On the estimation of location parameters in the multivariate one sample and two sample problems

Summary In this paper we consider the problem of estimating the vectors of location parameters in the multivariate one sample and two sample problems. These estimators are obtained through the use of the multivariate rank order statistics such as theWilcoxon or the normal scores statistic considered by the authors inPuri, Sen [1966] andSen, Puri [1967] for the corresponding testing problems. The distribution of these estimators is shown to be symmetric with respect to the parameters being estimated. These estimators are translation invariant, robust and asymptotically normal. Their asymptotic relative efficiencies with respect to the estimators based on the vector of means and medians are discussed by applying the criterion ofWilks generalized variance [Anderson, p. 166]. In particular, it is shown that the estimators based on the multivariate normal scores statistics are asymptotically as efficient as the ones based on the method of least squares when the parent distributions are normal. Research sponsored by National Science Foundation Grant No. GP-12462, and by Research Grant, GM-12868 from the N.I.H., Public Health Service.     

Statistical inference in simplicially contoured sample distributions

For generalizations of the n-dimensional two parameter exponential distribution with identical marginals with threshold and dispersion parameters the exact distributions of estimators and test statistics are given. Under certain conditions the consistency of the estimators and the rate of convergence is shown. Therefore generalized Gamma- and F-distributions are defined.     

On parameter estimation of partly observed bilinear discrete-time stochastic systems

The parameter estimation problem of a partly observed nonlinear discrete-time stochastic system is considered. The unobserved component of the system is a q-dimensional stable autoregressive process of the pth order with random parameters, observed in the presence of multiplicative and additive noises. The distributions of all the noises of the system are supposed to be unknown. The problem is to estimate the mean of the drifting parameters of the object and variances of the additive noises of the system. Asymptotic correlation estimators of all these parameters are investigated and sequential estimators with given mean square accuracy of the mean of the drifting autoregressive parameters are obtained.     

Estimation of spatial autoregressive panel data models with fixed effects

This paper establishes asymptotic properties of quasi-maximum likelihood estimators for SAR panel data models with fixed effects and SAR disturbances. A direct approach is to estimate all the parameters including the fixed effects. Because of the incidental parameter problem, some parameter estimators may be inconsistent or their distributions are not properly centered. We propose an alternative estimation method based on transformation which yields consistent estimators with properly centered distributions. For the model with individual effects only, the direct approach does not yield a consistent estimator of the variance parameter unless T is large, but the estimators for other common parameters are the same as those of the transformation approach. We also consider the estimation of the model with both individual and time effects.     

The UMVUE ofP(X<Y) based on type-II censored samples from Weinman multivariate exponential distributions

Based on two independent samples from Weinman multivariate exponential distributions with unknown scale parameters, uniformly minimum variance unbiased estimators ofP(X<Y) are obtained for both, unknown and known common location parameter. The samples are permitted to be Type-II censored with possibly different numbers of observations. Since sampling from two-parameter exponential distributions is contained in the model as a particular case, known results for complete and censored samples are generalized. In the case of an unknown common location parameter with a certain restriction of the model, the UMVUE is shown to have a Gauss hypergeometric distribution, which is further examined. Moreover, explicit expressions for the variances of the estimators are derived and used to calculate the relative efficiency.     

Sequential estimation of shape parameters in multivariate dynamic models

Sequential maximum likelihood and GMM estimators of distributional parameters obtained from the standardised innovations of multivariate conditionally heteroskedastic dynamic regression models evaluated at Gaussian PML estimators preserve the consistency of mean and variance parameters while allowing for realistic distributions. We assess their efficiency, and obtain moment conditions leading to sequential estimators as efficient as their joint ML counterparts. We also obtain standard errors for VaR and CoVaR, and analyse the effects on these measures of distributional misspecification. Finally, we illustrate the small sample performance of these procedures through simulations and apply them to analyse the risk of large eurozone banks.     

Percentile estimators in location-scale parameter families under absolute loss

Estimators of percentiles of location-scale parameter families are optimized based on median unbiasedness and absolute risk. Median unbiased estimators and minimum absolute risk estimators are shown to exist within a class of equivariant estimators and depend upon medians of two completely specified distributions. This work extends earlier findings to a larger class of equivariant estimators. These estimators are illustrated in the normal and exponential distributions.     

Estimation of order-restricted means of two normal populations under the LINEX loss function

In this paper, the estimation of order-restricted means of two normal distributions is studied under the LINEX loss function, when the variances are unknown and possibly unequal. Under certain sufficient conditions to be described in this paper, the proposed plug-in estimators uniformly perform better than the existing unrestricted maximum likelihood estimators. Further, the restricted maximum likelihood estimators are compared with the unrestricted maximum likelihood estimators under the Pitman nearness criterion. A simulation study is conducted and it is shown that our proposed plug-in estimators perform better than the unrestricted maximum likelihood estimators. An illustrative example of real data analysis is also given to compare the estimators.     

Estimation in generalized bivariate Birnbaum–Saunders models

In this paper, we propose two moment-type estimation methods for the parameters of the generalized bivariate Birnbaum–Saunders distribution by taking advantage of some properties of the distribution. The proposed moment-type estimators are easy to compute and always exist uniquely. We derive the asymptotic distributions of these estimators and carry out a simulation study to evaluate the performance of all these estimators. The probability coverages of confidence intervals are also discussed. Finally, two examples are used to illustrate the proposed methods.     

Predictors for the first-order autoregressive process

The error made in predicting a first-order autoregressive process with unknown parameters is investigated. It is shown that the least squares predictor is unbiased for symmetric error distributions. Alternative predictors for stationary and non-stationary processes are studied using the Monte Carlo method. The ordinary least squares statistics perform reasonably well for one period predictions with samples as small as ten for both stationary and non-stationary processes. It is demonstrated that there is a considerable loss in efficiency when outdated estimators are used to construct predictors.     

Instrumental variables estimation of a nearly nonstationary,heterogeneous error component model

This paper studies instrumental variables (IV) estimation for an error component model with stationary and nearly nonstationary regressors. It is assumed that the numbers of cross section and time series observations are infinite. Furthermore, autoregressive disturbances are assumed for the error component model, the structure of which may vary with individuals. The estimators considered are the Within-IV-OLS, Within-IV-GLS and IV-GLS estimators. The GLS estimators use Gohberg's formula, which is particularly useful when autoregressive structures are imposed on the disturbance terms. Sequential limit theories for the estimators are derived, and it is shown that all of the estimators have normal distributions in the limit. Additionally, Wald tests for coefficient vectors are shown to have chi-square distributions in the limit. Simulation results regarding the estimator efficiency and the size of the Wald tests are also reported. The results show that the Within-IV-GLS and IV-GLS estimators are more efficient than the Within-IV-OLS estimator in most cases and that the Wald tests keep nominal size reasonably well. The relation between the trade and budget deficits of 23 OECD nations is examined using the panel IV estimators. The empirical results support the view that the budget and trade deficits move in the same direction.     

Smooth estimators for estimating order restricted scale parameters of two gamma distributions

We consider the problem of component-wise estimation of ordered scale parameters of two gamma populations, when it is known apriori which population corresponds to each ordered parameter. Under the scale equivariant squared error loss function, smooth estimators that improve upon the best scale equivariant estimators are derived. These smooth estimators are shown to be generalized Bayes with respect to a non-informative prior. Finally, using Monte Carlo simulations, these improved smooth estimators are compared with the best scale equivariant estimators, their non-smooth improvements obtained in Vijayasree, Misra & Singh (1995), and the restricted maximum likelihood estimators. Acknowledgments. Authors are thankful to a referee for suggestions leading to improved presentation.     

Concomitants of Record Values Arising from Morgenstern Type Bivariate Logistic Distribution and Some of their Applications in Parameter Estimation

In this paper we have obtained the joint probability density function of concomitants of two record values and hence obtained an explicit expression for the product moment of concomitants of two record values arising from Morgenstern family of distributions. Appling this expression for the product moments of concomitants of record values we have derived the best linear unbiased estimators based on concomitants of record values of some parameters involved in Morgenstern type bivariate logistic distribution which is a subfamily of the Morgenstern family of distributions. The efficiencies of these estimators based on the first n concomitants of record values for n≤10 are also obtained.     

Simultaneous estimation of hazard rates of several exponential populations

Suppose independent random samples are drawn from k (2) populations with a common location parameter and unequal scale parameters. We consider the problem of estimating simultaneously the hazard rates of these populations. The analogues of the maximum likelihood (ML), uniformly minimum variance unbiased (UMVU) and the best scale equivariant (BSE) estimators for the one population case are improved using Rao‐Blackwellization. The improved version of the BSE estimator is shown to be the best among these estimators. Finally, a class of estimators that dominates this improved estimator is obtained using the differential inequality approach.     

Best unbiased estimators for the parameters of a two-parameter Pareto distribution

Summary For a two-parameter Pareto distributionMalik [1970] has shown that the maximum likelihood estimators of the parameters are jointly sufficient. In this article the maximum likelihood estimators are shown to be jointly complete. Furthermore, unbiased estimators for the two parameters are obtained and are shown to be functions of the jointly complete sufficient statistics, thereby establishing them as the best unblased estimators of the two parameters.This research is a part of the first author's Ph.D. dissertation. The authors wish to thank Dr. Kenny S. Crump, for many helpful suggestions and a referee for improvements in the proofs.     

On universal admissibility of scale parameter estimators

A characterization result of Kushary (1998) regarding universal admissibility of equivariant estimators in the one parameter gamma distribution is generalized to a scale family of distributions with monotone likelihood ratio. New examples are given, among them the F-distribution with a scale parameter. In particular, universal admissibility is characterized within the class of location-scale equivariant estimators of the ratio of the variances of two normal distributions with unknown means. In this context the maximum likelihood estimator is shown to be universally inadmissible by virtue of a general sufficient condition for universal inadmissibility of a scale equivariant estimator. Received: January 2000     

Simplified estimation of parameters in a cauchy distribution

Simplified estimators of the location and scale parameters of a Cauchy distribution are constructed along the lines developed by D ixon [3, 4]. Symmetrically censored samples are considered. The efficiency of these estimators is shown to be high enough to make them useful in practice.     

Inference regarding multiple structural changes in linear models with endogenous regressors

This paper considers the linear model with endogenous regressors and multiple changes in the parameters at unknown times. It is shown that minimization of a Generalized Method of Moments criterion yields inconsistent estimators of the break fractions, but minimization of the Two Stage Least Squares (2SLS) criterion yields consistent estimators of these parameters. We develop a methodology for estimation and inference of the parameters of the model based on 2SLS. The analysis covers the cases where the reduced form is either stable or unstable. The methodology is illustrated via an application to the New Keynesian Phillips Curve for the US.     

The numerical values of some key parameters in econometric models

In the case of two endogenous variables, exogenous predetermined variables, and normally distributed disturbances, the distributions of the Two-Stage Least Squares (TSLS) and Limited Information Maximum Likelihood (LIML) estimators can be compared on the basis of three key parameters: the non-centrality parameter, a standardization of the structural coefficient, and the number of excluded exogenous variables. In this paper the values of these parameters are estimated in eleven structural equations from various actual econometric models. The distribution functions of the normalized TSLS and LIML estimators are given for the first two key parameters set at approximately their trimmed means, and the third at its median.     

Method of moments estimation and identifiability of semiparametric nonlinear errors-in-variables models

This paper deals with a nonlinear errors-in-variables model where the distributions of the unobserved predictor variables and of the measurement errors are nonparametric. Using the instrumental variable approach, we propose method of moments estimators for the unknown parameters and simulation-based estimators to overcome the possible computational difficulty of minimizing an objective function which involves multiple integrals. Both estimators are consistent and asymptotically normally distributed under fairly general regularity conditions. Moreover, root-n consistent semiparametric estimators and a rank condition for model identifiability are derived using the combined methods of the nonparametric technique and Fourier deconvolution.