A Nonnegative density estimation based on the restricted Sobolev space H'_(0)(a,b)

Abstract

본인은 GOOD-GASKINS 에 의해 제안된 1st penalty functional을 이용하여 H^(1)_(O)(a,b)공간 위에서 true density에 대하여 공부하였다. 그리하여 general penalty functional을 이용하여 TAPIA - THOMPSON 이 얻어낸 density 에 비해 전구간 (a,b)에서 positive하고 degree 2인 Polynomisl spline을 얻게되었다.;We want to estimate a densty function with finite support (a, b). We use the 1st GOOD -GASKINS penalty function to resolve the numerical difficulty when we use the usual maximum likelihood method. The fundamental mathematical structure used for the problem is the restricted SOBOLEV SPACE, H?(a, b). Similar result was obtained by TAPIA and THOMPSON with different penalty functional. Our result is as follows : 1. The density estimator based on the 1st GOOD - GASKINS penalty functional exits and it is unique on H?(a, b). 2. The density estimator is nonnegative on (a, b) and a polynomial spline of degree 2. The main advantage of this estimator is that it is nonnegative on the whole support (a, b), which was not the case in general of the TAPIA and THOMPSION's Roughly speaking, it is smoother than the TAPIA and THOMPSON's. Nevertheless we may control the smoothness by the penalty parameter α