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Based on arbitrarily right-censored observations from a probability density function f deg the existence and uniqueness of the maximum penalized likelihood estimator (MPLE) of f deg is proven. In particular, the first MPLE of Good and Gaskins of a density defined on (0, infinity) is shown to exist and to be unique under arbitrary right-censorship. Furthermore, the MPLE is in the form of a solution to a linear integral equation. (Author).
This book deals with parametric and nonparametric density estimation from the maximum (penalized) likelihood point of view, including estimation under constraints. The focal points are existence and uniqueness of the estimators, almost sure convergence rates for the L1 error, and data-driven smoothing parameter selection methods, including their practical performance. The reader will gain insight into technical tools from probability theory and applied mathematics.
This volume covers an area of statistics dealing with complex problems in the production of goods and services, maintenance and repair, and management and operations. The opening chapter is by W. Edwards Deming, pioneer in statistical quality control, who was involved in the quality control movement in Japan and helped the country in its rapid industrial development. He gives a 14-point program for management to keep a country in an ascending path of industrial development.
Estimators for the probability density function, cumulative distribution function, and hazard function are proposed in the random censorship setting. The estimators are derived from the Kaplan-Meier product limit estimator by maximum penalized likelihood methods. The authors establish the existence and uniqueness of the estimates, which are exponential splines with knots at the uncensored observations, and provide an efficient algorithm for their numerical evaluation. They prove the consistency, in probability and almost surely, of the density estimates in the Hellinger distance, the L sub p norms for p =1, 2, infinity, and the Sobolev norm. The corresponding hazard rate estimator converges uniformly on bounded intervals. (Author).
The purpose of this article is to present the different types of nonparametric density estimates that have been proposed for the situation that the sample data are censored or incomplete. This type of data arises in many life testing situations and is common in survival analysis problems. Many of the methods of nonparametric density and hazard rate estimation from right-censored observations are discussed. These include histogram and kernel-type procedures, likelihood methods, Fourier series methods, and Bayesian nonparametric approaches. Examples of kernel density estimates are given for mechanical switch life data where data-based choices of the bandwidth values are used. Originator-supplied keywords included: Nonparametric density estimation; Random censorship; Failure rate; Kernel density estimator; Likelihood methods.