Mean intergrated squared error properties and optimal kernels when estimating a diatribution function

Let X₁,X₂,… X_n be a sample of independent identically distributed (i.i.d)random variables having an unknown absolutely continuous distribution function f with density f the twofold aim of his paper consists in, firstly deriving asymptotic expressions of the mean intergrated squared error (MISE) of a kernel estimator of F when f is either assumed to be continuous everywhere or problem of finding optimal kernels in these two cases is studied in detail.     

A linear regression with unobserved dependent variables

Let (?,X) be a random vector such that E(X|?) = ? and Var(x|?) a + b? + c?² for some known constants a, b and c. Assume X₁,…,X_n are independent observations which have the same distribution as X. Let t(X) be the linear regression of ? on X. The linear empirical Bayes estimator is used to approximate the linear regression function. It is shown that under appropriate conditions, the linear empirical Bayes estimator approximates the linear regression well in the sense of mean squared error.     

Nonparametric estimator for mean residual life and vitality function

In this paper, we study the estimation of the vitality function(v(x)=E(X|X>x) and mean residual life function(e(x)=E(X-x|X>x) from a sample ofX using the empirical estimator and kernel estimator. Under suitable conditions of regularity, the asymptotic normality of the kernel estimator is obtained. Partially supported by Consejeria de Cultura y Ed. (C.A.R.M.), under Grant PIB 95/90.     

Estimating population size with truncated sampling

Let X₁, X₂,…,X_n be independent, indentically distributed random variables with density f(x,θ) with respect to a σ-finite measure μ. Let R be a measurable set in the sample space X. The value of X is observable if X ? (X?R) and not otherwise. The number J of observable X’s is binomial, N, Q, Q = 1?P(X ? R). On the basis of J observations, it is desired to estimate N and θ. Estimators considered are conditional and unconditional maximum likelihood and modified maximum likelihood using a prior weight function to modify the likelihood before maximizing. Asymptotic expansions are developed for the [Ncirc]’s of the form [Ncirc] = N + α√N + β + o_p(1), where α and β are random variables. All estimators have the same α, which has mean 0, variance σ² (a function of θ) and is asymptotically normal. Hence all are asymptotically equivalent by the usual limit distributional theory. The β’s differ and Eβ can be considered an “asymptotic bias”. Formulas are developed to compare the asymptotic biases of the various estimators. For a scale parameter family of absolutely continuous distributions with X = (0,∞) and R = (T,∞), special formuli are developed and a best estimator is found.     

Variances of UMVU Estimators for Means and Variances After Using a Normalizing Transformation

Suppose that the function f is of recursive type and the random variable X is normally distributed with mean μ and variance α². We set C = f(x). Neyman & Scott (1960) and Hoyle (1968) gave the UMVU estimators for the mean E(C) and for the variance Var(C) from independent and identically distributed random variables X₁,…, X_n(n ≧ 2) having a normal distribution with mean μ and variance σ², respectively. Shimizu & Iwase (1981) gave the variance of the UMVU estimator for E(C). In this paper, the variance of the UMVU estimator for Var(C) is given.     

Exponential Family Techniques for the Lognormal Left Tail

Let X be lognormal(μ,σ²) with density f(x); let θ > 0 and define . We study properties of the exponentially tilted density (Esscher transform) f_θ(x) = e^?θxf(x)/L(θ), in particular its moments, its asymptotic form as θ→∞ and asymptotics for the saddlepoint θ(x) determined by . The asymptotic formulas involve the Lambert W function. The established relations are used to provide two different numerical methods for evaluating the left tail probability of the sum of lognormals S_n=X₁+?+X_n: a saddlepoint approximation and an exponential tilting importance sampling estimator. For the latter, we demonstrate logarithmic efficiency. Numerical examples for the cdf F_n(x) and the pdf f_n(x) of S_n are given in a range of values of σ²,n and x motivated by portfolio value‐at‐risk calculations.     

On multivariate variable-kernel density estimates for time series

Let X₁ be a strictly stationary multiple time series with values in R^d and with a common density f. Let X₁,.,.,X_n, be n consecutive observations of X₁. Let k = k_n, be a sequence of positive integers, and let H_ni be the distance from X_i to its kth nearest neighbour among X_j, j i. The multivariate variable-kernel estimate f_n, of f is defined by where K is a given density. The complete convergence of f_n, to f on compact sets is established for time series satisfying a dependence condition (referred to as the strong mixing condition in the locally transitive sense) weaker than the strong mixing condition. Appropriate choices of k are explicitly given. The results apply to autoregressive processes and bilinear time-series models.     

New difference-based estimator of parameters in semiparametric regression models

The paper introduces a new difference-based Liu estimator β?_Ldiff=([Xtilde]′[Xtilde]+I)^?1([Xtilde]′[ytilde]+η β?_diff) of the regression parameters β in the semiparametric regression model, y=Xβ+f+?. Difference-based estimator, β?_diff=([Xtilde]′[Xtilde])^?1[Xtilde]′[ytilde] and difference-based Liu estimator are analysed and compared with respect to mean-squared error (mse) criterion. Finally, the performance of the new estimator is evaluated for a real data set. Monte Carlo simulation is given to show the improvement in the scalar mse of the estimator.     

Generation of multivariate distributions by vertical density representation

Troutt (1991,1993) proposed the idea of the vertical density representation (VDR) based on Box-Millar method. Kotz, Fang and Liang (1997) provided a systematic study on the multivariate vertical density representation (MVDR). Suppose that we want to generate a random vector X[d]Rⁿthat has a density function ?(x). The key point of using the MVDR is to generate the uniform distribution on [D]_?(v) = {x :?(x) = v} for any v > 0 which is the surface in RⁿIn this paper we use the conditional distribution method to generate the uniform distribution on a domain or on some surface and based on it we proposed an alternative version of the MVDR(type 2 MVDR), by which one can transfer the problem of generating a random vector X with given density f to one of generating (X, X_n+i) that follows the uniform distribution on a region in Rⁿ⁺¹defined by ?. Several examples indicate that the proposed method is quite practical.     

Moments of suprema of random variables

The supremum of random variables representing a sequence of rewards is of interest in establishing the existence of optimal stopping rules. Necessary and sufficient conditions are given for existence of moments of sup_n(X_n ? c_n) and sup_n(S_n ? c_n) where X₁, X₂, … are i.i.d. random variables, S_n = X₁ + … + X_n, and c_n = (nL(n))^1/r, 0 < r < 2, L = 1, L = log, and L = log log. Following Cohn (1974), “rates of convergence” results are used in the proof.     

The number of records within a random interval of the current record value

LetX ₁,X ₂, … be a sequence of i.i.d. random variables with some continuous distribution functionF. LetX(n) be then-th record value associated with this sequence and μ _n ⁻ , μ _n ⁺ be the variables that count the number of record values belonging to the random intervals(f−(X(n)), X(n)), (X(n), f+(X(n))), wheref−, f+ are two continuous functions satisfyingf−(x)<x, f+(x)>x. Properties of μ _n ⁻ , μ _n ⁺ are studied in the present paper. Some statistical applications connected with these variables are also provided.     

Variants of the graph dependent model in extreme value theory

With a set X₁, X₂, .... X_n n random variables, a graph is associated whose vertices are the integers 1,2,..., n and whose edges represent those pairs i and j for which the events {X_i>X} and {X_j>X} do not become “almost independent” for “large X”. With a variety of assumption on the edge set of the graph, the asymptotic distribution of the extremes of the X_j, when properly normalized, is determined. This refines the earlier result of the present author on this kind of dependence, and extends and unifies several known dependent extreme value models.     

L'effet de l'erreur d'arrondi sur l'estimateur d'une densité par la méthode du noyau

In a model for rounded data suppose that the random sample X₁,.,.,X_n,. i.i.d., is transformed into an observed random sample X_1Δ,.,.,X_nΔ, where X_iΔ = 2vΔ if X_i, ∈ (2vΔ - Δ, 2vΔ + Δ), for i = 1,.,.,n. We show that the precision Δ of the observations has an important effect on the shape of the kernel density estimator, and we identify important points for the graphical display of this estimator. We examine the IMSE criteria to find the optimal window under the rounded-data model.     

Zur abschätzung von lösungen linearer integralgleichungen mit hilfe der monte-carlo-methode

Let X₁, X₂, … be i.i.d.r.v. and write (X₁+…X_n?A_n)/B_n?F_n, where B_n >0.A_nER₁, n≥1. It is known that solely one–sided asymptotic assumptions imposed on F_n imply F_n0. In the present note we show that stronger one–sided assumptions lead even to the existence of EX₁ ³ so that the BERRY-ESSEEN inequalities hold true.     

Optimality of blue's in a general linear model with incorrect design matrix

We consider the Gauss-Markoff model (Y,X₀β,σ²V) and provide solutions to the following problem: What is the class of all models (Y,Xβ,σ²V) such that a specific linear representation/some linear representation/every linear representation of the BLUE of every estimable parametric functional p'β under (Y,X₀β,σ²V) is (a) an unbiased estimator, (b) a BLUE, (c) a linear minimum bias estimator and (d) best linear minimum bias estimator of p'β under (Y,Xβ,σ²V)? We also analyse the above problems, when attention is restricted to a subclass of estimable parametric functionals.     

Goodness-of-fit test for density estimation

It is often necessary to test whether X,…, X_n are from a certain density f(x) or not. Most test statistics such as the Kolmogorov-Smirnov, Cramer-von Mises, and Anderson-Darling statistics are based on the empirical distribution function F(x). In this paper we suggest a test statistic based on the integrated squared error of the kernel density estimator. We derive the asymptotic distribution of the statistic under the null and alternative hypothesis. Some simulation results for power comparisons are also given.     

Optimal Nonparametric Quantile Estimators. Towards a General Theory. A Survey

“Nonparametric” in the title is used to say that observations X ₁,…,X _n come from an unknown distribution F ∈ ? with ? being the class of all continuous and strictly increasing distribution functions. The problem is to estimate the quantile of a given order q ∈ (0,1) of the distribution F. The class ? of distributions is very large; it is so large that even X _nq:n , where nq is an integer, may be very poor estimator of the qth quantile. To assess the performance of estimators no properties based on moments may be used: expected values of estimators should be replaced by their medians, their variances—by some characteristics of concentration of distributions around the median. If an estimator is median-biased for one of distributions, the bias of the estimator may be infinitely large for other distributions. In the note optimal estimators with respect to various criteria of optimality are presented. The pivotal function F(T) of the estimator T is introduced which enables us to apply the classical statistical approach.     

On the weak convergence of kernel density estimators in Lp spaces

Since its introduction, the pointwise asymptotic properties of the kernel estimator f?_n of a probability density function f on ?^d, as well as the asymptotic behaviour of its integrated errors, have been studied in great detail. Its weak convergence in functional spaces, however, is a more difficult problem. In this paper, we show that if f_n(x)=(f?_n(x)) and (r_n) is any nonrandom sequence of positive real numbers such that r_n/√n→0 then if r_n(f?_n?f_n) converges to a Borel measurable weak limit in a weighted L^p space on ?^d, with 1≤p<∞, the limit must be 0. We also provide simple conditions for proving or disproving the existence of this Borel measurable weak limit.     

Quasi-universal bandwidth selection for kernel density estimators

Let f?_n, h denote the kernel density estimate based on a sample of size n drawn from an unknown density f. Using techniques from L₂ projection density estimators, the author shows how to construct a data-driven estimator f?_n, h which satisfies This paper is inspired by work of Stone (1984), Devroye and Lugosi (1996) and Birge and Massart (1997).     

On the rate of convergence of recursive kernel estimates of probability densities

Recursive estimates f_n^r(x)of the rth derivative f^r(x)(r=0,1)of the univariate probability density f(x) for strictly stationary processes {X_j,} are considered. The asymptotic variance-covariance of f_n^r(x)is established for stationary triangular arrays of random variables satisfying various asymptotic independence-uncorrelatedness conditions.