Consistent estimation of the structural distribution function (Q2711687)

An i.i.d. sample of size \(n\) is considered for a discrete random variable with cell probabilities \(p_{iN}\), where \(i=1,\dots,N\), \(N\) is the number of possible values of the variable (cells). Let \(\nu_{in}\) be the sample absolute frequencies. The authors consider the case when \(N=N_n\) and \(\lim_{n\to\infty} n/N_n=\lambda\), \(0<\lambda<\infty\). Denote NEWLINE\[NEWLINEf_N(t)=\sum_{i=1}^N Np_{iN}{\mathbf I}\{(i-1)/N<t\leq i/N\},\qquad G_{f_N}(x)=\int_0^1g(f_N(t),x)dt,NEWLINE\]NEWLINE where \(g\) is an arbitrary Borel-measurable function. (In the case \(g(f,x)={\mathbf I}\{f<x\}\) this function is called the structural distribution function.) The authors consider the problem of \(G_{f_N}(x)\) estimation. It is demonstrated that a naive plug in estimator which uses \(\nu_{in}\) instead of \(p_{in}\) is inconsistent for \(\lambda< \infty\).NEWLINENEWLINENEWLINEThe authors propose two new estimators, one based on grouping of the cells and the other uses an inverse Laplace transform to correct the bias of the naive estimator. Consistency of these estimators is demonstrated. An example of linguistics data analysis is considered.