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, ϑ) which are continuous with a cusp at the point ϑ. In this connection, it is worth noting that Daniels [2] has obtained a modified MLE for the family of densities f(X, ϑ) = C(λ) exp {−|X − ϑ|λ}, for λ such that ½ < λ ½ 1, where C(λ) is a constant depending on λ and he has shown that this estimator is asymptotically efficient. In this paper, we shall show that the MLE of ϑ is hyper-efficient for the family of densities f(X, ϑ) given by (1.1) log f(X,ϑ) = ε(X,ϑ)|X − ϑ)|X−ϑ| ^λ + g(X,ϑ) for |X| ≤ A = g(X,ϑ) for |X| > A where A is a constant greater than zero, (1.2) ε(X,ϑ) = β(ϑ) if X < ϑ = γ(ϑ) if X > ϑ, (1.3) 0 < λ < ½ and ϑ ε(a, b) where -A < a < b < A, under some regularity conditions on f(X, ϑ) 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, ϑ). 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.