Wald Type Research Articles

Tests for serial correlation and overdispersion in a count data regression model∗

Lagrange multiplier (LM) and Wald type tests for overdispersion and serial correlation using the generalized method of moment approach are proposed for a time series count data regression model. The small sample properties of the tests are compared by Monte Carlo experimentation. It is found that a traditional test for overdispersion has both better size and power properties than the LM and Wald tests when the sample size is small. The Wald and LM tests of serial correlation have both better size and power than traditional portmanteau tests.

MANOVA type tests under a convex discrepancy function for the standard multivariate linear model

We provide the M-theory for the standard multivariate linear model Y = XB + E , where Y is n × p matrix of observations, X is n × m design matrix, B is m × p matrix of unknown parameters and E is n × p matrix of errors with the row vectors independently distributed. Two test criteria based on the roots of determinantal equations are proposed for testing linear hypotheses of the form P ′ B = C 0 , where P is a matrix of rank q . The tests are similar to those considered in MANOVA using least squares techniques. One of them is the Wald type statistic and another is the Rao's score type statistic. The asymptotic distributions of these test statistics are derived. Consistent estimates of nuisance parameters are obtained for use in computing the test statistics. The M-method of estimation considered is the minimization of Σϱ( e i ), where ϱ is a convex function and e i is the i -th row vector in ( Y − XB ). All results are derived under a minimal set of conditions.

Testing exogeneity in overidentified models

In this paper we analyse the consequences of model overidentification on testing exogeneity, when maximum likelihood techniques for estimation and inference are used. This situation is viewed as a particular case of the more general problem of considering how restrictions on nuisance parameters could help in making inference on the parameters of interest. At first a general model is considered. A suitable likelihood function factorization is used which allows a simple derivation of the information matrix and others tools useful for building up joint tests of exogeneity and overidentifying restrictions both of Wald and Lagrange Multiplier type. The asymptotic local power of the exogeneity test in the justidentified model is compared with that in the overidentified one, when we assume that the latter is the true model. Then the pseudo-likelihood framework is used to derive the consequences of working with a model where overidentifying restrictions are erroneously imposed. The inconsistency introduced by imposing false restrictions is analysed and the consequences of the misspecification on the exogeneity test are carefully examined.

Testing and estimation of equal variances for correlated variables

For testing the equality of variances for correlated variables, Harris (1985) suggested Wald type test statistics. In this note we show that this test coincides with the generalized least squares statistic employed in the analysis of covariance structures. We derive the generalized least squares estimator of the common variance. This estimator is compared with the average sample variances which is the ordinary least squares estimator for this problem.

Risk Analysis in Cohort Studies with Heterogeneous Strata. A Global χ2-Test for Dose-Response Relationship, Generalizing the Mantel-Haenszel Procedure

In cohort studies the Mantel-Haenszel estimator ORMH is computed from sample data and is used as a point estimator of relative risk. Test-based confidence intervals are estimated with the help of the asymptotic chi-squared distributed MH-statistic chi 2MHS. The Mantel-extension-chi-squared is used as a test statistic for a dose-response relationship. Both test statistics--the Mantel-Haenszel-chi as well as the Mantel-extension-chi--assume homogeneity of risk across strata, which is rarely present. Also an extended nonparametric statistic, proposed by Terpstra, which is based on the Mann-Whitney-statistics assumes homogeneity of risk across strata. We have earlier defined four risk measures RRkj (k = 1,2,...,4) in the population and considered their estimates and the corresponding asymptotic distributions. In order to overcome the homogeneity assumption we use the delta-method to get "test-based" confidence intervals. Because the four risk measures RRkj are presented as functions of four weights gik we give, consequently, the asymptotic variances of these risk estimators also as functions of the weights gik in a closed form. Approximations to these variances are given. For testing a dose-response relationship we propose a new class of chi 2(1)-distributed global measures Gk and the corresponding global chi 2-test. In contrast to the Mantel-extension-chi homogeneity of risk across strata must not be assumed. These global test statistics are of the Wald type for composite hypotheses.(ABSTRACT TRUNCATED AT 250 WORDS)

A specification strategy for order determination in arma models

In this paper a specification strategy is proposed for the determination of the orders in ARMA models. The strategy is based on two newly defined concepts: the q-conditioned partial auto-regressive function and the p-conditioned partial moving average function. These concepts are similar to the generalized partial autocorrelation function which has been recently suggested for order determination. The main difference is that they are defined and employed in connection with an asymptotically efficient estimation method instead of the rather inefficient generalized Yule-Walker method. The specification is performed by using sequential Wald type tests. In contrast to the traditional testing of hypotheses, these tests use critical values which increase with the sample size at an appropriate rate

Hypothesis tests for linear functional relationships

The analysis of linear functional relationships is considered. Expressions useful for the estimation of both unconstrained and con¬strained parameters are presented. The testing of hypotheses which consist of linear constraints on the parameters is discussed. Wald type and Wilks type test statistics and their asymptotic null dis¬tributions are derived. It appears that these test statistics are not asymptotically equivalent.

Sequential estimation and asymptotic properties in birth-and-death processes

The problem determining and characterizing efficient sequential estimation procedures in linear birth-and-death processes with immigration is considered. Basing on the likelihood function in the case of random observation time some equations of WALD type and an inequality of RAO-CRAMER-WOLFOWITZ type are obtained. Sequential procedures with minimal variance estimators are characterized by linear equations in the sufficient statistic. Using this fact efficiently estimable functions and the associated estimators are determined. With respect to increasing sequences of stopping times the efficient estimators are strongly consistent and asymptotically normal.

Locally most powerful sequential tests for stochastic processes

For a continuous time stochastic process with distribution P ϑ depending on a one-dimensional parameter ϑ the problem of sequentially testing ϑ = 0 against ϑ > 0 is treated. We assume that the process of likelihood ratios has a certain representation which allows to obtain identities of the Wald type for stopping times. These identities are then used to derive a result on locally most powerful tests for which a problem of optimal stopping is solved.

Note—A Replacement Problem using a Wald Identity for Discounted Variables

A fundamental model is presented for determining optimal replacement times for stochastically deteriorating equipment. It is an elementary Markov decision model in which an optimal policy is to replace an item when and only when its operating cost reaches a certain level Wald type identities for discounted random variables are introduced to derive a tractable method for computing these replacement levels. The analysis herein simplifies and corrects earlier work in this area.

The choice of a Wald test on the mean of a normal population

SUMMARY A figure is given that will allow the boundaries corresponding to fixed probabilities of error of a Wald type sequential test on the mean of a normal population with known standard deviation to be found, with reasonable accuracy, without difficulty. Another figure can be used to obtain, quite accurately, the average sample numbers corresponding to the two hypotheses used in constructing the test.

The Choice of a Wald Test on the Mean of a Normal Population

A figure is given that will allow the boundaries corresponding to fixed probabilities of error of a Wald type sequential test on the mean of a normal population with known standard deviation to be found, with reasonable accuracy, without difficulty. Another figure can be used to obtain, quite accurately, the average sample numbers corresponding to the two hypotheses used in constructing the test.

A Note on the Variance of the Distribution of Sample Number in Sequential Probability Ratio Tests

Little published information is available about moments, other than the first, of the distribution of the sample number in Wald type sequential probability ratio tests. Empirical evidence from Monte Carlo studies is presented for the inference that the variance of the sample number is approximately proportional to the square of the average sample number.

Quanta phenomena

This paper proposes a generalised Wald type tests to test the hypothesis of nonlinear restrictions. We circumvent the problem of singularity of the covariance matrix associated with the usual Wald test by proposing a generalised inverse procedure, and an alternative simple procedure which can be approximated by a suitable chi-square distribution. New threshold values are derived to estimate the rank of the covariance matrix. We also propose an operational procedure to test the hypothesis of the Granger non-causality in cointegrated systems.