2.1.1 The Theory

The maximum likelihood estimation (MLE) is undoubtedly the most popular (at least until now) procedure for estimation of reliability R=P(X 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

Assume that a random vector (X,Y) has the probability density function (pdf) f(x, y ?) with an unknown scalar or vector-valued parameter ? ? ?. The aim is to estimate R on the basis of observations ( X ₁, Y ₁), ,( 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 ₁ =n ₂ if X and Y are dependent.

Let f( X , Y ; ?) denote the joint pdf of the data, i.e.

(2.3)

Note that if X and Y are independent, (2.3) becomes

(2.4)