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Limiting Behavior of A Sequence of Density Ratios

Sunardi Wirjosudirdjo

Data & Software Engineering Research Group
STEI-ITB, Jl. Ganeca No.10, Bandung 40132, Indonesia


Let X1, X2,.. be a sequence of random variables and Р= {Pθ,θ Є (-)} be a family of distributions of the sequence.  For each n, An is the σ-field generated by X1,, Xn.If θ12 Є (-), we define Rn 12) as the density ratio of Pθ1,Pθ2 on An.

The main purpose of the paper is to investigate limiting behavior of the sequence Rn with respect to any Pθ. 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 Eθ {Yi}>0 or <0. For any θ, say θ0, for which Eθ0{Yi}=0 we have lim inf Rn="0" and lim sup Rn=~ a.e. Pθ0.

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 θ1 < θ2 then Rn converges a.e. to 0 if θ≤θ1 and to ~ if 0 ≥ θ2. For θ between θ1and θ2, except perhaps for one θ0, 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 θ. There is no example known to show  that there may exist a θ0 for which  the sequence density ratios has a positive lim inf and a finite lim sup.

Keywords: mathematics

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