of independence as well as symmetry. The above results have been extended also in Block and Basu (19874) model.

Modeling survival data using frailty models

Contents List of Tables List of Figures Preface About the Author Basic Concepts in Survival Analysis Introduction to Survival Analysis Introduction Bone Marrow Transplantation (BMT) for Leukemia Remission Duration from a Clinical Trial for Acute Leukemia Times of Infection of Kidney Dialysis Patients Kidney Infection Data Litters of Rats Data Kidney Dialysis (HLA) Patients Data Diabetic Retinopathy Data Myeloma Data Definitions and Notations .Survival Function .Failure (or Hazard) Rate Censoring Some Parametric Methods Introduction Exponential Distribution Weibull Distribution Extreme Value Distributions Lognormal Gamma Loglogistic Maximum Likelihood Estimation Parametric Regression Models Nonparametric and Semiparametric Models Empirical Survival Function Graphical Plotting Graphical Estimation Empirical Model Fitting: Distribution Free (Kaplan-Meier) Approach Comparison between Two Survival Functions Coxs Proportional Hazards Model Univariate and Shared Frailty Models for Survival Data The Frailty Concept Introduction The Definition of Shared Frailty The Implications of Frailty The Conditional Parametrization The Marginal Parametrization Frailty as a Model for Omitted Covariates Frailty as a Model of Stochastic Hazard Identifiability of Frailty Models Various Frailty Models Introduction Gamma Frailty Positive Stable Frailty Power Variance Function Frailty Compound Poisson Frailty Compound Poisson Distribution with Random Scale Frailty Models in Hierarchical Likelihood Frailty Models in Mixture Distributions Estimation Methods for Shared Frailty Models Introduction Inference for the Shared Frailty Model The EM Algorithm The Gamma Frailty Model The Positive Stable Frailty Model The Lognormal Frailty Model Application to Seizure Data Modified EM (MEM) Algorithm for Gamma Frailty Models Application Discussion Analysis of Survival Data in Shared Frailty Models Introduction Analysis for Bone Marrow Transplantation (BMT) Data Analysis for Acute Leukemia Data Analysis for HLA Data Analysis for Kidney Infection Data Analysis of Litters of Rats Analysis for Diabetic Retinopathy Data Tests of Hypotheses in Frailty Models Introduction Tests for Gamma Frailty Based on Likelihood Ratio and Score Tests Logrank Tests for Testing I = 0 Test for Heterogeneity in Kidney Infection Data Shared Frailty in Bivariate Exponential and Weibull Models Introduction Bivariate Exponential Distributions Gamma Frailty in BVW Models Positive Stable Frailty in BVW Models Power Variance Function Frailty in BVW Models Weibull Extension of BVE Models Lognormal and Weibull Frailties in BVW Models Compound Poisson Frailty in BVW Models Compound Poisson (with Random Scale) Frailty in BVW Models Estimation and Tests for Frailty under BVW Baseline Frailty Models Based on Levy Processes Introduction Levy Processes and Subordinators Proportional Hazards Derived from Levy Processes Other Frailty Process Constructions Hierarchical Levy Frailty Models Bivariate Frailty Models for Survival Data Bivariate Frailty Models and Estimation Methods Introduction Bivariate Frailty Models and Laplace Transforms Proportional Hazard Model for Covariate Effects The Problem of Confounding A General Model of Covariate Dependence Pseudo-Frailty Model Likelihood Construction Semiparametric Representations Estimation Methods in Bivariate Frailty Models Correlated Frailty Models Introduction Correlated Gamma Frailty Model Correlated Power Variance Function Frailty Model Genetic Analysis of Duration General Bivariate Frailty Model Correlated Compound Poisson Frailty for the Bivariate Survival Applications Additive Frailty Models Introduction Modeling Multivariate Survival Data Using the Frailty Model Correlated Frailty Model Relations to Other Frailty Models Additive Genetic Gamma Frailty Additive Genetic Gamma Frailty for Linkage Analysis of Diseases Identifiability of Bivariate Frailty Models Introduction Identifiability of Bivariate Frailty Models Identifiability of Correlated Frailty Models Non-Identifiability of Frailty Models without Observed Covariates Discussion Appendix Bibliography Index

Large sample tests of independence for absolutely continuous bivariate exponential distribution

This paper obtains a test based on MLE Of λ3 for testing the parameter λ3=0 in the absolutely continuous bivariate exponential distribution formulated by Block and Basu(1974) which is a modification of Marshall and 01kin(1967) model. The hypothesis λ3=0 is equivalent to independence of the two components. The asymptotic distribution of MLE which is univariate normal is used to construct the test. We compare the power of the above test with likelihood ratio test(LRT) given by Gupta, Mehrotra and Michalek(1984) and in the simulated study we observe that for large samples the test based on MLE has much better power performance.

A multivariate pareto distribution

In this paper, we introduce a new multivariate pareto (MVP) distribution with many interesting properties. we extend the results of characterization of univariate and bivariate pareto distributions given by Krishnaji (1970) and veenus and Nair (1994) respectively. We also extend the property of dullness of univariate pareto distribution given by Talwalkar (1980) to the multivariate pareto case. We obtain the maximum likelihood estimate (MLE) of the parameters and their asymptotic multivariate normal (AMVN) distrioutions. We propose large sample studentized test for testing independence and identical marginals of the components.

Large sample tests of λ3 in the bivariate exponential distribution

In this paper, we compare the power of three different tests for testing zero and non-zero values of the parameter [lambda]3 which measures the degree of dependence between the two components in bivariate exponential distribution (BVED) of the Marshall-Olkin (1967) model. We also compare the power of the UMPU test proposed by Bhattacharyya and Johnson (1973) with the test based on MLE 3 for [lambda]3=0 and observed that the UMPU test performs better.

Estimation of system reliability

In this paper, we estimate the reliability of a system with k components. The system functions when at least s (1≤s≤k) components survive a common random stress. We assume that the strengths of these k components are subjected to a common stress which is independent of the strengths of these k components. If (X1,X2,…,Xk) are strengths of k components subjected to a common stress (Y), then the reliability of the system or system reliability is given byR=P[Y<X(k−s+1)] whereX(k−s+1) is (k−s+1)-th order statistic of (X1,…,Xk). We estimate R when (X1,…,Xk) follow an absolutely continuous multivariate exponential (ACMVE) distribution of Hanagal (1993) which is the submodel of Block (1975) and Y follows an independent exponential distribution. We also obtain the asymptotic normal (AN) distribution of the proposed estimator.

Large sample tests for testing symmetry and independence in some bivariate exponential models

Bivariate Exponential Distribution (BVED) were introduced by Freund (1961), Marshall and Olkin (1967) and Block and Basu (1974) as models for the distributions of (X,Y) the failure times of dependent components (C1,C2). We study the structure of these models and observe that Freund model leads to a regular exponential family with a four dimensional orthogonal parameter. Marshall-Olkin model involving three parameters leads to a conditional or piece wise exponential family and Block-Basu model which also depends on three parameters is a sub-model of the Freund model and is a curved exponential family. We obtain a large sample tests for symmetry as well as independence of (X,Y) in each of these models by using the Generalized Likelihood Ratio Tests (GLRT) or tests basesd on MLE of the parameters and root n consistent estimators of their variance-covariance matrices.

Note on estimation of reliability under bivariate pareto stress-strength model

In this paper, we discuss the problem of estimating reliability (R) of a component based on maximum likelihood estimators (MLEs). The reliability of a component is given byR=P[Y<X]. Here X is a random strength of a component subjected to a random stress(Y) and (X,Y) follow a bivariate pareto(BVP) distribution. We obtain an asymptotic normal(AN) distribution of MLE of the reliability(R).

Estimation of system reliability from stress-strength relationship

In this paper, we estimate the reliability of parallel system with two components. We assume that strengths of these components follow a bi-variate exponential(BVE) distribution. These two components are subjected to a common stress which is independent of the strength of the components. If the strengths (X1,X2) are subjected to a common random stress(Y), then the reliability of a system or system reliability (R) is given by R = P[Y < Max(X1,X2)]- We estimate R when (X1,X2) have different BVE models proposed by Marshall-01kin(1967), Block-Basu(1974), Freund(1961) and Proschan-Sullo(1974). The distribution of Y is assumed to be either exponential or gamma. The asymptotic normal(AN) distributions of these estimates are obtained. We present a numerical study for obtaining MLE of R in all four BVE models when the common stress (Y) is exponentially distributed.

Bayesian Estimation of Parameters and Comparison of Shared Gamma Frailty Models

In this article, we introduce shared gamma frailty models with three different baseline distributions namely, Weibull, generalized exponential and exponential power distributions. We develop Bayesian estimation procedure using Markov Chain Monte Carlo(MCMC) technique to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. Also we apply these three models to a real life bivariate survival dataset of McGilchrist and Aisbett (1991) related to kidney infection data and a better model is suggested for the data.