SOME PROPERTIES OF A FOLDOVER DESIGN

The foldover design of the 12-run Plackett-Burman design is shown to give a 2⁴and 2^4-1design in every set of four factors, and to be resolution V in every set of five factors. In addition, the design allows the search and estimation of up to two non-zero interactions.     

ASYMPTOTIC PROPERTIES OF LINEAR REGRESSION ESTIMATORS UNDER A FIXED CENSORSHIP MODEL

Modifications to the usual least squares normal equations have been proposed by Buckley and James (1979) when using distribution-free linear regression modelling under fixed right-censorship of the response. In this paper we consider large-sample distributional properties of the modified normal equation for the slope parameter and of estimators which ‘satisfy’ this equation.     

PROPERTIES OF THE MINIMUM POINT OF AN UNBALANCED TWO-SIDED RANDOM WALK

In this paper, properties of minimum point of a unbalanced two-sided random walk are investigated. Under the condition that the parameters at both sides tend to zero at the same order, probabilities that the minimum point is on which side, and the second order expansions for the first two moments of the minimum point are obtained. Applications of these results are very promising. First, they can be used to study the properties of the maximum likelihood estimator for the change point in the large sample case; second, they can be used to study inference problems after CUSUM test.     

SOME PROPERTIES OF PROFILE BOOTSTRAP CONFIDENCE INTERVALS

We consider the problem of finding an equi-tailed confidence interval, with coverage probability (1-α), for a scalar parameter θ₀ in the presence of a (possibly infinite dimensional) nuisance parameter ψ₀. It is supposed that the value taken by θ₀ does not restrict the value that ψ₀ may take and vice-versa. Given a sensible estimate ψ_n of ψ₀, profile bootstrap confidence interval for θ₀ is defined to be the exact equi-tailed confidence interval with coverage probability (1-α) assuming that ψ₀=ψ_n. We compare the properties of the profile bootstrap confidence interval and the ordinary bootstrap confidence interval when they are based on studentised and unstudentised quantities. Under mild regularity conditions the profile bootstrap confidence interval is always a subset of the set of allowable values of θ₀ and is transformation-respecting when based on either an unstudentised quantity or a studentised quantity satisfying certain restrictions. As a confidence interval for the autoregressive parameter of an AR(1) process, the profile bootstrap confidence interval has important advantages over the ordinary bootstrap confidence interval based on a studentised quantity.     

RANDOM PERMUTATION TESTING IN MULTIPLE LINEAR REGRESSION

We developed an alternative random permutation testing method for multiple linear regression, which is an improvement over the existing one proposed by [1] Kennedy, P. E. 1995. Randomization tests in econometrics. Journal of Business and Economic Statistics, 13: 85–94. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar] or [2] Freedman, D. and Lane, D. 1983. A nonstochastic interpretation of reported significance levels. Journal of Business and Economic Statistics, 1: 292–298. [Taylor & Francis Online] , [Google Scholar].     

SOME ASYMPTOTIC PROPERTIES OF THE SIMPLE LOGLINEAR MODEL

Consistency and asymptotic normality of the maximum likelihood estimator of β in the loglinear model E(y_i) = e^α+βXi, where y_i are independent Poisson observations, 1 iaan, are proved under conditions which are near necessary and sufficient. The asymptotic distribution of the deviance test for β=β₀ is shown to be chi-squared with 1 degree of freedom under the same conditions, and a second order correction to the deviance is derived. The exponential model for censored survival data is also treated by the same methods.     

MAXIMUM LIKELIHOOD ESTIMATION IN LINEAR MODELS WITH EQUI-CORRELATED RANDOM ERRORS

Necessary and sufficient conditions for the existence of maximum likelihood estimators of unknown parameters in linear models with equi‐correlated random errors are presented. The basic technique we use is that these models are, first, orthogonally transformed into linear models with two variances, and then the maximum likelihood estimation problem is solved in the environment of transformed models. Our results generalize a result of Arnold, S. F. (1981) [The theory of linear models and multivariate analysis. Wiley, New York]. In addition, we give necessary and sufficient conditions for the existence of restricted maximum likelihood estimators of the parameters. The results of Birkes, D. & Wulff, S. (2003) [Existence of maximum likelihood estimates in normal variance‐components models. J Statist Plann. Inference. 113 , 35–47] are compared with our results and differences are pointed out.     

THE EXAMINATION AND ANALYSIS OF RESIDUALS FOR SOME BIASED ESTIMATORS IN LINEAR REGRESSION

In the presence of collinearity certain biased estimation procedures like ridge regression, generalized inverse estimator, principal component regression, Liu estimator, or improved ridge and Liu estimators are used to improve the ordinary least squares (OLS) estimates in the linear regression model. In this paper new biased estimator (Liu estimator), almost unbiased (improved) Liu estimator and their residuals will be analyzed and compared with OLS residuals in terms of mean-squared error.     

A GEOMETRIC CHARACTERIZATION OF LINEAR REGRESSION

This paper concerns the asymptotic properties of the maximum likelihood estimators of the parameters in a non regular Cox model involving a change-point in the regression on time-dependent covariates. The global consistency derives from the uniform convergence of the partial log-likelihood. We prove that the estimator of the change-point is n -consistent and the estimator of the regression parameter n 1/2 -consistent, and their asymptotic distributions are established.     

ON SOME OPTIMALITY PROPERTIES OF FISHER-RAO SCORE FUNCTION IN TESTING AND ESTIMATION

The score function is associated with some optimality features in statistical inference. This review article looks on the central role of the score in testing and estimation. The maximization of the power in testing and the quest for efficiency in estimation lead to score as a guiding principle. In hypothesis testing, the locally most powerful test statistic is the score test or a transformation of it. In estimation, the optimal estimating function is the score. The same link can be made in the case of nuisance parameters: the optimal test function should have maximum correlation with the score of the parameter of primary interest. We complement this result by showing that the same criterion should be satisfied in the estimation problem as well.     

ASYMPTOTIC AND SMALL SAMPLE STATISTICAL PROPERTIES OF RANDOM FRAILTY VARIANCE ESTIMATES FOR SHARED GAMMA FRAILTY MODELS

This paper concerns maximum likelihood estimation for the semiparametric shared gamma frailty model; that is the Cox proportional hazards model with the hazard function multiplied by a gamma random variable with mean 1 and variance θ. A hybrid ML-EM algorithm is applied to 26 400 simulated samples of 400 to 8000 observations with Weibull hazards. The hybrid algorithm is much faster than the standard EM algorithm, faster than standard direct maximum likelihood (ML, Newton Raphson) for large samples, and gives almost identical results to the penalised likelihood method in S-PLUS 2000. When the true value θ₀ of θ is zero, the estimates of θ are asymptotically distributed as a 50–50 mixture between a point mass at zero and a normal random variable on the positive axis. When θ₀ > 0, the asymptotic distribution is normal. However, for small samples, simulations suggest that the estimates of θ are approximately distributed as an x ? (100 ? x)% mixture, 0 ≤ x ≤ 50, between a point mass at zero and a normal random variable on the positive axis even for θ₀ > 0. In light of this, p-values and confidence intervals need to be adjusted accordingly. We indicate an approximate method for carrying out the adjustment.     

ON A SHARED ALLELE TEST OF RANDOM MATING

A simple random sample is observed from a population with a large number‘K’ of alleles, to test for random mating. Of n couples, n_ijkl have female genotype ij and male genotype kl (i, j, k, l{1,…, A‘}). The large contingency table is collapsed into three counts, n₀, n₁ and n₂ where n_p is the number of couples with s alleles in common (s = 0,1, 2). The counts are estimated by np?_o where n₀, is the estimated probability of a couple having s alleles in common under the hypothesis of random mating. The usual chi-square goodness of fit statistic X² compares observed (n_s) with expected (np?) over the three categories, s = 0,1,2. An empirical observation has suggested that X² is close to having a chi-square distribution with two degrees of freedom (X) despite a large number of parameters implicitly estimated in _e. This paper gives two theorems which show that x is indeed the approximate distribution of X² for large n and K₁“, provided that no allele type over-dominates the others.     

ASYMPTOTIC PROPERTIES OF RESTRICTED MAXIMUM LIKELIHOOD (REML) ESTIMATES FOR HIERARCHICAL MIXED LINEAR MODELS

This paper explores the asymptotic distribution of the restricted maximum likelihood estimator of the variance components in a general mixed model. Restricting attention to hierarchical models, central limit theorems are obtained using elementary arguments with only mild conditions on the covariates in the fixed part of the model and without having to assume that the data are either normally or spherically symmetrically distributed. Further, the REML and maximum likelihood estimators are shown to be asymptotically equivalent in this general framework, and the asymptotic distribution of the weighted least squares estimator (based on the REML estimator) of the fixed effect parameters is derived.