Derrick S. Tracy

Sampling properties of estimators of the log‐logistic distribution with application to Canadian precipitation data

We consider the probability-weighted moment and the maximum-likelihood estimators of two parameters in the log-logistic distribution. Quantile estimators are obtained using both methods. The distributional properties of these estimators are studied in large samples, via asymptotic theory, and in small and moderate samples, via Monte Carlo simulation. The distribution is shown to be appropriate for a wide variety of meteorological data.

Multivariate Maxima and Minima with Matrix Derivatives

Abstract The purpose of this paper is the presentation of formulae for obtaining matrix derivatives of the second order to use in making tests for maxima and minima. The theory of such second order derivatives is presented. These formulae require the rearrangment of the parameter elements in vector form and the transformed results feature Kronecker products, which have certain desirable properties. Application is made to several types of problems.

An alternative to the ratio-cum-product estimator in sample surveys

Abstract In this paper, using a transformation (Srivenkataramana and Tracy, Ann. Inst. Statist. Math. 32A (1980), 111–120; Austral. J. Statist. 23 (1981), 95–100), an alternative to the usual ratio-cum-product estimator, considered by Singh (Metrika 12 (1967), 34–42), is proposed. Exact expressions for bias and mean square error of the proposed estimator are derived. Under a simple random sampling without replacement (SRSWOR) scheme, an exactly unbiased estimator is obtained with its approximate variance formula. The conditions for the proposed unbiased estimator to be more efficient than the conventional unbiased estimator y , usual ratio ( y R ) and product ( y P ) estimators, Singth (Metrika 12 (1967), 34–42) and Srivenkataramana and Tracy (Ann. Inst. Statist. Math. 32A (1980), 111–120) estimators are derived. An empirical study is carried out to demonstrate the efficiency of the proposed estimator over the other estimators.

Comparison of operational variants of best homogeneous and heterogeneous estimators in linear regression

Several adaptive versions of the minimum mean squared error estimator of the coefficient vector in a linear regression model are proposed in the literature. Some of these are compared here, and another estimator is also proposed.

An improved two stage randomized response strategy

Mangat and Singh (1990) have suggested a two stage randomized response technique to estimate the proportion of population possessing a sensitive attribute. The procedure was shown to be more efficient than the procedure due to Warner (1965). Recently, Tracy and Osahan (1993) have suggested a modification to the Mangat and Singh (1990) procedure which results in a more efficient strategy in practice. In this paper we propose a modification to the Tracy and Osahan (1993) procedure. The modified procedure is a generalization of Tracy and Osahan (1993) and is always more efficient than their strategy. An empirical study has also been undertaken to find the extent of relative efficiency.

Recurrence relations for the moments of truncated multinormal distribution

In this paper some recurrence relations of moments of doubly truncated multivariate normal distribution are obtained. The bivariate case is given as an example and some applications are indicated.

A Class of Almost Unbiased Estimators for Finite Populations Mean Using Two Auxiliary Variables

For estimating finite population mean Y 0 of study character y 0 , a class of almost unbiased estimators applying jackknife technique envisaged by QUENOUILLE (1956) is derived. Optimum unbiased estimator (OUE) is also investigated with its variance formula. An empirical study is carried out to demonstrate the performance of the constructed estimator over the usual unbiased estimator, SRIVASTAVA (1965), SINGH (1967), SINGH and BIRADAR (1992), TRACY, SINGH, and SINGH (1996) and other almost unbiased estimators.

Partitioned kronecker products of matrices and applications

Some generalized commutation matrices are defined and used to establish relationships between π-products and Kronecker products. These are applied to obtain expectations of π-products of random vectors and matrices. On definit des generalisations de matrices de “commutation” et on les utilise pour etablir des relations entre des π-produits et des produits de Kronecker. A partir de ces relations, on obtient des esperances de π-produit, de vecteurs et matrices aleatoires.

Moments of the complex multivariate normal distribution

Abstract Moments of the complex multivariate normal distribution are obtained by differentiating its characteristic function, applying the differential operators for the differentiation of functions of complex vectors. A recurrence relation for the derivatives of the characteristic function is derived, and explicit expressions for the moments are obtained. Moments up to order 6 are expressed in terms of Hermitian matrices. Moments of Hermitian quadratic forms up to order 3 are also obtained.

Higher order moments of multivariate normal distribution using matrix derivatives

A general formula for the central moments of multivariate normal distribution is derived by differentiating its characteristic function using matrix derivatives. An explicit expression for the moments is obtained. Two applications of these results are given. The sixth order moments are arranged in a square matrix using the properties of commutation matrices and vec operators