Investigation of The Limiting Behavior of The Sequence of Density Ratios in A Family of Non-Central t-Distribution

Sunardi Wirjosudirdjo,

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

Let X₁, X₂, …..be a sequence of noncentral t-distributed random variables with parameter θ. In this paper we investigate the limiting behavior of density ratios of X₀. This is an example of the family of distributions discussed in [4].Here, we derive the results of David and Kurskall [2] in a slightly more general form and a stronger result is obtained, namely:

If θ₁ < θ₂ are two parameters and R_n(θ₁,θ₂) is the density ratio of X_n, then there is  θ₀ such that the limit of R_n is 0 with respect  to θ<θ₀ and is ∞  with respect  to θ>θ₀. With respect to θ=θ₀, the limit infimum is 0 and the limit supermum is ∞. It is also shown that θ₀ is between 0 and ½ (θ₁+θ₂).

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