Estimating a restricted normal mean

Let X ₁, X ₂, ..., X _n be a random sample from a normal distribution with unknown mean μ and known variance σ ². In many practical situations, μ is known a priori to be restricted to a bounded interval, say [−m, m] for some m > 0. The sample mean , then, becomes an inadmissible estimator for μ. It is also not minimax with respect to the squared error loss function. Minimax and other estimators for this problem have been studied by Casella and Strawderman (Ann Stat 9:870–878, 1981), Bickel (Ann Stat 9:1301–1309, 1981) and Gatsonis et al. (Stat Prob Lett 6:21–30, 1987) etc. In this paper, we obtain some new estimators for μ. The case when the variance σ ² is unknown is also studied and various estimators for μ are proposed. Risk performance of all estimators is numerically compared for both the cases when σ ² may be known and unknown.     

On characterizing discrete distributions via conditional expectations of weak record values

Let (W _n ,n ≥ 0) denote the sequence of weak records from a distribution with support S = { α₀,α₁,...,α _N }. In this paper, we consider regression functions of the form ψ _n (x) = E(h(W _n ) |W _n+1 = x), where h(·) is some strictly increasing function. We show that a single function ψ _n (·) determines F uniquely up to F(α₀). Then we derive an inversion formula which enables us to obtain F from knowledge of ψ _n (·), ψ _n-1(·), h(·) and F(α₀).     

Blockwise bootstrap wavelet in nonparametric regression model with weakly dependent processes

In this paper, we suggest a blockwise bootstrap wavelet to estimate the regression function in the nonparametric regression models with weakly dependent processes for both designs of fixed and random. We obtain the asymptotic orders of the biases and variances of the estimators and establish the asymptotic normality for a modified version of the estimators. We also introduce a principle to select the length of data block. These results show that the blockwise bootstrap wavelet is valid for general weakly dependent processes such as α-mixing, φ-mixing and ρ-mixing random variables.     

Estimating the conditional mode of a stationary stochastic process from noisy observations

Let {(X _i, Y _i,)}, i≥1, be a strictly stationary process from noisy observations. We examine the effect of the noise in the response Y and the covariates X on the nonparametric estimation of the conditional mode function. To estimate this function we are using deconvoluting kernel estimators. The asymptotic behavior of these estimators depends on the smoothness of the noise distribution, which is classified as either ordinary smooth or super smooth. Uniform convergence with almost sure convergence rates is established for strongly mixing stochastic processes, when the noise distribution is ordinary smooth. Received: April 1998     

Iterative Density Estimation from Contaminated Observations

We introduce an iterative procedure for estimating the unknown density of a random variable X from n independent copies of Y=X+ɛ, where ɛ is normally distributed measurement error independent of X. Mean integrated squared error convergence rates are studied over function classes arising from Fourier conditions. Minimax rates are derived for these classes. It is found that the sequence of estimators defined by the iterative procedure attains the optimal rates. In addition, it is shown that the sequence of estimators converges exponentially fast to an estimator within the class of deconvoluting kernel density estimators. The iterative scheme shows how, in practice, density estimation from indirect observations may be performed by simply correcting an appropriate ordinary density estimator. This allows to assess the effect that the perturbation due to contamination by ɛ has on the density to be estimated. We also suggest a method to select the smoothing parameter required by the iterative approach and, utilizing this method, perform a simulation study.     

Inference by the <Emphasis Type="Italic">m</Emphasis> out of <Emphasis Type="Italic">n</Emphasis> bootstrap in nonparametric frontier models

It is well-known that the naive bootstrap yields inconsistent inference in the context of data envelopment analysis (DEA) or free disposal hull (FDH) estimators in nonparametric frontier models. For inference about efficiency of a single, fixed point, drawing bootstrap pseudo-samples of size m < n provides consistent inference, although coverages are quite sensitive to the choice of subsample size m. We provide a probabilistic framework in which these methods are shown to valid for statistics comprised of functions of DEA or FDH estimators. We examine a simple, data-based rule for selecting m suggested by Politis et al. (Stat Sin 11:1105–1124, 2001), and provide Monte Carlo evidence on the size and power of our tests. Our methods (i) allow for heterogeneity in the inefficiency process, and unlike previous methods, (ii) do not require multivariate kernel smoothing, and (iii) avoid the need for solutions of intermediate linear programs.     

Moment consistency of estimators in partially linear models under NA samples

Consider the heteroscedastic regression model Y ^(j)(x _in , t _in ) = t _in β + g(x _in ) + σ _in e ^(j)(x _in ), 1 ≤ j ≤ m, 1 ≤ i ≤ n, where s_in²=f(u_in){\sigma_{in}^{2}=f(u_{in})}, (x _in , t _in , u _in ) are fixed design points, β is an unknown parameter, g(·) and f(·) are unknown functions, and the errors {e ^(j)(x _in )} are mean zero NA random variables. The moment consistency for least-squares estimators and weighted least-squares estimators of β is studied. In addition, the moment consistency for estimators of g(·) and f(·) is investigated.     

Nonparametric inference on multivariate versions of Blomqvist’s beta and related measures of tail dependence

We consider nonparametric estimation of multivariate versions of Blomqvist’s beta, also known as the medial correlation coefficient. For a two-dimensional population, the sample version of Blomqvist’s beta describes the proportion of data which fall into the first or third quadrant of a two-way contingency table with cutting points being the sample medians. Asymptotic normality and strong consistency of the estimators are established by means of the empirical copula process, imposing weak conditions on the copula. Though the asymptotic variance takes a complicated form, we are able to derive explicit formulas for large families of copulas. For the copulas of elliptically contoured distributions we obtain a variance stabilizing transformation which is similar to Fisher’s z-transformation. This allows for an explicit construction of asymptotic confidence bands used for hypothesis testing and eases the analysis of asymptotic efficiency. The computational complexity of estimating Blomqvist’s beta corresponds to the sample size n, which is lower than the complexity of most competing dependence measures.      

Kernel estimation of a smooth distribution function based on censored data

The problem of estimating a smooth distribution functionF at a pointτ based on randomly right censored data is treated under certain smoothness conditions onF. The asymptotic performance of a certain class of kernel estimators is compared to that of the Kaplan-Meier estimator ofF(τ). It is shown that the relative deficiency of the Kaplan-Meier estimator ofF(τ) with respect to the appropriately chosen kernel type estimator tends to infinity as the sample sizen increases to infinity. Strong uniform consistency and the weak convergence of the normalized process are also proved. Research Surported in part by NIH grant 1R01GM28405.     

Berry–Esseen bounds for density estimates under NA assumption

Let {X _j } be a strictly stationary sequence of negatively associated random variables with the marginal probability density function f(x). The recursive kernel estimators of f(x) are defined by

Estimation of quantities determined as implicit functions of unknown parameters

Consider a family of distribution functions ${\{F(x, \theta),\,\theta \in \Theta\}}Consider a family of distribution functions {F(x, q), q ? Q}{\{F(x, \theta),\,\theta \in \Theta\}} . Suppose that there exists an estimator of the unknown parameter vector θ based on given data set. Then it is readily to obtain an estimator of any quantity given as an explicit function g(θ). Particularly, it is the case when the maximum likelihood estimator of θ is available. However, often some quantities of interest can not be expressed as an explicit function, rather it is determined as an implicit function of θ. The present article studies this problem. Sufficient conditions are given for deriving estimators of these quantities. The results are then applied to estimate change point of failure rate function, and change point of mean residual life function.     

Response of double-auction markets to instantaneous Selling�CBuying signals with stochastic Bid�CAsk spread

Statistical properties of order-driven double-auction markets with Bid–Ask spread are investigated through the dynamical quantities such as response function. We first attempt to utilize the so-called Madhavan–Richardson–Roomans model (MRR for short) to simulate the stochastic process of the price-change in empirical data sets (say, EUR/JPY or USD/JPY exchange rates) in which the Bid–Ask spread fluctuates in time. We find that the MRR theory apparently fails to simulate so much as the qualitative behaviour (‘non-monotonic’ behaviour) of the response function R(l) (l denotes the difference of times at which the response function is evaluated) calculated from the data. Especially, we confirm that the stochastic nature of the Bid–Ask spread causes apparent deviations from a linear relationship between the R(l) and the auto-correlation function C(l), namely, R(l) μ -C(l){R(l) \propto -C(l)}. To make the microscopic model of double-auction markets having stochastic Bid–Ask spread, we use the minority game with a finite market history length and find numerically that appropriate extension of the game shows quite similar behaviour of the response function to the empirical evidence. We also reveal that the minority game modeling with the adaptive (‘annealed’) look-up table reproduces the non-linear relationship R(l) μ -f(C(l)){R(l) \propto -f(C(l))} (f(x) stands for a non-linear function leading to ‘λ-shapes’) more effectively than the fixed (‘quenched’) look-up table does.     

One optional observation inflates α by $${100/\sqrt{n}}$$ per cent

For one-sample level α tests ψ _m based on independent observations X ₁, . . . , X _m , we prove an asymptotic formula for the actual level of the test rejecting if at least one of the tests ψ _n , . . . , ψ _n+k would reject. For k = 1 and usual tests at usual levels α, the result is approximately summarized by the title of this paper. Our method of proof, relying on some second order asymptotic statistics as developed by Pfanzagl and Wefelmeyer, might also be useful for proper sequential analysis. A simple and elementary alternative proof is given for k = 1 in the special case of the Gauss test.     

Estimation of the location parameter under LINEX loss function: multivariate case

The Baysian estimation of the mean vector θ of a p-variate normal distribution under linear exponential (LINEX) loss function is studied when as a special restricted model, it is suspected that for a p × r known matrix Z the hypothesis θ = Zβ, ${\beta\in\Re^r}The Baysian estimation of the mean vector θ of a p-variate normal distribution under linear exponential (LINEX) loss function is studied when as a special restricted model, it is suspected that for a p × r known matrix Z the hypothesis θ = Zβ, b ? ?^r{\beta\in\Re^r} may hold. In this area we show that the Bayes and empirical Bayes estimators dominate the unrestricted estimator (when nothing is known about the mean vector θ).     

Estimation and testing for a partially linear single-index spatial regression model

ABSTRACT

Observations recorded on ‘locations’ usually exhibit spatial dependence. In an effort to take into account both the spatial dependence and the possible underlying non-linear relationship, a partially linear single-index spatial regression model is proposed. This paper establishes the estimators of the unknowns. Moreover, it builds a generalized F-test to determine whether or not the data provide evidence on using linear settings in empirical studies. Their asymptotic properties are derived. Monte Carlo simulations indicate that the estimators and test statistic perform well. The analysis of Chinese house price data shows the existence of both spatial dependence and a non-linear relationship.     

Estimation of multiple poisson means: A compromise between maximum likelihood and empirical bayes estimators

For estimatingp(⩾ 2) independent Poisson means, the paper considers a compromise between maximum likelihood and empirical Bayes estimators. Such compromise estimators enjoy both good componentwise as well as ensemble properties. Research supported by the NSF Grant Number MCS-8218091.     

Bounds for the mean residual life function of a <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> system

In the reliability studies, k-out-of-n systems play an important role. In this paper, we consider sharp bounds for the mean residual life function of a k-out-of-n system consisting of n identical components with independent lifetimes having a common distribution function F, measured in location and scale units of the residual life random variable X _t  = (X−t|X > t). We characterize the probability distributions for which the bounds are attained. We also evaluate the so obtained bounds numerically for various choices of k and n.     

Social networks and labour productivity in Europe: an empirical investigation

This paper uses firm-level data recorded in the Amadeus database to investigate the distribution of labour productivity in different European countries. We find that the upper tail of the empirical productivity distributions follows a decaying power-law, whose exponent α is obtained by a semi-parametric estimation technique recently developed by Clementi et al. [Physica A 370(1):49–53, 2006]. The emergence of “fat tails” in productivity distribution has already been detected in Di Matteo et al. [Eur Phys J B 47(3):459–466, 2005] and explained by means of a model of social network. Here we show that this model is tested on a broader sample of countries having different patterns of social network structure. These different social attitudes, measured using a social capital indicator, reflect in the power-law exponent estimates, verifying in this way the existence of linkages among firms’ productivity performance and social network.     

Estimation of the parameters of log-gamma distribution using order statistics

In this work we propose a technique of estimating the location parameter μ and scale parameter σ of log-gamma distribution by U-statistics constructed by taking best linear functions of order statistics as kernels. The efficiency comparison of the proposed estimators with respect to maximum likelihood estimators is also made.     

Bayesian estimation and prediction based on Rayleigh sample quantiles

Ordered data arise naturally in many fields of statistical practice. Often some sample values are unknown or disregarded due to various reasons. On the basis of some sample quantiles from the Rayleigh distribution, the problems of estimating the Rayleigh parameter, hazard rate and reliability function, and predicting future observations are addressed using a Bayesian perspective. The construction of β-content and β-expectation Bayes tolerance limits is also tackled. Under squared-error loss, Bayes estimators and predictors are deduced analytically. Exact tolerance limits are derived by solving simple nonlinear equations. Highest posterior density estimators and credibility intervals, as well as Bayes estimators and predictors under linear loss, can easily be computed iteratively.