ABSTRACT
Usually it is convenient to work with the natural logarithm, denoted by /{ /( .) ; y} and called the log likelihood
In most applications, we consider families 3 in which the func tional form of the p.d.f. is specified but in which a finite number of unknown parameters 0 = (0l5 . . . , 0„) are unknown. Then the p.d.f. of Y for given 0 can be written / ( y ; 0) or, where desirable, as f y ( y ; 0)- The set o f allowable values for 0, denoted by f l , or some times by £le , is called the parameter space. Then the likelihood (1) is a function of 0 and we can write, always for 0 E J2 ,
lik {/(.) ;y} = / (y ) . (1)
l { f ( . ) ; y } = log /(y ) . (2)
lik(0 ;y) = / ( y ; 0 ), lid ;y ) = l o g / ( y ; 0 ). (3)
If it is required to stress the particular random variable for which the likelihood is calculated we add a suffix, as in liky(0 ;y). It is crucial that in these definitions 6 is the argument of a function and can take any value in £2. A very precise notation would distinguish between this use of 6 and the particular value that happens to be true, but fortunately this is unnecessary, at least in the present chapter.
