Legendre polynomial kernel estimation of a density function with censored observations and an application to clinical trials

Let f(x), x ∈ ?^M, M ≥ 1, be a density function on ?^M, and X₁, …., X_n a sample of independent random vectors with this common density. For a rectangle B in ?^M, suppose that the X's are censored outside B, that is, the value X_k is observed only if X_k ∈ B. The restriction of f(x) to x ∈ B is clearly estimable by established methods on the basis of the censored observations. The purpose of this paper is to show how to extrapolate a particular estimator, based on the censored sample, from the rectangle B to a specified rectangle C containing B. The results are stated explicitly for M = 1, 2, and are directly extendible to M ≥ 3. For M = 2, the extrapolation from the rectangle B to the rectangle C is extended to the case where B and C are triangles. This is done by means of an elementary mapping of the positive quarter‐plane onto the strip {(u, v): 0 ≤ u ≤ 1, v > 0}. This particular extrapolation is applied to the estimation of the survival distribution based on censored observations in clinical trials. It represents a generalization of a method proposed in 2001 by the author 2]. The extrapolator has the following form: For m ≥ 1 and n ≥ 1, let K_{m, n}(x) be the classical kernel estimator of f(x), x ∈ B, based on the orthonormal Legendre polynomial kernel of degree m and a sample of n observed vectors censored outside B. The main result, stated in the cases M = 1, 2, is an explicit bound for E|K_{m, n}(x) ? f(x)| for x ∈ C, which represents the expected absolute error of extrapolation to C. It is shown that the extrapolator is a consistent estimator of f(x), x ∈ C, if f is sufficiently smooth and if m and n both tend to ∞ in a way that n increases sufficiently rapidly relative to m. © 2006 Wiley Periodicals, Inc.