NONNEGATIVE ESTIMATORS FOR THE ONE-WAY RANDOM EFFECTS MODEL

The relative merits of ten estimators for the variance component of the balanced and unbalanced one-way random effects models are compared. Six of the estimators are nonnegative, two of which are obtained by modifying the Minimum Variance Quadratic Unbiased Estimator (MIVQUE) and the Weighted Least Square Estimator (WLS), and two more from the positive parts of these estimators. The Minimum Norm Quadratic Estimator (MINQE), which is nonnegative, is adjusted for reducing its bias. The nonnegative Minimum Mean Square Error Estimator (MIMSQE), the Analysis of Variance (ANOVA) and Unweighted Sums of Squares (USS) estimator are also included.     

VARIANCE ESTIMATION IN THE ERROR COMPONENTS REGRESSION MODEL

ABSTRACT

In a regression model with a random individual and a random time effect explicit representations of the nonnegative quadratic minimum biased estimators of the corresponding variances are deduced. These estimators always exist and are unique. Moreover, under normality assumption of the dependent variable unbiased estimators of the mean squared errors of the variance estimates are derived. Finally, confidence intervals on the variance components are considered.     

POWER ASPECTS OF ANALYSIS OF VARIANCE IN FIXED EFFECT MODEL ONE-WAY CLASSIFICATION

The sensitivity of the power in analysis of variance to the departure from the in-built assumptions other than the normality of errors is discussed in Kanji (1975). To obtain the power values he considered the general linear hypothesis model.
Kanji (1976a) discussed a particular case of the above situation, that is fixed effect model two-way classification. In this paper another particular case, that of the fixed effect model one-way classification is discussed, the main purpose of which is to investigate whether it could show a different picture to the two-way classification, especially for unequal replication. The results so obtained are presented in Tables 1A, 1B and 1C. They indicate that the power value is greatly affected by the inequality of error variances and unequal group sizes.     

A NOTE ON COMPETING VARIANCE ESTIMATORS IN RANDOMISED RESPONSE SURVEYS

When gathering randomised rather than direct responses on a variable of interest relating to sensitive issues, one may use a modified version of the well-known generalised regression predictor of a finite population total. To construct confidence intervals, this paper proposes four alternative variance estimators – modifications to those usable with direct responses – and examines their relative efficiencies through simulations from simple super-population models.     

AN ESTIMATION PROCEDURE FOR ERROR VARIANCE INCORPORATING PTS FOR RANDOM EFFECTS MODEL UNDER LINEX LOSS FUNCTION

In the present paper an estimator of the error variance for a three-way layout in random effects model incorporating two preliminary tests of significance has been proposed. It has been well recognized that estimation of parameters, of interest under asymmetric loss function (ASL) is generally better than that under squared error loss function (SELF), particularly where overestimation and underestimation are not equally penalised. As neither overestimation nor underestimation of error variance is desirable, with this motivation, the proposed estimator for the error variance has been studied under LINEX loss function. It is claimed that, with proper choice of degree of asymmetry and level of significance, proposed the sometimes pool estimator performs fairly better than unbiased estimator. Recommendations regarding its application have been attempted.     

SOME PROPERTIES OF A RANDOM LINEAR DIFFERENCE EQUATION1

Some simple conditions are given for the absolute continuity of the limiting distribution of a random linear difference equation. These results are applied to the super-critical Bellman-Harris branching process with immigration. When the coefficients of the difference equation are non-negative and there is no limiting distribution, it is shown that the asymptotic behaviour of the solutions is the same as that of the partial sums of a divergent random power series. A number of limit theorems are given for the latter situation.     

ON THE NONCENTRAL DISTRIBUTION OF A RANDOM MATRIX USEFUL IN CLASSIFICATION THEORY

To prove the optimality properties of the maximum likelihood (and also minimum distance) discriminant rule Rogers (1980, p. 98) embeds the maximum likelihood discriminant function in a Cauchy-Schwartz inequality. This embedding procedure of Rogers (1980) may be used to derive a new distribution for Anderson's (1958) classification statistic.     

ON COMPONENTS OF VARIANCE and MODELLING EXCEEDANCES OVER A THRESHOLD

Variance component estimates from a simple hierarchical model are used to model the distribution of large data values exceeding a threshold. The advantages of interrelating the components of variance and the exceedances are stressed. The theory is outlined and an analysis of some data on blood pressure is discussed.     

THE SAMPLING PERFORMANCE OF INEQUALITY RESTRICTED AND PRE-TEST ESTIMATORS IN A MIS-SPECIFIED LINEAR MODEL

The literature on the sampling properties of the inequality restricted and pre-test estimators typically assumes a properly specified model and focuses on the estimation of the regression coefficient vector. In this paper, we derive and evaluate the risk functions of these estimators for both the prediction vector and the disturbance variance in a model which is mis-specified through the exclusion of relevant regressors. The results suggest that unrestricted estimation is generally preferable to pre-testing or naively imposing restrictions.     

APPLICATIONS OF ROBUST ESTIMATION FOR VARIANCE COMPONENTS MODELS:THE DETECTION OF MAJOR GENE EFFECTS IN FINGER RIDGE COUNTS IN NORMAL AND FRAGILE X FAMILIES

A diagnostic technique is proposed to detect major gene effects and other systematic departures from a model for the trait means in the presence of outliers. The technique is based on the examination of residuals from fitting variance components models to quantitative pedigree data using robust statistical procedures. The approach is demonstrated using the total ridge count and ridge count of the middle finger from 54 extended families affected with the Fragile X syndrome, and a sample of 217 normal pedigrees.     

A LINEAR MODEL WITH ERRORS LACKING A VARIANCE II1

Numerical results are presented for estimates of the parameters in the linear model Y =βX +ε in which X is normally distributed and ε is symmetric stable. The study complements an earlier paper of the same title and the main concern is with numerical comparisons between four estimates of β; the least squares estimate, the minimum absolute deviations estimate, and two moment estimates of the form derived in Chambers and Heathcote (1975). The generation of fifty independent sets of observations (X_j, Y_j), j = 1,2, …, n for each of n = 100, 500 and selected combinations of parameter values provided the basis of the results. It is indicated that the moment estimators and the minimum absolute deviation estimator performed comparably, and are a significant improvement on the least squares estimator. The main conclusion is that one of the moment estimates, based on a two stage adaptive procedure and denoted by β¯_n(t_a) below, is generally the most useful of the four.     

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.     

APPLYING THE SAVAGE-DICKEY DENSITY RATIO TO DEFAULT BAYES FACTORS,WITH AN ILLUSTRATION TO OUTLIER DETECTION IN RANDOM EFFECTS MODELS

When no information is available and hence improper noninformative priors should be used, Bayes factor includes the unspecified constants and can not be calibrated. To solve this problem, we modify the intrinsic Bayes factor (IBF) of Berger and Pericchi 1-2 Berger, J. O. and Pericchi, L. R. 1996. The Intrinsic Bayes Factor for Model Selection and Prediction. Journal of the American Statistical Association, 91: 109–122. Berger, J. O. and Pericchi, L. R. 1998. Accurate and Stable Bayesian Model Selection: The Median Intrinsic Bayes Factor. Sankhya, Series B, 60: 1–18.   and the fractional Bayes factor (FBF) of O'Hagan [3] O'Hagan, A. 1995. Fractional Bayes Factors for Model Comparison. Journal of the Royal Statistical Society, Series B, 57: 99–138.  [Google Scholar] with the generalized Savage-Dickey density ratio of Verdinelli and Wasserman [4] Verdinelli, I. and Wasserman, L. 1995. Computing Bayes Factors Using a Generalization of Savage-Dickey Density Ratio. Journal of the American Statistical Association, 90: 614–618. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]. These modified IBF and FBF are applied to detecting outliers in random effects models with a mean-shift structure. The proposed methodology is exemplified by a simulation experiment with a generated data set and also applied to a real data set, Dyestuff data in Box and Tiao [5] Box, G. E.P. and Tiao, G. C. 1973. Bayesian Inference in Statistical Analysis U.S.A.: Addison-Wesley Publishing Co..  [Google Scholar]     

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.     

ON SOME PRODUCT TYPE ESTIMATORS IN VARYING PROBABILITY1

It is shown that varying probability in product method estimation increases the efficiency of estimate under some usual conditions. Further, a necessary condition, for the product estimate from a sample drawn with pps and without replacement to be more efficient than the product estimate from a sample drawn with pps and replacement, has been obtained; incidentally, it is the same as obtained by Narain (1951) for varying probability without replacement to be more efficient than varying probability with replacement. The use of the R.H.C. sampling scheme has also been suggested, and its variance and bias has been shown to be always less than that of pps with replacement.     

√n-CONSISTENT ESTIMATION IN A RANDOM COEFFICIENT AUTOREGRESSIVE MODEL

This paper deals with √n-consistent estimation of the parameter μ in the RCAR(l) model defined by the difference equation X_j=(μ+U_j)X_j-l+ej (jε Z), where {e_j: jε Z} and {U_j: jε Z} are two independent sets of i.i.d. random variables with zero means, positive finite variances and E[(μ+U₁)²] < 1. A class of asymptotically normal estimators of μ indexed by a family of bounded measurable functions is introduced. Then an estimator is constructed which is asymptotically equivalent to the best estimator in that class. This estimator, asymptotically equivalent to the quasi-maximum likelihood estimator derived in Nicholls & Quinn (1982), is much simpler to calculate and is asymptotically normal without the additional moment conditions those authors impose.     

THE EFFECTS OF MISCLASSIFICATION COSTS AND SKEWED DISTRIBUTIONS IN TWO-GROUP CLASSIFICATION

ABSTRACT

In this study, Monte Carlo simulation experiments were employed to examine the performance of four statistical two-group classification methods when the data distributions are skewed and misclassification costs are unequal, conditions frequently encountered in business and economic applications. The classification methods studied are linear and quadratic parametric, nearest neighbor and logistic regression methods. It was found that when skewness is moderate, the parametric methods tend to give best results. Depending on the specific data condition, when skewness is high, either the linear parametric, logistic regression, or the nearest-neighbor method gives the best results. When misclassification costs differ widely across groups, the linear parametric method is favored over the other methods for many of the data conditions studied.     

A COMPONENT OF VARIANCE DUE TO COMPETITION1

The existence of a component of variance for competition among sampling units or among individuals in a group was discussed by Yates and Zacopany in 1935. No procedure was suggested for estimating this component of variance. It is the purpose of this paper to give a procedure for estimating the component of variance due to competition and to apply the procedure to a set of data on weaning weights of pigs with 116 litters of various sizes and for Yorkshire, Chester-White, and Berkshire breeds. The first problem was to define litter size. Within this definition then, litters sizes of 3 to 14 pigs per litter were obtained. The variation among pigs within a litter of size h was considered to have an expected value equal to V_s+ V_ch where V_s is the sampling variance component and V_ch is the competition variance component for a litter of size h. In order to obtain an estimate of V_ch, a polynomial relation between h and V_ch was postulated. In particular, it was postulated that where E(.) denotes expected value. This form states that V_ch goes to zero for one pig per litter; it may be appropriate as long as small litter sizes (say 1 and 2 at least) are omitted from the analysis as was done in the present instance. Using an iterative procedure of reestimating the weights at each stage, a form of weighted least squares analysis was performed. The procedure appears to converge after three to four steps of iteration. Solutions for some or all of the parameters V_p, V_s, β₁ and β₂ for h even, h odd, and all h = 3, 4, …, 14, were obtained using among litter mean squares, A_h, only, using within litter mean squares, W_h, only, and using both A_h, and W_h values. The A_h values for h = 3, 4, and 5 appeared to form a different group than for the other values of h. Both the A_h and W_h mean squares were from 56 day weights adjusted for birth weight. The maximum value of V_ch for odd h, was nine whereas it was six for even h. Using all h the maximum value for V_ch occurred when h was equal to nine. It appeared that expressing V_ch as a quadratic function of litter size was satisfactory for these litter sizes and mean squares.