Limiting Behavior of A Sequence of Density Ratios - Sunardi Wirjosudirdjo (Journal of Mathematical and Fundamental Sciences, 1963)
Let X1, X2,"¦.. be a sequence of random variables and Ð = {PÎ¸,Î¸ Ð„ (-)} be a family of distributions of the sequence. For each n, An is the Ïƒ-field generated by X1,"¦, Xn.If Î¸1,Î¸2 Ð„ (-), we define Rn (Î¸1,Î¸2) as the density ratio of PÎ¸1,PÎ¸2 on An.The main purpose of the paper is to investigate limiting behavior of the sequence Rn with respect to any PÎ¸. This has applications in sequential analysis, where it is desired to know whether a sequential probability ratio test terminates with probability one. The same conclusion can be drawn in the case of generalized sequential probability ratio test, under some restriction as to how the stopping bounds vary with n.If the Xi are independent and identically distributed, then we can write in Rn as where the Yi are independent and identically distributed. We have then that converges to ~ or to -~ a.e. according as EÎ¸ Yi>0 or <0. For any Î¸, say Î¸0, for which EÎ¸0Yi=0 we have lim inf Rn="0" and lim sup Rn=~ a.e. PÎ¸0.A sequence of non-independent nor identically distributed random variables Xi may arise in tests of composite hypotheses in the presence of nuisance parameters. An example of the situation is the sequential t-test, by some authors called the WAGR test. In this example we have the same qualitative result as if the Xi are independent and identically distributed.The foregoing example suggested the more general problem with the assumption A and B (see chapters 2 and 3). The result can be described as follows: If Î¸1 < Î¸2then Rn converges a.e. to 0 if Î¸â‰¤Î¸1 and to ~ if 0 â‰¥ Î¸2. For Î¸ between Î¸1and Î¸2, except perhaps for one Î¸0, then lim inf is 0 or lim sup is ~ a.e. So that a sequential probability ratio test terminates with probability one, except perhaps for one value of Î¸. There is no example known to show that there may exist a Î¸0 for which the sequence density ratios has a positive lim inf and a finite lim sup.