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Limiting Behavior of A Sequence of Density Ratios
Sunardi Wirjosudirdjo
Data & Software Engineering Research Group STEIITB, Jl. Ganeca No.10, Bandung 40132, Indonesia
Abstract. Let X_{1}, X_{2},….. be a sequence of random variables and Р= {P_{θ},θ Є ()} be a family of distributions of the sequence. For each n, An is the σfield generated by X_{1},…, X_{n}.If θ_{1},θ_{2} Є (), we define R_{n }(θ_{1},θ_{2}) as the density ratio of P_{θ1},P_{θ2} on An.
The main purpose of the paper is to investigate limiting behavior of the sequence R_{n} with respect to any P_{θ}. 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 X_{i }are independent and identically distributed, then we can write in Rn as where the Y_{i} are independent and identically distributed. We have then that converges to ~ or to ~ a.e. according as E_{θ} {Y_{i}}>0 or <0. For any θ, say θ_{0}, for which Eθ_{0}{Yi}=0 we have lim inf Rn="0" and lim sup R_{n}=~ a.e. Pθ_{0.}
A sequence of nonindependent 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 ttest, 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 θ_{1} < θ_{2} then R_{n} converges a.e. to 0 if θ≤θ_{1} and to ~ if 0 ≥ θ_{2}. For θ between θ_{1}and θ_{2}, except perhaps for one θ_{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 θ. There is no example known to show that there may exist a θ_{0} for which the sequence density ratios has a positive lim inf and a finite lim sup.
Keywords: mathematics
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