Optimum multiple- and single-stress accelerated life tests

Abstract

<p>We consider the optimal design of Accelerated Life Tests (ALT) with Type II censored data. It is assumed that the time-to-failure, or a transformation of it, follows a location scale Gumbel distribution. The location parameter of this distribution is assumed to be of the form;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);where the (beta)(,j)'s are unknown and the f(,j)'s are functions of the stresses x. The scale parameter, (sigma), of the distribution is assumed to be independent of the stresses x;We estimate the parameters in the model using a linear model inwhich the dependent variables are observed order statistics at thepoints of the design. We give general formulae for the Best LinearUnbiased Estimator (BLUE), Y(,p), of the 100pth percentile, Y(,p), of the(' )Gumbel distribution at the design stress x(,D). Also, a general expression for Var(Y(,p)) is derived.(' );For designs in k+1 points, we use asymptotic theory to express^Var(Y(,p)) as a function of the number of allocated units and the(' )^proportion of censoring at each of the k+1 stress levels in the^design. The primary objective is to find designs that minimize^Var(Y(,p)). However, this minimization is complicated by the lack of(' )^closed form expressions for the expected values of the order^statistics and the variance-covariance matrix of the error term in^the linear model. Thus, we propose to restrict the optimization to^designs that minimize one of the components of Var(Y(,p)); in practical(' )situations, this component is usually the dominant term of Var(Y(,p)).(' )We prove that the proposed minimization is equivalent to a problemof optimal extrapolation under a linear model with uncorrelated errors;In the case of a single accelerating stress, we present a characterization of optimal designs in k+1 points when the functions f(,0),...,f(,k) are a T-system in a finite interval a,b;In the case of two or more accelerating stresses and uniform proportion of censoring at each point in the design, we present optimal designs for diverse forms of the regression function;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);In particular, these forms include: regression in two variables with first order terms and the cross product, polynomials in r dimension of degree less than or equal to s, and polynomials in three variables with first order terms and cross products of second order.</p>