Poisson Process with Linear Drift and Related Function Series

Consider the random process $Y(t)=at-\nu_+(pt)+\nu_-(-qt)$, $t\in(-\infty,\infty)$, where $\nu_{\pm}(t)$ are independent standard Poisson processes for $t\geqslant 0$ and $\nu_{\pm}(t)=0$ for $t<0$. The parameters $a$, $p$, and $q$ are such that $\mathbf{E}Y(t)<0$, $t\neq0$. We evaluate the sums $\varphi_m(z,r)=\sum_{k\geq0}(re^{-r})^{k}(z+k)^{m+k-1}/k!$, $m=1,2,\dots$, $z\geq0$, of function series with parameter $ r\in(0,1) $. These series are used for recursive evaluation of the moments $\mathbf{E}(t^*)^m$, $m\geq 1$, for the time $t^*$ when the trajectory of the process $Y(t)$ attains its maximum value. The results obtained are applied to the problem of estimating the parameter $\theta$ from $n$ observations with density $f(x,\theta)$, which has a jump at the point $x=x(\theta)$, $x'(\theta)\neq 0$. If $\widehat\theta_n$ is a maximum likelihood estimator for the true parameter $\theta_0$, then the limit distribution as $n\to\infty$ for the normalized estimators $n(\widehat\theta_n-\theta_0)$ is the distribution of the argument of the maximum $t^*_{\theta_0}$ of the trajectory of the process $Y(t)$ with parameters $a$, $p$, and $q$, which depend on both the one-sided limits of the density at the point $x(\theta_0)$ and the derivative $x'(\theta_0)$. In this case, by evaluating the moments $\mathbf{E}(t^*_{\theta_0})^m$, $m=1, 2$, one can estimate both the asymptotic bias for the maximum likelihood estimator and its efficiency.