ABSTRACT
When the origin is at an unknown point -y such that x > -y, then -y becomes an additional (threshold) parameter that must be estimated from sample data. Although in many applications the origin is known to be at zero, there are numerous situations in which it is unknown. In this more general three-parameter case, the random variable X is replaced by (X - -y). In equations (5.2.3) we now have
E(X) = -y + f3f 1, Me(X) = -y + l3(ln 2)w', (5.5.1) (8-1) 115 Mo(X) = -y + 13 - 8 - ,
but V(X), a 3(X), and aiX) are invariant under changes of the origin. With the addition of the threshold parameter, the loglikelihood function of a
singly right censored sample as given by (5.3.3) now becomes I II
In L = n In 8 - nIne - - ~ (x; - -y)8 e i= 1
(5.5.2)
