The maximum likelihood estimation (MLE) is undoubtedly the most popular (at least until now) procedure for estimation of reliability R = P(X < Y) due to its flexibility and generality. The technique can always be used if the joint distribution of the stress X and the strength Y is a known function with some unknown parameters. A detailed description of the MLE method is presented, for example, in Casella and Berger (1990) and Lehmann and Casella (1998). Here, we shall concentrate on a discussion of the MLE of the reliability R = P(X < Y).Assume that a random vector (X, Y) has the probability density function (pdf) f(x, y\&) with an unknown scalar or vector-valued parameter 6 € ©. The aim is to estimate R on the basis of observations {X\, Y\), • • •, (X n , Y n ) Note that if X and Y are independent with the pdf of the form (2.1) the number of observations for X and Y need not be the same. In general, the data is of the form (X_, Y_) (2.2) with n\ =ni'\iX and Y are dependent. 11 The Stress-Strength Model and Its Generalizations Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM LIBRARY -INFORMATION SERVICES on 03/21/15. For personal use only. The Theory and Some Useful Approaches Let f(2L, Yji 8) denote the joint pdf of the data, i.e. Note that if X and Y are independent, (2.3) becomes n\ ri2 f(X,Y\8) = Ylfx{Xi\B) J ] MYj\0). (2.4) Definition 2.1 Given that (X, Y) is observed, the function of 9 defined by L(9\X_,Y_) = f{X_,Y_\8) is called the likelihood function. Definition 2.2 The maximum likelihood estimator (MLE) 8 = 9(X_, Y_) of the parameter 8 based on the sample (X_, Y_) is the parameter value at which the likelihood function L(8\X_, Y) attains its maximum as a function of 9. Theorem 2.1 (Invariance property of the MLEs.) If 8 is the MLE of 8, then for any function 0,is the Laplace transform of fy(y\9) calculated at the point a.If the pdf of (X,...
