On A strongly consistent nonparametric density estimator for the deconvolution problem

The problem of nonparametric estimation of a probability density function is studied when the sample observations are contaminated with random noise. Previous authors have proposed estimators which use kernel density and deconvolution techniques. The appearance and properties of the previously proposed estimators are affected by constants M_n and h_n which the user may choose. However, the optimal choices of these constants depend on the sample size n, the noise distribution and the unknown distribution which is being estimated. Hence, in practice, M_n and h_n are optimally selected as functions of the data. In this paper it is shown that a class of the proposed estimators are uniformly, strongly consistent when M_n and h_n are allowed to be random variables. Even when M_n and h_n are constants, these results are new findings.     

Estimators of shift based on statistics of the Kolmogorov-Smirnov type

This paper is concerned with the estimation of a shift parameter δ_o, based on some nonnegative functional Hg₁ of the pair (D^δ_N(x), f?^δ_N(x)), where D^δ_N(x) = K_N/b {F_2,n(x)—F_1,m (x + δ)}, +^δ_N(x) = {mF_1,m (x + δ) + nF_2,n(x)}/N, where F_1,m and F_2,n are the empirical distribution functions of two independent random samples (N = m + n), and where K²_N = mn/N. First an estimator δ_N, is defined as a value of δ minimizing a functional H of the type of H₁. A second estimator δ¹_N is also defined which is a linearized version of the first. Finite and asymptotic properties of these estimators are considered. It is also shown that most well-known test statistics of the Kolmogorov-Smirnov type are particular cases of such functionals H₁. The asymptotic distribution and the asymptotic efficiency of some estimators are given.     

Spectral analysis with replicated time series

A doubly stochastic process {x(b,t);b?B,t?Z} is considered, with (B,β,P_β) being a probability space so that for each b, {X(b,t);t ? Z} is a stationary process with an absolutely continuous spectral distribution. The population spectrum is defined as f(ω) = E_B[Q(b,ω)] with Q(b,ω) being the spectral density function of X(b,t). The aim of this paper is to estimate f(ω) by means of a random sample b₁,…,b_r from (B,β,P_β). For each b₁? B, the processes X(b₁,t) are observed at the same times t=1,…,N. Thus, r time series (x(b₁,t)} are available in order to estimate f(ω). A model for each individual periodogram, which involves f(ω), is formulated. It has been proven that a certain family of linear stationary processes follows the above model In this context, a kernel estimator is proposed in order to estimate f(ω). The bias, variance and asymptotic distribution of this estimator are investigated under certain conditions.     

Simultaneous estimation of several CDF’s: homogeneity constraint

Let {x_ij(1 ? j ? n_i)|i = 1, 2, …, k} be k independent samples of size n_j from respective distributions of functions F_j(x)(1 ? j ? k). A classical statistical problem is to test whether these k samples came from a common distribution function, F(x) whose form may or may not be known. In this paper, we consider the complementary problem of estimating the distribution functions suspected to be homogeneous in order to improve the basic estimator known as “empirical distribution function” (edf), in an asymptotic setup. Accordingly, we consider four additional estimators, namely, the restricted estimator (RE), the preliminary test estimator (PTE), the shrinkage estimator (SE), and the positive rule shrinkage estimator (PRSE) and study their characteristic properties based on the mean squared error (MSE) and relative risk efficiency (RRE) with tables and graphs. We observed that for k ? 4, the positive rule SE performs uniformly better than both shrinkage and the unrestricted estimator, while PTEs works reasonably well for k < 4.     

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.     

MINIMUM MSE ESTIMATION BY LINEAR COMBINATIONS OF ORDER STATISTICS1

In this paper we assume that in a random sample of size ndrawn from a population having the pdf f(x; θ) the smallest r₁ observations and the largest r₂ observations are censored (r₁0, r₂0). We consider the problem of estimating θ on the basis of the middle n-r₁-r₂ observations when either f(x;θ)=θ^-1f(x/θ) or f(x;θ) = (aθ)¹f(x-θ)/aθ) where f(·) is a known pdf, a (<0) is known and θ (>0) is unknown. The minimum mean square error (MSE) linear estimator of θ proposed in this paper is a “shrinkage” of the minimum variance linear unbiased estimator of θ. We obtain explicit expressions of these estimators and their mean square errors when (i) f(·) is the uniform pdf defined on an interval of length one and (ii) f(·) is the standard exponential pdf, i.e., f(x) = exp(–x), x0. Various special cases of censoring from the left (right) and no censoring are considered.     

Robust M-estimators of scale: Minimax bias versus maximal variance

In the location-scale estimation problem, we study robustness properties of M-estimators of the scale parameter under unknown ?-contamination of a fixed symmetric unimodal error distribution F₀. Within a general class of M-estimators, the estimator with minimax asymptotic bias is shown to lie within the subclass of α-interquantile ranges of the empirical distribution symmetrized about the sample median. Our main result is that as ? → 0, the limiting minimax asymptotic bias estimator is sometimes (e.g., when F_o is Cauchy), but not always, the median absolute deviation about the median. It is also shown that contamination in the neighbourhood of a discontinuity of the influence function of a minimax bias estimator can sometimes inflate the asymptotic variance beyond that achieved by placing all the ?-contamination at infinity. This effect is quantified by a new notion of asymptotic efficiency that takes into account the effect of infinitesimal contamination of the parametric model for the error distribution.     

Estimation of the derivative of a density and related functions

Let Y₁,…,Y _n, (Y₁ <Y₂<…<Y _n) be the order statistics of a random sample from a distribution F with density f on the realline. This paper gives a class of estimators of the derivativef'(x) of the density f at points x for which f has

a continuoussecond derivative. These estimators are based on spacings inthe order statistics Y_j+kn -y _j j = 1,…,n-k_n,k_n<n.     

New estimators of distribution functions

ABSTRACT

This article considers the estimation of a distribution function F_X(x) based on a random sample X₁, X₂, …, X_n when the sample is suspected to come from a close-by distribution F₀(x). The new estimators, namely the preliminary test (PTE) and Stein-type estimator (SE) are defined and compared with the “empirical distribution function” (edf) under local departure. In this case, we show that Stein-type estimators are superior to edf and PTE is superior to edf when it is close to F₀(x). As a by-product similar estimators are proposed for population quantiles.     

Approximation of density functions by orthogonal series with grouped data

Summary Letg(x) andf(x) be continuous density function on (a, b) and let {ϕ_j} be a complete orthonormal sequence of functions onL ₂(g), which is the set of squared integrable functions weighted byg on (a, b). Suppose that over (a, b). Given a grouped sample of sizen fromf(x), the paper investigates the asymptotic properties of the restricted maximum likelihood estimator of density, obtained by setting all but the firstm of the ϑ_j’s equal to0. Practical suggestions are given for performing estimation via the use of Fourier and Legendre polynomial series. Research partially supported by: CNR grant, n. 93. 00837. CT10.     

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.     

Sequential estimation in two-truncation parameter family of distributions under type II censoring

Summary This paper deals with the sequential estimation ofq(ϑ₁, ϑ₂) when the underlying density function is of the formf(x)=q(ϑ₁, ϑ₂)h(x), where ϑ₁ and ϑ₂ are unknown truncation parameters. We study the sequential properties of the stopping rule and the sequential estimator ofq(ϑ₁, ϑ₂). In this study we assume that the sample is type II censored.     

Kolmogorov-type tests of goodness-of-fit against stochastic ordering

This article addresses the problem of testing the null hypothesis H₀ that a random sample of size n is from a distribution with the completely specified continuous cumulative distribution function F_n(x). Kolmogorov-type tests for H₀ are based on the statistics C⁺ _n = Sup[F_n(x)?F₀(x)] and C^? _n=Sup[F₀(x)?F_n(x)], where F_n(x) is an empirical distribution function. Let F(x) be the true cumulative distribution function, and consider the ordered alternative H₁: F(x)≥F₀(x) for all x and with strict inequality for some x. Although it is natural to reject H₀ and accept H₁ if C ⁺ _n is large, this article shows that a test that is superior in some ways rejects F₀ and accepts H₁ if C^mdash _n is small. Properties of the two tests are compared based on theoretical results and simulated results.     

Conditional estimation of average on the basis of weighting data

LetF(x,y) be a distribution function of a two dimensional random variable (X,Y). We assume that a distribution functionF _x(x) of the random variableX is known. The variableX will be called an auxiliary variable. Our purpose is estimation of the expected valuem=E(Y) on the basis of two-dimensional simple sample denoted by:U=[(X ₁, Y₁)…(X_n, Y_n)]=[X Y]. LetX=[X ₁…X _n]andY=[Y ₁…Y _n].This sample is drawn from a distribution determined by the functionF(x,y). LetX _(k)be the k-th (k=1, …,n) order statistic determined on the basis of the sampleX. The sampleU is truncated by means of this order statistic into two sub-samples: % MathType!End!2!1! and % MathType!End!2!1!.Let % MathType!End!2!1! and % MathType!End!2!1! be the sample means from the sub-samplesU _k,1 andU _k,2, respectively. The linear combination % MathType!End!2!1! of these means is the conditional estimator of the expected valuem. The coefficients of this linear combination depend on the distribution function of auxiliary variable in the pointx _(k).We can show that this statistic is conditionally as well as unconditionally unbiased estimator of the averagem. The variance of this estimator is derived. The variance of the statistic % MathType!End!2!1! is compared with the variance of the order sample mean. The generalization of the conditional estimation of the mean is considered, too.     

The uniform convergence of the nadaraya-watson regression function estimate

If (X₁,Y₁), …, (X_n,Y_n) is a sequence of independent identically distributed R_d × R-valued random vectors then Nadaraya (1964) and Watson (1964) proposed to estimate the regression function m(x) = ? {Y₁|X₁ = x{ by where K is a known density and {h_n} is a sequence of positive numbers satisfying certain properties. In this paper a variety of conditions are given for the strong convergence to 0 of ess_Xsup|m_n (X)-m(X)| (here X is independent of the data and distributed as X₁). The theorems are valid for all distributions of X₁ and for all sequences {h_n} satisfying h_n → 0 and nh/log n→0.     

Randomly weighted sums and their maxima with heavy-tailed increments and dependence structure

Consider the randomly weighted sums S_m(θ) = ∑^m_{i = 1}θ_iX_i, 1 ? m ? n, and their maxima M_n(θ) = max?_{1 ? m ? n}S_m(θ), where X_i, 1 ? i ? n, are real-valued and dependent according to a wide type of dependence structure, and θ_i, 1 ? i ? n, are non negative and arbitrarily dependent, but independent of X_i, 1 ? i ? n. Under some mild conditions on the right tails of the weights θ_i, 1 ? i ? n, we establish some asymptotic equivalence formulas for the tail probabilities of S_n(θ) and M_n(θ) in the case where X_i, 1 ? i ? n, are dominatedly varying, long-tailed and subexponential distributions, respectively.     

Do bootstrap confidence procedures behave well uniformly in P?

An important statistical problem is to construct a confidence set for some functional T(P) of some unknown probability distribution P. Typically, this involves approximating the sampling distribution J_n(P) of some pivot based on a sample of size n from P. A bootstrap procedure is to estimate J_n(P) by J_n(&Pcirc;_n), where P?_n is the empirical measure based on a sample of size n from P. Typically, one has that J_n(P) and J_n(P?_n) are close in an appropriate sense. Two questions are addressed in this note. Are J_n(P) and J_n(P?_n) uniformly close as P varies as well? If so, do confidence statements about T(P) possess a corresponding uniformity property? In the case T(P) = P, the answer to the first questions is yes; the answer to the second is no. However, bootstrap confidence statements about T(P) can be made uniform over a restricted, though large, class of P. Similar results apply to other functional T(P).     

Super efficient frequency estimation

In this paper we propose a modified Newton-Raphson method to obtain super efficient estimators of the frequencies of a sinusoidal signal in presence of stationary noise. It is observed that if we start from an initial estimator with convergence rate O_p(n⁻¹) and use Newton-Raphson algorithm with proper step factor modification, then it produces super efficient frequency estimator in the sense that its asymptotic variance is lower than the asymptotic variance of the corresponding least squares estimator. The proposed frequency estimator is consistent and it has the same rate of convergence, namely O_p(n^−3/2), as the least squares estimator. Monte Carlo simulations are performed to observe the performance of the proposed estimator for different sample sizes and for different models. The results are quite satisfactory. One real data set has been analyzed for illustrative purpose.     

Expected Absolute Error of the Usual Estimator of the Binomial Parameter

For X with binomial (n, p) distribution the usual measure of the error of X/n as an estimator of p is its standard error S_n(p) = √{E(X/n – p)²} = √{p(1 – p)/n}. A somewhat more natural measure is the average absolute error D_n(p) = E‖X/n – p‖. This article considers use of D_n(p) instead of S_n(p) in a student's first introduction to statistical estimation. Exact and asymptotic values of D_n(p), and the appearance of its graph, are described in detail. The same is done for the Poisson distribution.     

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.