Stochastic Comparison Of Conditional Order Statistics And Sample Spacings With Applications

In this thesis,we will carry out stochastic comparison based on conditional order statistics and sample spacings from i.i.d,and non-i.i.d,samples,which has wide applications in various areas of applied probability and statistics including life testing,survival analysis,operations research,economics,insurance,actuarial sciences,reliability theory.Firstly,we put our attention to GRL(general residual life) and GIT(general inactive time) of a(n-k+1)-out-of-n(or equivalently,conditional order statistics).We carry out stochastic comparison and examine aging properties based on GITs and GRLs.Further, the study is extended in two directions,one direction is based on the conditional generalized order statistics and hence some related results involved in other ordered random variables such as record values,progressively typeâ…¡censored order statistics,are established as well;the other direction is based on the order statistics from independent but not identically distributed observations.Secondly,we focus on the topic about spacings.It is well-known that spacings of record values inter-epoch intervals of nonhomogeneous Poisson processes,which can be regarded as spacings of record values(see Belzunce 1999).We will further carry out stochastic comparison of spacings of record values from one sample and two samples.Let X₁,…,X_p be a simple random sample from an exponential distribution with failure rateλ;Let X_p+1,…,X_p+q be another simple random sample from the exponential distribution with failure rateλ^*,where p+q=n.Under this exponential framework,the likelihood ratio order between m-spacings of the combined sample is developed.Thirdly,we establish characterization results in terms of parameters for stochastic orders of sample range and the second order statistics involved in heterogeneously exponential random variables.Under the heterogeneously exponential setup,we obtain the equivalent characterization in terms of parameters for the stochastic order between the sample range from nonidentical independent exponential observations and that from i.i.d,exponential ones;also,we derive the characterization for the likelihood ratio order between the second order statistics from heterogeneous exponential random variables. Finally,we introduce a mixture model based on the proportional odds.We build the TP2 dependence between the population variable and the unobserved mixing variable and present some preservation properties.Relations on aging characteristics,such as odds function and(reversed) hazard rate are discussed.Stochastic comparisons on population variables are conducted as well.