Abstract
In this note, stochastic comparisons and results for weighted and Lindley models are presented. Approximation of weighted distributions via Lindley distribution in the class of increasing failure rate (IFR) and decreasing failure rate (DFR) weighted distributions with monotone weight functions are obtained including approximations via the length-biased Lindley distribution. Some useful bounds and moment-type inequality for weighted life distributions and applications are presented. Incorporation of covariates into Lindley model is considered and an application to illustrate the usefulness and applicability of the proposed Lindley-Cox model is given.
Highlights
Weighted distributions occur naturally in probability and statistics, and provides an approach to dealing with model specification and data interpretation problems
Zelen and Feinleib (1969) introduced weighted distribution to discuss length biased sampling and showed that cases of chronic diseases identified by early detection screening programs constitute a length-biased sample due to the fact that individuals with a long pre-clinical disease phase have greater probability of being identified
In Theorems 2.1 and 2.2, we present useful ageing properties namely Used better than aged (UBA) and Harmonic used better than aged in expectation (HUBAE) for Lindley distribution as well as stochastic comparisons of proportional hazards Lindley distribution and weighted distributions with monotone weight function, respectively
Summary
Weighted distributions occur naturally in probability and statistics, and provides an approach to dealing with model specification and data interpretation problems. Oluyede (1999) developed and presented useful results on inequalities and selection of experiments for length-biased distributions. Brown (1983) among others obtained measures of departure from exponentiality within the class of completely monotone distributions and the class of increasing mean residual life (IMRL) distribution functions. These measures of departure are given in terms of ρ. Approximations via Lindley distribution in the class of weighted life distributions with monotone weight functions including distributions generated under proportional hazards transforms are obtained. The connection between weighted distributions and distributions generated under proportional hazards transforms and results on stochastic dominance and comparisons are given.
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