Bayesian approach to prediction in the presence of outliers for Weibull distribution

The predictive distribution of ther-th order statistics is obtained for the future sample based on the original sample from Weibull distribution in the presence ofk outliers. Next, in the presence ofk outliers two sample case is considered where prediction can be on ther ₂-th order statistics in the second sample based on ther ₁-th order statistics in the first sample. Finally, extension top-sample case is made for a particular case of predicting minimum in thep-th sample based on minimum in earlier samples. An illustration is provided with simulated samples where minimum is actually predicted in one and two sample cases.     

Some large-concentration-parameter asymptotics for the k-class estimators

A sufficient condition is derived in this paper for the consistency and asymptotic normality of the k-class estimators (k-stochastic or nonstochastic) as the concentration parameter increases indefinitely, with the sample size either staying fixed or also increasing. It is further shown that the limited-information maximum likelihood estimator satisfies this condition. Since large sample size implies a large concentration parameter, but not vice versa, the usual conditions for consistency and asymptotic normality of the k-class estimators as the sample size increases can be inferred from the results given in this paper. But more importantly, the results in this paper shed further light on the small-sample properties of the stochastic k-class estimators and can serve as a starting point for the derivation of asymptotic approximations for these estimators as the concentration parameter goes to infinity, while the sample size either stays fixed or also goes to infinity.     

On the distribution of a distance function on the sphere

Summary The distribution of the square of the distance between a random point and a fixed point on ap-dimensional unit sphere when (i) the two points lie on the whole sphere and (ii) the two points lie in the positive quadrant, has been derived, assuming that the random point is distributed proportionally to exp (ky ₁), wherek is a concentration parameter. Then-th order moment in both cases is also obtained.     

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.     

On maximum likelihood prediction based on Type II doubly censored exponential data

In this paper, the maximum likelihood predictor (MLP) of the kth ordered observation, t _k, in a sample of size n from a two-parameter exponential distribution as well as the predictive maximum likelihood estimators (PMLE's) of the location and scale parameters, θ and β, based on the observed values t _r, …, t _s (1≤r≤s<k≤n), are obtained in closed forms, contrary to the belief they cannot be so expressed. When θ is known, however, the PMLE of β and MLP of t _k do not admit explicit expressions. It is shown here that they exist and are unique; sharp lower and upper bounds are also provided. The derived predictors and estimators are reasonable and also have good asymptotic properties. As applications, the total duration time in a life test and the failure time of a k-out-of-n system may be predicted. Finally, an illustrative example is included. Received: August 1999     

Sequential estimation of a linear function of normal means under asymmetric loss function

The problem of estimating a linear function of k normal means with unknown variances is considered under an asymmetric loss function such that the associated risk is bounded from above by a known quantity. In the absence of a fixed sample size rule, sequential stopping rules satisfying a general set of assumptions are considered. Two estimators are proposed and second-order asymptotic expansions of their risk functions are derived. It is shown that the usual estimator, namely the linear function of the sample means, is asymptotically inadmissible, being dominated by a shrinkage-type estimator. An example illustrates the use of different multistage sampling schemes and provides asymptotic expansions of the risk functions. Received: August 1999     

On selecting the best ofk lognormal distributions

Summary Fork lognormal populations, which differ only in one certain parameter Ϙ, the problem of finding the population with the largest value ofϑ is considered. For two-parameter lognormal families, several natural choices ofϑ are treated, where the problem can be solved, through logarithmic transformation of the observations, within the framework of estimating parameters ink, possibly restricted, normal populations. For three-parameter lognormal families, this standard approach of selecting in terms of natural estimators fails to work ifϑ is the “guaranteed lifetime”. For this case, a selection procedure is derived which is based on anL-statistic which has the smallest asymptotic variance. Of importance here is that it is location equivariant, whereas it does not matter what it actually estimates. Comparisons are made with other suitable selection rules, through the asymptotic relative efficiencies, as well as in an example of intermediate sample sizes. It is shown that only in the latter, the selection rule, which is based on the sample minima, compares favorably. The research of this author was supported by the Office of Naval Research Contract N00014-88-K-0170 and NSF Grant Number DMS-8606964 at Purdue University. Reproduction in whole or in part is permitted for any purpose of the United States Government. The research of this author was supported by the Air Force Office of Scientific Research Grant 85-0347 at the University of Illinois at Chicago.     

Asymptotic expansions of the distributions of the structural variance estimators in a simultaneous equations system

Asymptotic expansions of three alternative classes of structural variance estimators associated with the k-class estimators of structural coefficients are derived for two parameter sequences: a sequence in which the non-centrality parameter increases while the sample size stays fixed (called large-μ or small-disturbance sequence), and that in which the number of observations increases. The accuracy of approximations to small-sample distributions are numerically examined with help of Monte Carlo studies. Properties of the sum of squared residuals of an estimated structural equation are also found from our study.     

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.     

Characterization of Normal Distribution Related to Two Samples Based on Regression

Characterization of normal distribution related to two samples based on second conditional moments has been obtained. This characterization has been transformed to a characterization based on the UMVU estimators of the density function. These results are generalized to k samples from normal distributions. Finally applications of these characterization results to goodness-of-fit test are discussed.     

On the existence of moments of partially restricted reduced form coefficients

In this paper ridgelike Bayesian estimators of structural coefficients have been used to form the partially restricted reduced form estimators. These partially restricted reduced form estimators are simple in form and possess finite sampling moments and risk in contrast to other restricted reduced form estimators that possess no finite moments and have infinite risk relative to quadratic loss functions. The usual k-class implied partially restricted reduced form estimators with 0≦k≦1 do not posses finite moments unless the degree of overidentification (or the excess of sample size over the number of coefficients) of the structural equation being estimated is suitably restricted.     

Estimation of Fisher information using model selection

In the paper the problem of estimation of Fisher information I _f for a univariate density supported on [0, 1] is discussed. A starting point is an observation that when the density belongs to an exponential family of a known dimension, an explicit formula for I _f there allows for its simple estimation. In a general case, for a given random sample, a dimension of an exponential family which approximates it best is sought and then estimator of I _f is constructed for the chosen family. As a measure of quality of fit a modified Bayes Information Criterion is used. The estimator, which is an instance of Post Model Selection Estimation method is proved to be consistent and asymptotically normal when the density belongs to the exponential family. Its consistency is also proved under misspecification when the number of exponential models under consideration increases in a suitable way. Moreover we provide evidence that in most of considered parametric cases the small sample performance of proposed estimator is superior to that of kernel estimators.     

Bootstrap refinements for QML estimators of the GARCH(1,1) parameters

This paper reconsiders a block bootstrap procedure for Quasi Maximum Likelihood estimation of GARCH models, based on the resampling of the likelihood function, as proposed by Gonçalves and White [2004. Maximum likelihood and the bootstrap for nonlinear dynamic models. Journal of Econometrics 119, 199–219]. First, we provide necessary conditions and sufficient conditions, in terms of moments of the innovation process, for the existence of the Edgeworth expansion of the GARCH(1,1) estimator, up to the k$k$-th term. Second, we provide sufficient conditions for higher order refinements for equally tailed and symmetric test statistics. In particular, the bootstrap estimator based on resampling the likelihood has the same higher order improvements in terms of error in the rejection probabilities as those in Andrews [2002. Higher-order improvements of a computationally attractive k$k$-step bootstrap for extremum estimators. Econometrica 70, 119–162].     

K-matrix-class estimators and the full-information maximum-likelihood estimator as a special case

This article presents a unified treatment of simultaneous system estimation. A general class of full-information estimators is proposed, called K-matrix-class (KMC). It is shown that the K-matrix-class includes both full-information maximum-likelihood and three-stage least- squares estimators as special cases and that the k-class can be regarded as a subclass of the K-matrix-class. Conditions under which KMC estimators are consistent (similar to those of the k-class estimators) are given. Furthermore, as a full information-generalization of the double k-class estimators, the double K-matrix-class estimators (DKMC) are proposed.     

On the efficiency of least squares estimators in non-linear models

Summary The identity of least squares estimators å and maximum likelihood estimators â is studied in non-linear models of the type z=g(a), where z are observable quantities with a probability density function pr(z). This identity was proved for independent random variables z and for distributions pr(z), of which the arithmetic sample mean is an optimal estimate.     

Combining information for prediction in linear regression

This article deals with the prediction problem in linear regression where the measurements are obtained using k different devices or collected from k different independent sources. For the case of k=2, a Graybill-Deal type combined estimtor for the regression parameters is shown to dominate the individual least squares estimators under the covariance criterion. Two predictors ŷ _c and ŷ _p are proposed. ŷ _c is based on a combined estimator of the regression coefficient vector, and ŷ _p is obtained by combining the individual predictors from different models. Prediction mean square errors of both predictors are derived. It is shown that the predictor ŷ _p is better than the individual predictors for k≥2 and the predictor ŷ _c is better than the individual predictors for k=2. Numerical comparison between ŷ _c and ŷ _p shows that the former is superior to the latter for the case k=2.     

The existence of moments of some simple Bayes estimators of coefficients in a simultaneous equation model

In this paper a simple modification of the usual k-class estimators has been suggested so that for 0 ≦ k ≦ 1 the problem of the non-existence of moments disappears. These modified estimators can be interpreted either as Bayes estimators or as constrained estimators subject to the restriction that the squared length of the coefficient vector is less than or equal to a given number.     

Het rekenkundig gemiddelde stijgingspercentage – een ondoelmatige groei-indicator

Summary In business practice it happens that annual growth is measured by the ordinary (arithmetic) average of yearly growth percentages. It is argued that there are better methods in any case. A method requiring relatively much time is least squares. Methods requiring very little time are a) determining growth graphically, using semi-logarithmic graph paper b) computing the geometrical average, which is nothing else than the n-th root (if there are n+l observations) of the ratio between the last and first observation. In a table, a survey of bias and dispersion of three estimators is given for a few numerical cases.     

k‐Nearest neighbors local linear regression for functional and missing data at random

We combine the k‐Nearest Neighbors (kNN) method to the local linear estimation (LLE) approach to construct a new estimator (LLE‐kNN) of the regression operator when the regressor is of functional type and the response variable is a scalar but observed with some missing at random (MAR) observations. The resulting estimator inherits many of the advantages of both approaches (kNN and LLE methods). This is confirmed by the established asymptotic results, in terms of the pointwise and uniform almost complete consistencies, and the precise convergence rates. In addition, a numerical study (i) on simulated data, then (ii) on a real dataset concerning the sugar quality using fluorescence data, were conducted. This practical study clearly shows the feasibility and the superiority of the LLE‐kNN estimator compared to competitive estimators.     

Multivariate Hill Estimators

We propose two classes of semi‐parametric estimators for the tail index of a regular varying elliptical random vector. The first one is based on the distance between a tail probability contour and the observations outside this contour. We denote it as the class of separating estimators. The second one is based on the norm of an arbitrary order. We denote it as the class of angular estimators. We show the asymptotic properties and the finite sample performances of both classes. We also illustrate the separating estimators with an empirical application to 21 worldwide financial market indexes.