Parameter Estimation for a Discontinuous Signal in White Gaussian Noise

It is shown that for a discontinuous and quasidiscontinuous signal $S(t-\theta)$ the quadratic risk of the estimate of the parameter $\theta$ in white Gaussian noise of spectral density $\varepsilon^2$ is proportional to $\varepsilon^4$ when $\varepsilon\to 0$. The minimum attainable coefficient for $\varepsilon^4$ is found, as well as estimates for which this minimum is attained. It is shown that the maximum-likelihood estimate in this sense is inferior to the optimum one by roughly a factor of 1.3 when $\varepsilon\to 0$. The limiting distributions of the estimates are also found; they are non-Gaussian but general for all $S(t)$ with discontinuities of the first kind. The only parameter that appears in these distributions is the number $r^2$, this being equal to the sum of squares of the discontinuities of $S(t-\theta)$ in the observation interval.