Lan of thinned empirical process with an application to fuzzy set density (Q1294768)

This article is the development of the first author's previous work [Ann. Stat. 26, No. 2, 692-718 (1998; Zbl 0930.62017)], where properties of the truncated empirical process were investigated. Now assume that a random element \(X\), with values in some arbitrary sample space \(S\) equipped with a \(\sigma\)-field \(\mathcal{D}\), has distribution \(P_{\theta}\), \({\theta}\in {\Theta}\in R^1\). Let \(U\) be a random variable with values in \(\{0,1\}\) such that \(P(U=1 |X=x)=\phi(x)\), where \(X\in S\) and \(\phi\) is a Borel measurable function. Denote by \(\varepsilon_x(B)=I_B(x)\) the pertaining Dirac measure. For \((X_i,U_i)\) , \(i=1,\dots,n\), being independent copies of \((X,U)\) a thinned empirical process \(N_n^{\phi}(\cdot)=\sum_{i=1}^n U_i\varepsilon_{X_i}(\cdot)\) models the sampling scheme according to which an observation \(X_i\), if falling on \(x\in S\), enters the data set only with probability \(\phi(x)\). Such a setup can be utilized for modeling truncated, censored and missing observations. The authors established the LAN-property (local asymptotic normality) of a thinned empirical process \(N_n^{\phi_n}\) for an arbitrary sequence \(\phi_n\) of thinning functions satisfying \(\alpha_n=E(\phi_n(x))\to 0\), \(n\alpha_n\to\infty\) as \(n\to\infty\). They give a characterization of the central sequence in the LAN-expansion of \(N_n^{\phi_n}\). In particular, the central sequence depends only on the total number \(\tau(n)\) of nonthinned observations iff the limit of the coefficient of variation of the tangent function is 1 or -1. In this case, under additional regularity conditions, an asymptotically efficient estimator \(\theta_n\) of the unknown true value \(\theta_0\) of the parameter \(\theta\) can be based on \(\tau(n)\) only. These results are applied to the investigation of a fuzzy set density estimator both in parametric and nonparametric models.