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.     

Exact finite sample properties of double k-class estimators in simultaneous equations

In our earlier paper [Srivastava, Agnihotri and Dwivedi (1980)] the dominance of double k-class over k-class with respect to exact mean squared error matrix criteria is established. It is observed that given a member of k-class, one can pick up a member of double k-class that will provide an improved estimator of the coefficients. This result prompted us to study the exact finite sample properties of the double k-class estimator. For this, we have considered a structural equation containing two endogenous variables and have investigated the properties of double k-class estimators of the coefficients of explanatory endogenous variables assuming characterizing scalars to be non-stochastic.     

Improved stein-rule estimator for regression problems

Stein-Rule estimator for regression problems has been studied by several authors including Sclove (1968) and others listed in Vinod's (1978) survey. Ullah and Ullah (1978) provide the expressions for the mean squared error (MSE) of a double k-class (KK) estimator with parameters k₁ and k₂. When k₂=1 the Stein-Rule estimator becomes a special case of KK and an optimal choice of k₁ is known. This paper explores optimal theoretical choice of k₁ and k₂. We note that negative choices of k₂ are permissible, and that thereis a large range of choices for K₁ and k₂ where the MSE of the Stein-Rule estimator can be reduced for regression problems based on multicollinear data. A simulation experiment is included.     

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.     

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     

On the product of gamma random variables

We give expressions for the distribution and density of a product of gamma or equivalently chi-square random variables. In particular, we give the distribution of the product of two independent gamma variables of mean k in terms of the Bessel functions K ₁, … , K _k .     

Improved Stein-rule estimator for regression problems

This note extends the asymptotic expansion of the risk of the double k-class estimator of Ullah and Ullah (1978, 1981) and discusses the k₁ and k₂ values which minimize it. An error in Vinod (1980) is also corrected.     

Relations for product moments and covariances of <Emphasis Type="Italic">k</Emphasis>th records from discrete distributions

The aim of this paper is to establish recurrence relations satisfied by product moments and covariances of kth records arising from discrete distributions. They will be evaluated for geometric underlying distribution. Then we use these results to obtain formulas for correlation coefficients of geometric kth records. We consider all three known types of kth records: strong, ordinary, and weak.     

Exact moments of the sample autocorrelations from series generated by general arima processes of order (p, d, q), d=0 or 1

Formulae for the numerical computation of the first four exact moments of the sample autocorrelations, given a time series realisation from a general autoregressive moving average process of order (p, d, q) with d=0 or 1, are presented. The exact mean and variance of the sample autocorrelations are computed for various sample sizes and several time series models. The evaluated results are compared with those obtained from approximate formulae for the mean and variance of the sample autocorrelations. A specification of the numerical accuracy of the first two exact moments is included.     

The graph of the k-class estimator: An algebraic and statistical interpretation

This paper presents an algebraic analysis of the graphs of the k-class estimator, its asymptotic standard error and asymptotic t-ratio as functions of k for a single structural equation containing one or more endogenous explanatory variables. These results are illustrated by the corresponding graphs of the second and fifth equations of the Girshick-Haavelmo (1947) Demand for Food Model.Tests of the rank condition for identification are also developed. They are found to involve the values of k which explode the k-class estimator.     

Using an approximate kiefer-weiss solution for testing insect population densities

Summary A sequential testing procedure called 2-SPRT for the meanμ of the negative binomial distribution with known exponentk is presented. For any fixedμ ₀(μ ₁<μ ₀<μ ₂), the 2-SPRT provides an asymptotic solution to the modified Kiefer-Weiss problem. Determination ofμ ₀ such that the test provides an approximate solution to the Kiefer-Weiss problem is described. The behavior of the 2-SPRT and the Sequential Probability Ratio Test is investigated using Monte Carlo methods. The error probabilities and the average sample numbers are compared. All the computations were carried out on the Alabama Supercomputer.     

How to poison Poisson (when approximating binomial tails)*

Summary In these days some sportsmen are given dangerous drugs in order to stimulate them into still higher efforts. The present paper wants to add some poison to the classical approximation of a binomial by a Poisson distribution. The relatively harmless drug of a little extra calculation shall be seen to result in a much better accuracy. This paper reports both on theoretical considerations, following from series expansions of Poisson-type parameters, and on extensive numerical investigations of accuracy by means of an Electrologica X8 computer. The main conclusion is that the probability of at most k successes in n Bernoulli trials with success probability p <.5 can be very closely approximated by the probability of at most k events in a Poisson distribution with expectation not np but (12–2p)n-7kζ= (12–8p)n-k+k/n^np. In the case p > 5 one should make p < .5 by a reversal, i.e. the interchanging of successes and failures, before the application of a Poisson-type approximation (with one exception mentioned in section 4). For the probability of at least j successes one should approximate the complementary probability of at most j—1 successes. After an introductory section 1 and a discussion of measures of accuracy in section 2, the existing Poisson-type approximations are treated in section 3. The fourth section introduces a new parameter and discusses the advantages of reversal in other situations than p > 5. In section 5 a series expansion for the exact Poisson parameter is derived. Two new parameters, among which is ζ, are introduced in section 6, where the series expansions for parameters given in table 4 lead to conclusions about the various approximations, summarized in table 6. The final section 7 gives some numerical results, which confirm to a large extent the conclusions from the asymptotic expansions.     

A reexamination of the equity-premium puzzle: A robust non-parametric approach

Recent tests of stochastic dominance of several orders, proposed by Linton, Maasoumi and Whang [Linton, O., Maasoumi, E., & Whang, Y. (2005). Consistent testing for stochastic dominance under general sampling schemes. Review of Economic Studies, 72(3), 735–765], are applied to reexamine the equity-premium puzzle. An advantage of this non-parametric approach is that it provides a framework to assess whether the existence of a premium is due to particular cardinal choices of either the utility function or the underlying returns distribution, or both. The approach is applied to the original Mehra–Prescott data and more recent data that include daily yields on Treasury bonds and daily returns on the S&P500 and the NASDAQ indexes. The empirical results show little evidence of stochastic dominance among the assets investigated. This suggests that the observed equity premium represents compensation for bearing higher risk, taking into account higher-order moments such as skewness and kurtosis. There is some evidence of a reverse puzzle, whereby Treasury bonds stochastically dominate equities at the third order, a result which potentially reflects insufficient compensation to investors for bearing the negative skewness associated with the S&P500 index.     

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.     

Estimation in the first-order moving average model through the finite autoregressive approximation: Some asymptotic results

To estimate α in the model y_t = u_t+αu_t?1, we consider a proposal by Durbin (Biometrika, 1969). It consists in fitting an autoregression of order k to the data, and deriving from there an estimate α^. The probability limit and the variance of the limiting normal distribution of α^ are presented and discussed in detail, when the sample size T → ∞, but k remains fixed. The differences between the resulting values and those corresponding to the maximum likelihood estimator are exponentially decreasing functions of k. Several modifications of the estimator are discussed and found consistent, but asymptotically inefficient.     

Efficient discriminating design for a class of nested polynomial regression models

This paper studies efficient designs for simultaneous model discrimination among polynomial regression models up to degree k. Based on the -optimality criterion proposed by Dette (Ann Stat 22:890–903, 1994), a maximin -optimal discriminating design is derived in terms of canonical moments for . Theoretical and numerical results show that the proposed design performs well for model discrimination in most of the considered models.     

Information and computation in simultaneous equations estimation

The complexity and size of simultaneous equations systems necessitates great care with computations for parameter estimation. In three-stage least-squares (3SLS) large matrix inversions are required, and because of the sensitivity of many economic systems to key parameters, accuracy in estimation is important. There are many numerical techniques available which yield accurate solutions to systems of equations. We make use of Householder transformations and recursive triangulation solutions in presenting numerical algorithms for the computation of 3SLS and k-class estimates. Another numerical technique, the singular value decomposition is valuable in providing additional information in k-class estimation. The values of k for which this estimator does not exist are accurately derived, their use being demonstrated by an example.     

Index

The Yule distribution is shown to have certain interesting properties in the area of regression analysis. In particular, it is shown that under certain conditions, a random variable Z will have linear regressions on another random variable X and on its observable part Y only when X has a Yule distribution. More generally, the regression on the observed part Y will be constant for a finite number of values of Y, say k, and linear otherwise, only when X has a Yule distribution with its first k frequencies truncated.     

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].