## Orthogonal Design of Life Testing with Replacement: Exponential Parametric Regression Model

##### Abstract

This paper describes how to plan an "optimal" life testing experiment when the lifetime is assumed to have an Exponential distribution. We further assume that the mean lifetime is equal to where the covariates xi form an orthogonal Hadamard-type matrix which depends on testing conditions, and �i are the unknown parameters. n0 devices are put on test. The period of testing, t0, is divided into k stages of length ti , i=1,...,k, and on each of these stages all devices operate under a fixed testing regime. (The number of different testing regimes, k , equals to the number of parameters to be estimated). Each device which fails is immediately restored and continues to operate. A closed maximum likelihood solution is given for estimates of �i which exists if and only if at least one failure has been observed on each of the testing stages. Also the approximate optimal duration of the i-th testing stage, ti*, which would provide the minimum of is derived. It is shown that the near-optimal testing policy is obtained when ti* is proportional to the square root of the mean lifetime for the corresponding testing regime. Finally, the expression for the Fisher information matrix is derived and the optimality criterion (which is the trace of its inverse) is expressed as a function of model parameters �i, the duration of testing stages ti , and the number of devices operating on each of the testing stages.