Estimators of the regression parameters of the zeta distribution

The zeta distribution with regression parameters has been rarely used in statistics because of the difficulty of estimating the parameters by traditional maximum likelihood. We propose an alternative method for estimating the parameters based on an iteratively reweighted least-squares algorithm. The quadratic distance estimator (QDE) obtained is consistent, asymptotically unbiased and normally distributed; the estimate can also serve as the initial value required by an algorithm to maximize the likelihood function. We illustrate the method with a numerical example from the insurance literature; we compare the values of the estimates obtained by the quadratic distance and maximum likelihood methods and their approximate variance–covariance matrix. Finally, we calculate the bias, variance and the asymptotic efficiency of the QDE compared to the maximum likelihood estimator (MLE) for some values of the parameters.     

Asymptotically efficient rank MANOVA tests for restricted alternatives in randomized block designs

In a randomized block design MANOVA model, for intrablock as well as aligned rank tests for homogeneity of treatment effects against some restricted alternatives, asymptotic optimality is studied by reference to the corresponding restrieted likelihood ratio tests. Tests based on aligned ranks are better than intra-block rank tests when the error distributions are homogeneous aeross the blocks.Work of this author was partially supported by the Office of Naval Research, Contract No. N00014-83-K-0987.     

Optimal Threshold for the k-Out-Of-n Monitor with Dual Failure Modes

A monitor consists of n identical sensors working independently. Each sensor measures a variate of output or environment of a system, and is activated if a variate is over a threshold specified in advance for each sensor. The monitor alarms if at least k out of n sensors are activated. The performance of the monitor, the probabilities of failure to alarm and false alarming, depends on the number k, the threshold values and the probability distributions of the variate at normal and abnormal states of the system. In this paper, a sufficient condition on the pair of the distributions is given under which the same threshold values for all the sensors are optimal. The condition motivates new orders between probability distributions. Solving an optimization problem an explicit condition is obtained for maximizing or minimizing a symmetric function with the constraint of another symmetric function.     

Uniformly more powerful tests for hypotheses about linear inequalities when the variance is unknown

Let X be a -dimensional normal random vector with unknown mean and covariance matrix , where is a known matrix and an unknown parameter. This paper gives a test for the null hypothesis that lies either on the boundary or in the exterior of a closed, convex polyhedral cone versus the alternative hypothesis that lies in the interior of the cone. Our test is uniformly more powerful than the likelihood ratio test.

     

Inference in lognormal multidimensional diffusion processes with exogenous factors: Application to modelling in economics

The multidimensional lognormal diffusion process with exogenous factors is treated using the Kolmogorov equations, and the mean vector and covariance matrix are estimated using discrete sampling by the maximum-likelihood method. Also, this process is constructed as a solution of a multidimensional stochastic differential equation, and an estimation is made through the maximum-likelihood method to infer the parameters of the exogenous factors, this time using continuous sampling. Finally, a test for a hypothesis based on these parameters is constructed.     

An EM Algorithm for Batch Markovian Arrival Processes and its Comparison to a Simpler Estimation Procedure

Although the concept of Batch Markovian Arrival Processes (BMAPs) has gained widespread use in stochastic modelling of communication systems and other application areas, there are few statistical methods of parameter estimation proposed yet. However, in order to practically use BMAPs for modelling, statistical model fitting from empirical time series is an essential task. The present paper contains a specification of the classical EM algorithm for MAPs and BMAPs as well as a performance comparison to the computationally simpler estimation procedure recently proposed by Breuer and Gilbert. Furthermore, it is shown how to adapt the latter to become an estimator for hidden Markov models.     

ESTIMATING A DISTRIBUTION FUNCTION WITH TRUNCATED DATA

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The iterative instrumental variables method and the full information maximum likelihood method for estimating interdependent systems

The “iterative instrumental variables” (IIV) method for estimating interdependent systems, originally referred to as a symmetric counterpart to the “fix-point” (FP) method, shares its symmetry properties with Durbin's iterative method for performing the “full information maximum likelihood” (FIML) estimation. Classical interdependent systems are considered and identities may occur among the structural equations. Alternative symmetric procedures for obtaining FIML estimates are also dealt with, including the sequential maximization of the likelihood function with respect to the coefficients of one structural equation at a time.Two recent estimation methods developed by Brundy and Jorgenson (1971, Review of Economics and Statistics53, 207–224) as well as Dhrymes (1971, Austral. J. Statist.13, 168–175) can be considered the second approximation of the IIV method and Durbin's method respectively with the first approximation obtained by the “ordinary instrumental variables” (OIV) method. In practice the second approximation depends heavily on the choice of initial instrumental variables, although the asymptotic distribution is not changed by the continued iteration.     

On the use of the directional derivative in obtaining multivariate extreme values

Current use of the directional derivative appears, with notable exceptions such as Whittle (1971, 1973) and Vainberg (1973), to be limited largely to textbooks on advanced calculus, and to spaces of at most three dimensions. The present paper develops a calculus of the directional derivative for arbitrary finite dimensional vector spaces. Applications are made to classical maximum likelihood estimation in the case of the multivariate normal density and to other multivariate problems involving stationary points.