TAILIEUCHUNG - Ebook Mathematical statistics (2nd edition): Part 2
(BQ) Part 2 book "Mathematical statistics" has contents: Estimation in nonparametric models, hypothesis tests, confidence sets, distribution estimators, statistical functionals, variance estimation. | Chapter 5 Estimation in Nonparametric Models Estimation methods studied in this chapter are useful for nonparametric models as well as for parametric models in which the parametric model assumptions might be violated (so that robust estimators are required) or the number of unknown parameters is exceptionally large. Some such methods have been introduced in Chapter 3; for example, the methods that produce UMVUE’s in nonparametric models, the U- and V-statistics, the LSE’s and BLUE’s, the Horvitz-Thompson estimators, and the sample (central) moments. The theoretical justification for estimators in nonparametric models, however, relies more on asymptotics than that in parametric models. This means that applications of nonparametric methods usually require large sample sizes. Also, estimators derived using parametric methods are asymptotically more efficient than those based on nonparametric methods when the parametric models are correct. Thus, to choose between a parametric method and a nonparametric method, we need to balance the advantage of requiring weaker model assumptions (robustness) against the drawback of losing efficiency, which results in requiring a larger sample size. It is assumed in this chapter that a sample X = (X1 , ., Xn ) is from a population in a nonparametric family, where Xi ’s are random vectors. Distribution Estimators In many applications the .’s of Xi ’s are determined by a single . F on Rd ; for example, Xi ’s are . random d-vectors. In this section, we 319 320 5. Estimation in Nonparametric Models consider the estimation of F or F (t) for several t’s, under a nonparametric model in which very little is assumed about F . Empirical .’s in . cases For . random variables X1 , ., Xn , the empirical . Fn is defined in (). The definition of the empirical . based on X = (X1 , ., Xn ) in the case of Xi ∈ Rd is analogously given by Fn (t) = 1 n n I(−∞,t] (Xi ), i=1 t ∈ Rd .
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