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Inference in $α$-Brownian bridge based on Karhunen-Loève expansions

We study a simple decision problem on the scaling parameter in the $α$-Brownian bridge $X^{(α)}$ on the interval $[0,1]$: given two values $α_0, α_1 \geq 0$ with $α_0 + α_1 \geq 1$ and some time $0 \leq T \leq 1$ we want to test $H_0: α= α_0$ vs. $H_1: α= α_1$ based on the 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.