Inference for α-Brownian bridge based on Karhunen-Loève expansions
(English)Article in journal (Refereed) Submitted
We study a simple decision problem for the scaling parameter in the α-Brownian bridge X(α) on the interval [0,1]: given two values α0, α1 ≥ 0 with α0 + α1 ≥ 1 and some time 0 ≤ T ≤ 1 we want to test H0: α = α0 vs. H1: α = α1 based on an observation of X(α) until time T. The likelihood ratio can be written as a functional of a quadratic form ψ(X(α)) of X(α). In order to calculate the distribution of ψ(X(α)) under the null hypothesis, we generalize the Karhunen-Loève Theorem to positive finite measures on [0,1] and compute the Karhunen-Loève expansion of X(α) under such a measure. Based on this expansion, the distribution of ψ(X(α)) follows by Smirnov's formula.
Probability Theory and Statistics
Research subject Mathematical Statistics
IdentifiersURN: urn:nbn:se:uu:diva-232541OAI: oai:DiVA.org:uu-232541DiVA: diva2:748603