Fast Nonparametric Maximum Likelihood Density Deconvolution Using Bernstein Polynomial

Abstract

A new maximum approximate likelihood method for deconvoluting a continuous density on a finite interval in additive measurement error models with known error distribution using the approximate Bernstein polynomial model, a finite mixture of specific beta distributions, is proposed. The change-point detection method is used to choose an optimal model degree. Based on a contaminated sample of size n, under an assumption which is satisfied, among others, by the generalized normal error distribution, the optimal rate of convergence of the mean integrated squared error is proved to be O(n −1+5/k log n) if the underlying unknown density admits an approximate Bernstein polynomial model of degree m within χ 2-divergence of order O(m −k) with k > 5. Simulation shows that the small sample performance of the proposed estimator is better than the deconvolution kernel density estimator. The proposed method is illustrated by a real data application.

Maximum approximate likelihood estimation in accelerated failure time model for interval‐censored data

“…This model has been successfully applied to grouped, contaminated, multivariate, and interval censored data. [25][26][27] Particularly, Guan 28 applied this model to obtain maximum likelihood estimation in Cox's proportional hazards regression model. Due to a better approximation of the unknown underlying baseline density function, not only a smooth estimate of the survival function but improved estimates of regression coefficients can be resulted.…”

“…This Bernstein polynomial approximation is actually a mixture of some specified beta distributions with shapes related to the degree. This model has been successfully applied to grouped, contaminated, multivariate, and interval censored data 25‐27 . Particularly, Guan 28 applied this model to obtain maximum likelihood estimation in Cox's proportional hazards regression model.…”

Maximum approximate Bernstein likelihood estimation in proportional hazard model for interval‐censored data

“…This Bernstein polynomial model performs much better than the classical kernel method for estimating density even from grouped data 33 and data with measurement errors. 34 The maximum approximate Bernstein likelihood estimate can be viewed as a continuous version of the non-or semi-parametric maximum likelihood estimate. In this article, such estimates of the conditional survival function and density function given covariate are proposed by fitting interval censored data with Cox's proportional hazards model.…”

Maximum Approximate Likelihood Estimation in Accelerated Failure Time Model for Interval-Censored Data

“…This Bernstein polynomial approximation is actually a mixture of some specified beta distributions with shapes related to the degree. This model has been successfully applied to grouped, contaminated, multivariate, and interval censored data (Guan, 2017(Guan, , 2019aWang & Guan, 2019). This model shall be applied to find maximum likelihood estimates of the regression coefficients, and the density and survival functions in the AFT model.…”