#### Presentation Title

Mixtures of Linear Mean Residual Life Functions with Time Dependent Weights

#### Start Date

November 2016

#### End Date

November 2016

#### Location

HUB 302-#169

#### Type of Presentation

Poster

#### Abstract

The *mean residual life function* (MRL) is defined as the expected value of the residual value of a random variable after attaining some time t. This function has wide applications in the field of reliability engineering, medicine, and actuarial science, among others. This function is also interesting mathematically as it uniquely characterizes the survival function of the random variable through the inversion formula. Our interest lies in modeling the MRL function directly under the Bayesian framework. However, obtaining the likelihood through the inversion formula requires a complex integration. Thus far in literature, models that have closed form under the inversion formula, such as the family of *linear* mean residual life functions, are limited to monotonicity in the functional shape. We propose mixtures of linear mean residual life functions with time-dependent weights, allowing for more flexibility in modeling while achieving closed form of the likelihood. Here, we focus on a finite mixture model of constant MRL functions. We show our model satisfies the characterization criteria to be a valid mean residual life function, and that the corresponding survival function is a mixture of Exponential survival functions. We demonstrate the flexibility and simplicity of our model by obtaining inference for a simulated and a real data example. We obtain inference for the model parameters via Monte Carlo Markov Chain updating under conjugate priors.

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Mixtures of Linear Mean Residual Life Functions with Time Dependent Weights

HUB 302-#169

The *mean residual life function* (MRL) is defined as the expected value of the residual value of a random variable after attaining some time t. This function has wide applications in the field of reliability engineering, medicine, and actuarial science, among others. This function is also interesting mathematically as it uniquely characterizes the survival function of the random variable through the inversion formula. Our interest lies in modeling the MRL function directly under the Bayesian framework. However, obtaining the likelihood through the inversion formula requires a complex integration. Thus far in literature, models that have closed form under the inversion formula, such as the family of *linear* mean residual life functions, are limited to monotonicity in the functional shape. We propose mixtures of linear mean residual life functions with time-dependent weights, allowing for more flexibility in modeling while achieving closed form of the likelihood. Here, we focus on a finite mixture model of constant MRL functions. We show our model satisfies the characterization criteria to be a valid mean residual life function, and that the corresponding survival function is a mixture of Exponential survival functions. We demonstrate the flexibility and simplicity of our model by obtaining inference for a simulated and a real data example. We obtain inference for the model parameters via Monte Carlo Markov Chain updating under conjugate priors.