STRONG CONSISTENCY OF THE MAXIMUM PRODUCT OF SPACINGS ESTIMATES WITH APPLICATIONS IN NONPARAMETRICS AND IN ESTIMATION OF UNIMODAL DENSITIES

YONGZHAO SHAO^{1} AND MARJORIE G. HAHN^{2}

1Department of Statistics, Columbia University, New York, NY 10027, U.S.A. 2Department of Mathematics, Tufts University, Medford, MA 02155, U.S.A.

(Received November 14, 1996; revised June 16, 1997)

Abstract. Distributions with unimodal densities are among the most commonly used in practice. However, for many unimodal distribution families the likelihood functions may be unbounded, thereby leading to inconsistent estimates. The maximum product of spacings (MPS) method, introduced by Cheng and Amin and independently by Ranneby, has been known to give consistent and asymptotically normal estimators in many parametric situations where the maximum likelihood method fails. In this paper, strong consistency theorems for the MPS method are obtained under general conditions which are comparable to the conditions of Bahadur and Wang for the maximum likelihood method. The consistency theorems obtained here apply to both parametric models and some nonparametric models. In particular, in any unimodal distribution family the asymptotic MPS estimator of the underlying unimodal density is shown to be universally L^{1} consistent without any further conditions (in parametric or nonparametric settings).

Key words and phrases: Asymptotic MPS estimator, L^{1} consistency, monotone density, nonparametric, spacing, unimodal density.