Inference in $\alpha$-Brownian bridge based on Karhunen-Lo\`eve expansions

We study a simple decision problem on the scaling parameter in the $\alpha$-Brownian bridge $X^{(\alpha )}$ on the interval $[0,1]$: given two values ${\alpha }_0,{\alpha }_1\ge 0$ with ${\alpha }_0+{\alpha }_1\ge 1$ and some time $0\le T\le 1$ we want to test $H_0:\alpha ={\alpha }_0$ vs. $H_1: \alpha = \alpha_1$basedontheobservationof$X^{(\alpha)}$untiltime$T$. The likelihood ratio can be written as a functional of a quadratic form $\psi (X^{(\alpha )})$ of $X^{(\alpha )}$. In order to calculate the distribution of $\psi (X^{(\alpha )})$ under the null hypothesis, we generalize the Karhunen-Lo`eve Theorem to positive finite measures on $[0,1]$ and compute the Karhunen-Lo`eve expansion of $X^{(\alpha )}$ under such a measure. Based on this expansion, the distribution of $\psi (X^{(\alpha )})$ follows by Smirnov’s formula.