Asymptotic properties for estimates of nonparametric regression model with martingale difference errors

Consider the nonparametric regression model with martingale difference errors. Nonparametric estimator g _n (x) of regression function g(x) will be introduced, and its asymptotic properties are studied. In particular, the pointwise and uniform convergence of g _n (x) and its asymptotic normality will be investigated. This extends the earlier work on independent random errors.     

On the error term in bahadur's representation of an order statistic

Bahadur (1966) presented a representation of an order statistic, giving its asymptotic distribution and the rate of convergence, under weak assumptions on the density function of the parent distribution. In this paper we consider the mean(squared) deviation of the error term in Bahadur’s approximation of the q th sample quantile (q_n ). We derive a uniform bound on the mean (squared) deviation of q_n , not depending on the value of q. An application of the given result provides the corresponding result for a kernel type estimator of the q th quantile.     

Local asymptotic normality of intermediate and central order statistics

Consider n independent random variables Z_i,…, Z_n on R with common distribution function F, whose upper tail belongs to a parametric family F(t) = F_θ(t),t ≥ x₀, where θ ∈ ? ? R ^d. A necessary and sufficient condition for the family F_θ, θ ∈ ?, is established such that the k-th largest order statistic Z_n?k+1:n alone constitutes the central sequence yielding local asymptotic normality ( LAN ) of the loglikelihood ratio of the vector (Z_n?i+1:n)¹ _i=kof the k largest order statistics. This is achieved for k = k(n)→_n→∞∞ with k/n→_n→∞ 0.

In the case of vectors of central order statistics ( Z_r:n, Z_r+1:n,…, Z_s:n ), with r/n and s/n both converging to q ∈ ( 0,1 ), it turns out that under fairly general conditions any order statistic Z_m:n with r ≤ m ≤s builds the central sequence in a pertaining LAN expansion.These results lead to asymptotically optimal tests and estimators of the underlying parameter, which depend on single order statistics only     

Large deviation local limit theorems for ratio statistics

Let {T_n, n ≥ 1} be an arbitrary sequence of nonlattice random variables and let {S_n, n ≥ 1} be another sequence of positive random variables. Assume that the sequences are independent. In this paper we obtain asymptotic expression for the density function of the ratio statistic R_n = T_n/S_n based on simple conditions on the moment generating functions of T_n and S_n. When S_n = re, our main result reduces to that of Chaganty and Sethura-man[Ann. Probab. 13(1985):97-114]. We also obtain analogous results when T_n and S_n are both lattice random variables. We call our theorems large deviation local limit theorems for R_n, since the conditions of our theorems imply that R_n → c in probability for some constant c. We present some examples to illustrate our theorems.     

Uniformly strong consistency and Berry-Esseen bound of frequency polygons for α-mixing samples

In this article, the frequency polygon investigated by Scott is studied as a nonparametric estimator for α-mixing samples. By some known exponent and moment inequalities, we obtain the uniformly strong consistency and Berry-Esseen bound of the estimator. The present results relax the relevant conditions used by Carbon et al. Furthermore, the convergence rate of the uniformly asymptotic normality is derived, which is O(n^{? 1/11}) under the given conditions.     

Asymptotic Properties of Generalized Multivariate Rank Statistics

This article discusses generalization of the well-known multivariate rank statistics under right-censored data case. Empirical process representation used to get the generalization. The marginal distribution functions are estimated by Kaplan–Meier estimators. Sufficient conditions for asymptotic normality of the generalized multivariate rank statistics under independently right censored data are specified. Several auxiliary results on sup-norm convergence of Kaplan–Meier estimators in randomly exhausting regions are given too.     

On the LBI criterion for the multivariate one-way random effects model under non-normality

The asymptotic null distribution of the locally best invariant (LBI) test criterion for testing the random effect in the one-way multivariable analysis of variance model is derived under normality and non-normality. The error of the approximation is characterized as O(1/n). The non-null asymptotic distribution is also discussed. In addition to providing a way of obtaining percentage points and p-values, the results of this paper are useful in assessing the robustness of the LBI criterion. Numerical results are presented to illustrate the accuracy of the approximation.     

Jackknifing L-estimates

Linear functions of order statistics (“L-estimates”) of the form T_n =under jackknifing are investigated. This paper proves that with suitable conditions on the function J, the jackknifed version T_n of the L-estimate T_n has the same limit distribution as T_n. It is also shown that the jackknife estimate of the asymptotic variance of n^1/2 is consistent. Furthermore, the Berry-Esséen rate associated with asymptotic normality, and a law of the iterated logarithm of a class of jackknife L-estimates, are characterized.     

Asymptotic properties of the GMLE in the case 1 interval-censorship model with discrete inspection times

We consider the case 1 interval censorship model in which the survival time has an arbitrary distribution function F₀ and the inspection time has a discrete distribution function G. In such a model one is only able to observe the inspection time and whether the value of the survival time lies before or after the inspection time. We prove the strong consistency of the generalized maximum-likelihood estimate (GMLE) of the distribution function F₀ at the support points of G and its asymptotic normality and efficiency at what we call regular points. We also present a consistent estimate of the asymptotic variance at these points. The first result implies uniform strong consistency on [0, ∞) if F₀ is continuous and the support of G is dense in [0, ∞). For arbitrary F₀ and G, Peto (1973) and Tumbull (1976) conjectured that the convergence for the GMLE is at the usual parametric rate n½ Our asymptotic normality result supports their conjecture under our assumptions. But their conjecture was disproved by Groeneboom and Wellner (1992), who obtained the nonparametric rate ni under smoothness assumptions on the F₀ and G.     

Limits of experiments associated with sampling plans

Starting from Milbrodt (1985), the asymptotic behaviour of experiments associated with Poisson sampling, Rejective sampling and its Sampford-Durbin modification is investigated. As superpopulation models so-called L^r-generated regression parameter families (1⩽r⩽2) are considered, allowing also the presence of nuisance parameters. Under some assumptions on the first order probabilities of inclusion it can be shown that the sampling experiments converge weakly if the underlying shift parameter families do so. In case of convergence the limit of the sampling experiments is characterized in terms of its Hellinger transforms and its Lévy-Khintchine representation, leading to criteria for the limit to be a pure Gaussian or a pure Poisson experiment respectively. These results are then applied to the situation of sampling in the presence of random non-response, and to establish local asymptotic normality (LAN) under more restrictive conditions. Applications also include asymptotic optimality properties of tests based on Horvitz-Thompson-type statistics, and LAM bounds and criteria for adaptivity, when testing or estimating a continuous linear functional in LAN situations. They especially cover the case of sampling from an unknown symmetric distribution, which has been subject to detailed investigations in the i.i.d. case.     

Corrected p-values for tests based on estimated nuisance parameters

In some situations the asymptotic distribution of a random function T _n() that depends on a nuisance parameter is tractable when has known value. In that case it can be used as a test statistic, if suitably constructed, for some hypothesis. However, in practice, often needs to be replaced by an estimator S _n. In this paper general results are given concerning the asymptotic distribution of T _n(S _n) that include special cases previously dealt with. In particular, some situations are covered where the usual likelihood theory is nonregular and extreme values are employed to construct estimators and test statistics.     

Set estimation and nonparametric detection

The authors consider the estimation of a set S ? R^d from a random sample of n points. They examine the properties of a detection method, proposed by Devroye & Wise (1980), which relies on the use of a “naive” estimator of S defined as a union of balls centered at the sample points with common radius ?_n. They obtain the convergence rate for the probability of false alarm and show that the smoothing parameter ?_n can be used to incorporate some prior information on the shape of S. They suggest two general methods for selecting ?_n and illustrate them with a simulation study and a real data example.     

Asymptotic normality of the multiscale wavelet density estimator

A multiscale wavelet density estimator (MWDE) was recently introduced and was shown to have nice convergence properties as well as good simulation results (Wu, 1995). This paper studies the asymptotic normality of the MWDE. It is proved that, under mild conditions, the MWDE has the asymptotic normality in the support of the unknown density f.As by-products, the author establishes the asymptotic normality of the wavelet estimator and discovers several interesting statistical properties of the reproducing kernel q_m(x,t)ofV_m .     

The asymptotic distribution of the correlation coefficient in testing fit to the exponential distribution

The asymptotic distribution of certain tests of fit to the exponential distribution is obtained. The tests are based on regression of the order statistics on their expectations under a standard exponential distribution. Asymptotic normality at the rate (log n)^1/2 is obtained for a family of statistics including the correlation coefficient.     

SOME ASYMPTOTIC PROPERTIES OF THE SIMPLE LOGLINEAR MODEL

Consistency and asymptotic normality of the maximum likelihood estimator of β in the loglinear model E(y_i) = e^α+βXi, where y_i are independent Poisson observations, 1 iaan, are proved under conditions which are near necessary and sufficient. The asymptotic distribution of the deviance test for β=β₀ is shown to be chi-squared with 1 degree of freedom under the same conditions, and a second order correction to the deviance is derived. The exponential model for censored survival data is also treated by the same methods.     

Estimation and Simulation for the M-Wright Function

In this article, a structural form of an M-Wright distributed random variable is derived. The mixture representation then led to a random number generation algorithm. A formal parameter estimation procedure is also proposed. This procedure is needed to make the M-Wright function usable in practice. The asymptotic normality of the estimator is established as well. The estimator and the random number generation algorithm are then tested using synthetic data.     

Some Properties of the Pivotal Statistic Based on the Asymptotically Distribution-Free Theory in Structural Equation Modeling

For normally distributed data, the asymptotic bias and skewness of the pivotal statistic Studentized by the asymptotically distribution-free standard error are shown to be the same as those given by the normal theory in structural equation modeling. This gives the same asymptotic null distributions of the two pivotal statistics up to the next order beyond the usual normal approximation under normality. With an alternative hypothesis, the asymptotic variances of the two statistics under normality/non normality are also derived. It is, however, shown that the asymptotic variances of the non null distributions of the statistics are generally different even under normality.     

Asymptotic expaxsioxs for the joint distribution of cirrelated hotellings t2 statlstics under normality

Let T² _i=z′_iS^?1z_i, i==,…k be correlated Hotelling's T² statistics under normality. where z=(z′_i,…,z′_k)′ and nS are independently distributed as N_kp((O,ρ?∑) and Wishart distribution W_p(∑, n), respectively. The purpose of this paper is to study the distribution function F(x₁,…,x_k) of (T² _i,…,T² _k) when n is large. First we derive an asymptotic expansion of the characteristic function of (T² _i,…,T² _k) up to the order n^?2. Next we give asymptotic expansions for (T² _i,…,T² _k) in two cases (i)ρ=I_k and (ii) k=2 by inverting the expanded characteristic function up to the orders n^?2 and n^?1, respectively. Our results can be applied to the distribution function of max (T² _i,…,T² _k) as a special case.     

Two-stage generalized moment method approach for bidimensional random coefficient autoregressive models

ABSTRACT

A two-dimensionally indexed random coefficients autoregressive models (2D ? RCAR) and the corresponding statistical inference are important tools for the analysis of spatial lattice data. The study of such models is motivated by their second-order properties that are similar to those of 2D ? (G)ARCH which play an important role in spatial econometrics. In this article, we study the asymptotic properties of two-stage generalized moment method (2S ? GMM) under general asymptotic framework for 2D ? RCA models. So, the efficiency, strong consistency, the asymptotic normality, and hypothesis tests of 2S ? GMM estimation are derived. A simulation experiment is presented to highlight the theoretical results.     

Multivariate τ-estimators for location and scatter

We discuss the robustness and asymptotic behaviour of τ-estimators for multivariate location and scatter. We show that τ-estimators correspond to multivariate M-estimators defined by a weighted average of redescending ψ-functions, where the weights are adaptive. We prove consistency and asymptotic normality under weak assumptions on the underlying distribution, show that τ-estimators have a high breakdown point, and obtain the influence function at general distributions. In the special case of a location-scatter family, τ-estimators are asymptotically equivalent to multivariate S-estimators defined by means of a weighted ψ-function. This enables us to combine a high breakdown point and bounded influence with good asymptotic efficiency for the location and covariance estimator.