Estimation of the Location of the Cusp of a Continuous Density

Abstract

Chernoff and Rubin [1] and Rubin [5] investigated the problem of estimation of the location of a discontinuity in density. They have shown that the maximum likelihood estimator (MLE) is hyper-efficient under some regularity conditions on the density and that asymptotically the estimation problem is equivalent to that of a non-stationary process with unknown center of non-stationarity. We have obtained here similar results for a family of densities $f(x, \vartheta)$ which are continuous with a cusp at the point $\vartheta$. In this connection, it is worth noting that Daniels [2] has obtained a modified MLE for the family of densities $f(x, \vartheta) = C(\lambda) \exp \{-|x - \vartheta|^\lambda\}$, for $\lambda$ such that $\frac{1}{2} < \lambda < 1$, where $C(\lambda)$ is a constant depending on $\lambda$ and he has shown that this estimator is asymptotically efficient. In this paper, we shall show that the MLE of $\vartheta$ is hyper-efficient for the family of densities $f(x, \vartheta)$ given by \begin{equation*}\begin{align*}\tag{1.1}\log f(x, \vartheta) = \epsilon(x, \vartheta)|x - \vartheta|^\lambda + g(x, \vartheta) \text{for} |x| \leqq A \\ = g(x,\vartheta)\quad\text{for} |x| > A \\ \end{align*}\end{equation*} where $A$ is a constant greater than zero, \begin{equation*}\begin{align*}\tag{1.2}\epsilon (x,\vartheta) = \beta(\vartheta) \text{if} x < \vartheta \\ = \gamma(\vartheta) \text{if} x > \vartheta, \\ \end{align*}\end{equation*}\begin{equation*}\tag{1.3}0 < \lambda < \frac{1}{2} \text{and}\end{equation*}\begin{equation*}\tag{1.4}\vartheta\varepsilon (a, b) \text{where} -A < a < b < A,\end{equation*} under some regularity conditions on $f(x, \vartheta)$ and we shall derive the asymptotic distribution of the MLE implicitly. Section 2 contains the regularity conditions imposed on the family of densities $f(x, \vartheta)$. Section 3 contains some results related to the asymptotic properties of the MLE. The estimation problem is reduced to that of a stochastic process in Section 4. The asymptotic distribution of MLE is obtained in Section 5.