Finite Sample Theory Research Articles

Approximating the Exact Finite Sample Distribution of a Spectral Estimator

In this paper an explicit and computationally convenient expansion of the exact finite sample distribution function of a quasi-maximum likelihood spectral estimator is given. In the majority of practical situations it will be necessary to estimate certain nuisance parameters of the distribution. Therefore, a method of evaluating these parameters is suggested and some Monte-Carlo evidence concerning the practical implementation of the results is given. 1 INTRODUCTfON AN EXTENSIVE SET of asymptotic results relating complex statistical analysis to the problems that arise in estimating spectra from the discrete Fourier transforms of time series data has been well established; see, for example, Brillinger [3] and Goodman [6]; and Hannan [11] has recently extended these results to cover the modified Fourier coefficients, proposed by Bingham, Godfrey, and Tukey [1] and defined in (2.3) below. Unfortunately it seems likely that the sample sizes required for one to approach the asymptotic position will not be available when considering the analysis of many economic time series. In a recent article, however, Hatanaka [12] has shown that the elimination of leakage produced by the modified Fourier coefficients is effective for small finite realizations and that it may be possible to recover the loss of degrees of freedom associated with the familiar estimator obtained by averaging over the modified periodogram.2 The purpose of the present paper is to extend these results by using complex statistical analysis to derive expressions for both the form and exact finite sample distribution of a spectral estimator obtained from the modified Fourier coefficients. Thus in the following section a brief exposition of some basic theory is given and a quasi-maximum likelihood estimation procedure is suggested. In Section 3 an exact expression for the finite sample distribution of the proposed estimator is given and shown to incorporate an established distributional result as a particular special case. In the majority of practical situations it will be necessary to estimate certain nuisance parameters of this distribution if it is to be employed and a method of evaluating these parameters is also suggested. It is well known, however, that while the replacement of nuisance parameters by consistent estimates will generally leave asymptotic theory intact the consequences of estimating nuisance parameters are unlikely to be negligible in finite sample theory. Since it is not possible to determine analytically the effect that the estimation of these nuisance parameters will have, the results of some simple Monte-Carlo

Likelihood functions in finite population sampling theory

SUMMARY Both conventional randomization models and prediction, i.e. superpopulation, models for finite populations are discussed from the viewpoint of the likelihood axiom. It is argued that finite populations do fall within the scope of this axiom, and that the likelihood function, when properly defined and interpreted, can play the same fundamental role in finite population theory as it does elsewhere in statistical inference. Under a multivariate normal regression model, the relationship between the likelihood function for the population total and the probability distribution of the minimum variance unbiased estimator is studied. The anticipated result, that this estimator maximizes the likelihood function, is shown to hold under the most familiar models, but an example shows it to be false in general. The role of balanced samples in providing robust inferences is discussed briefly. Conditions are described under which the likelihood function is unchanged by the addition of a new regressor to the model.

The Asymptotic Bias and Mean-Squared Error of Double K-Class Estimators When the Disturbances are Small

Two STRUCTURAL ESTIMATION TECHNIQUES for the parameters of a single equation in a system of simultaneous linear equations were available at the time of the Cowles Commission work (Koopmans [13], Hood and Koopmans [8]): ordinary least squares (OLS) and limited-information maximum likelihood (LIML). Later Theil [19, 20, 21] and Basinann [2] developed two-stage least squares (2SLS). Theil [20, 21] invented the k-class, which includes OLS (k 0), 2SLS (k = 1), and LIML (k = 2, the minimum root of a certain determinental equation (Anderson and Rubin [1])). Because the k-class contains these three leading contenders among single-equation estimators, it has been a convenient class of estimators for theoretical study. Theil [21, (353-354)] also proposed the h-class, which contains OLS and 2SLS buLt not LIML. Nagar [14] proposed the double kclass, a two-parameter family which embraces both the hand k-classes. The double k-class allows considerable flexibility with regard to the choice of an estimation teclhique, as well as providing a useful sumllmary statement of previoLusly proposed single-equnation estimators. Comparisons among competing estimators are greatly facilitated if the estimators can all be regarded as special instances of a general procedure. This paper provides small-disturbance asymptotic moment expressions for the general case of the double k-class estimator. These expressions can be applied to make comparisons among any double k-class estimators. Previous results describing the behavior of special cases of the double k-class estimators have been derived by various methods, including large-sample asymptotic theory, exact finite-sample theory, Monte Carlo studies, and small-disturbance asymptotic theory. Kadane [10] gave a review of these methods and their results as applied to the (single) k-class. He also derived the small-disturbance asymptotic bias and mean-squared error of the k-class estimators. The only analytic results on the entire double k-class previously available are those of Nagar [14], who derived expressions for the large-sample asymptotic bias to order (lIT) and the asymptotic mean-squared error to order (I/T2) for all double k-class estimators with parameters of the form k, 1 + c1/T, k2 = 1 +a2/T for constant a, and a2. The method of analysis used in this paper to investigate the double k-class estimators is based upon the small-disturbance asymptotic approach introduced

On finite population sampling theory under certain linear regression models

SUMMARY Problems of estimating totals in finite populations, when auxiliary information regarding variate values is available, are considered under some linear regression, 'super-population', models. Optimal strategies involving linear estimators are derived under certain variance assumptions and compared under various assumptions. For a model which seems to apply in many practical problems, the conventional ratio estimator is shown to be, in a certain natural sense, optimal, but for all models considered, the optimal sampling plans are purposive, i.e. nonrandom. With a squared error loss function, the strategy of using a probability proportional to size sampling plan and the Horvitz-Thompson estimator is shown to be inadmissible in many models for which the strategy seems 'reasonable' and in a particular model for which it is, in one sense, optimal. Some of the results concerning purposive sampling and the ratio estimator are supported by an empirical study.