Estimation of regression coefficients after a preliminary test for homoscedasticity

When heteroscedasticity of the variances of disturbances in a regression model is suspected, we perform a preliminary test for homoscedasticity prior to estimation of regression coefficients. According to the result of the pre-test, we use either the ordinary least squares estimator or the two-stage Aitken estimator (2SAE). In this paper, using orthonormal regressors, we derive the mean square error (MSE) of the pre-test estimator and show that the 2SAE is inadmissible when the MSE is used as a criterion. Further, we seek the optimal critical value of the pre-test in the sense of minimizing the average relative risk which is based on the MSE.     

Estimating the error variance in regression after a preliminary test of restrictions on the coefficients

Maximum-likelihood estimation of the variance of the disturbances in a linear regression is considered in the context of exact linear restrictions on the coefficient vector. The exact risk function for the associated preliminary-test estimator of this variance is derived and evaluated numerically on the basis of relative quadratic loss. Our results indicate that in practical situations the risk of the pre-test estimator differs only slightly from that of the naive estimator which ignores the restrictions without testing. Applying the coefficient restrictions without testing their validity is not recommended.     

Estimation of a regression coefficient after two preliminary tests of significance

Summary The estimation of the regression coefficient of a population, defined byE (y)= +x, incorporating two preliminary tests of significance has been discussed. The experimenter has two random samples of different sizes from two such populations, as defined above, with regression coefficients ₁ and ₂ respectively, where ₂ may possibly be equal to ₁. Besides this, it is also conjectured that the common conditional variance ² of the two populations has a specified value ₀ ² . The two preliminary tests are used to resolve these two uncertainties.The author has rejoined Lucknow University, Lucknow India on Oct. 4, 1976 after availing leave for two years.     

On choosing the optimal level of significance for the Durbin-Watson test and the Bayesian alternative

This paper critically evaluates the usual ad hoc selection of the level of significance in the Durbin-Watson test and compares this procedure to the Bayesian alternative. The results of Monte Carlo experiments indicate that an α-level substantially larger than that normally used may be appropriate. The Bayesian estimator performed better than all preliminary test estimates in terms of MSE.     

On a test of homogeneity for samples from a negative binomial distribution

Summary An exact test is proposed for the equality of probabilities of an event in a series of independent trials, each drawn from a negative binomial distribution with a specified number of successes. The power function of this conditional test is also derived, as well as an approximate test for homogeneity.     

A two-stage test for one-way analysis of variance

A two-stage analogue of the Ftest for one-way analysis of variance is proposed. Critical values are given for selected cases and the method of obtaining critical values is described. Comparisons of the one-stage and two-stage tests are made in terms of the expected sample size required to achieve a given power.     

On the Kuiper test for normality with mean and variance unknown

Summary If one wants to test the hypothesis as to whether a set of observations comes from a completely specified continuous distribution or not, one can use the Kuiper test. But if one or more parameters have to be estimated, the standard tables for the Kuiper test are no longer valid. This paper presents a table to use with the Kuiper statistic for testing whether a sample comes from a normal distribution when the mean and variance are to be estimated from the sample. The critical points are obtained by means of Monte-Carlo calculation; the power of the test is estimated by simulation; and the results of the powers for several alternative distributions are compared with the estimated powers of the Kolmogorov-Smirnov test.     

On the Kuiper test for normality with mean and variance unknown

Summary A modified form of the Kuiper statistic V _n is developed for testing the composite hypothesis that a sample of size n comes from a normal population with unspecified mean and variance. Its distribution is derived using Monte Carlo methods. Power comparison with the adjusted Kuiper test proposed by L outer and K oerts [6] indicates that our test is superior with respect to certain alternatives.     

Estimation and optimal designs for linear Haar-wavelet models

This paper gives an analytical expression for the best linear unbiased estimator (BLUE) of the unknown parameters in the linear Haar-wavelet model. From the analytical expression, we solve for the eigenvalues of the covariance matrix of the BLUE in analytical form. Further, we use these eigenvalues to construct some conventional discrete optimal designs for the model. The equivalences among these optimal designs are demonstrated and some examples are also given.      

Asymptotically optimal nonparametric one- and two-way analysis of variance tests for the log linear model under censoring

We considerr ×c populations with failure ratesλ _ij(t) satisfying the condition

A test for the weights of the global minimum variance portfolio in an elliptical model

In this paper we consider the weights of the global minimum variance portfolio (GMVP). The returns are assumed to follow a matrix elliptically contoured distribution, i.e., the returns are assumed to be neither independent nor normally distributed. A test for the general linear hypothesis is given. The distribution of the test statistic is derived under the null and the alternative hypothesis. It turns out that its distribution is invariant with respect to the type of the matrix elliptical distribution, i.e., the stochastic properties of the GMVP do not depend either on the mean vector or on the distributional assumptions imposed on asset returns. In an empirical study we analyze an international diversified portfolio.     

A test of assumed cluster homogeneity for performance appraisal management in four Southeast Asian countries

This study examines the extent of divergence versus convergence effect on the design characteristics of performance appraisal systems across four Southeast Asian countries: Indonesia, Malaysia, the Philippines, and Thailand. The results indicate that there exist significant differences in prevailing managerial practices and behaviours pertinent to the design and conduct of performance appraisal. Contrary to an assertion of homogeneity in previous cross-cultural cluster research and regio-centric perspectives, managers of MNCs should not necessarily generalize the appropriateness of performance appraisal design and practice across countries in the same traditional cluster.     

Fourth-order autocorrelation: Further significance points for the Wallis test

This note provides tables of 0.5, 1.0 and 2.5 percent significance points of the bounds of Wallis's statistic for testing for fourth-order autocorrelation in regressions either including or excluding quarterly dummy variables.     

Experimental designs for estimating plateau-type production functions and economically optimal input levels

Estimation of nitrogen response functions has a long history and yet there is still considerable uncertainty about how much nitrogen to apply to agricultural crops. Nitrogen recommendations are usually based on estimation of agronomic production functions that typically use data from designed experiments. Nitrogen experiments, for example, often use equally spaced levels of nitrogen. Past agronomic research is mostly supportive of plateau-type functional forms. The question addressed is if one is willing to accept a specific plateau-type functional form as the true model, what experimental design is the best to use for estimating the production function? The objective is to minimize the variance of the estimated expected profit maximizing level of input. Of particular interest is how well does the commonly used equally-spaced design perform in comparison to the optimal design. Mixed effects models for winter wheat (Triticum aestivium L.) yield are estimated for both Mitscherlich and linear plateau functions. With three design points, one should be high enough to be on the plateau and one should be at zero. The choice of the middle design point makes little difference over a wide range of values. The optimal middle design point is lower for the Mitscherlich functional form than it is for the plateau function. Equally spaced designs with more design points have a similar precision and thus the loss from using a nonoptimal experimental design is small.     

Monitoring the mean and the variance of a stationary process

We deal with the problem of how deviations in the mean or the variance of a time series can be detected. Several simultaneous control charts are introduced which are based on EWMA (exponentially weighted moving average) statistics for the mean and the empirical variance. The combined X − S² EWMA chart is extended to time series. Further simultaneous charts are considered. The comparision of these schemes shows that the residual attempt must be favored if a variance change is present.