Wald-type Test Statistic Research Articles

On tests of rate ratio under standard inverse sampling

Inverse sampling suggests one continues to sample subjects until a pre-specified number of rare events of interest is observed. It is generally considered to be more appropriate than the usual binomial sampling when the subjects come sequentially, when the response probability is rare, and when maximum likelihood estimators of some epidemiological measures are undefined under binomial sampling. Reliable but conservative exact conditional procedure for the ratio of the response probabilities of subject without the attribute of interest has been studied. However, such a procedure is inapplicable to the risk ratio (i.e., ratio of the response probabilities of subject with the attribute of interest). In this paper, we investigate various test statistics (namely Wald-type, score and likelihood ratio test statistics) for testing non-unity risk ratio under standard inverse sampling scheme, which suggests one continue to sample until the predetermined number of index subjects with the attributes of interest is observed. Both asymptotic and numerical approximate unconditional methods are considered for P-value calculation. Performance of these test procedures are evaluated under different settings by means of Monte Carlo simulation. In general, the Wald-type test statistic is preferable for its satisfactory and stable performance with approximate unconditional procedures. The methodologies are illustrated with a real example from a heart disease study.

Comparing two samples by penalized logistic regression

Inference based on the penalized density ratio model is proposed and studied. The model under consideration is specified by assuming that the log–likelihood function of two unknown densities is of some parametric form. The model has been extended to cover multiple samples problems while its theoretical properties have been investigated using large sample theory. A main application of the density ratio model is testing whether two, or more, distributions are equal. We extend these results by arguing that the penalized maximum empirical likelihood estimator has less mean square error than that of the ordinary maximum likelihood estimator, especially for small samples. In fact, penalization resolves any existence problems of estimators and a modified Wald type test statistic can be employed for testing equality of the two distributions. A limited simulation study supports further the theory.

Comment on: Threshold Autoregressions With a Unit Root

In this paper we revisit the results in Caner and Hansen (2001), where the authors obtained novel limiting distributions of Wald type test statistics for testing for the presence of threshold nonlinearities in autoregressive models containing unit roots. Using the same framework, we obtain a new formulation of the limiting distribution of the Wald statistic for testing for threshold effects, correcting an expression that appeared in the main theorem presented by Caner and Hansen. Subsequently, we show that under a particular scenario that excludes stationary regressors such as lagged dependent variables and despite the presence of a unit root, this same limiting random variable takes a familiar form that is free of nuisance parameters and already tabulated in the literature, thus removing the need to use bootstrap based inferences. This is a novel and unusual occurrence in this literature on testing for the presence of nonlinear dynamics.

Lag‐augmented two‐ and three‐stage least squares estimators for integrated structural dynamic models

Summary We consider a lag-augmented two- or three-stage least-squares estimator for a structural dynamic model of non-stationary and possibly cointegrated variables without the prior knowledge of unit roots or rank of cointegration. We show that the conventional two-and three-stage least-squares estimators are consistent but contain non-standard distributions without the strict exogeneity assumption; hence the conventional Wald type test statistics may not be chi-square distributed. We propose a lag order augmented two- or three-stage least-squares estimator that is consistent and asymptotically normally distributed. Limited Monte Carlo studies are conducted to shed light on the finite sample properties of various estimators.

Lag-Augmented Two- and Three-Stage Least Squares Estimators for Integrated Structural Dynamic Models

We consider a lag-augmented two- or three-stage least squares estimator for a structural dynamic model of nonstationary and possibly cointegrated variables without the prior knowledge of unit roots or rank of cointegration. We show that the conventional two- and three-stage least squares estimators are consistent but contain nonstandard distributions without the strict exogeneity assumption, hence the conventional Wald type test statistics may not be chi-square distributed. We propose a lag order augmented two- or three-stage least squares estimator that is consistent and asymptotically normally distributed. Limited Monte Carlo studies are conducted to shed light on the finite sample properties of various estimators.

Inference in the indeterminate parameters problem

We face an indeterminate parameters problem when there are two sets of parameters, x and g, say, such that the null hypothesis H0:x=x0 makes the likelihood independent of g. A consequence of indeterminacy is the singularity of the information matrix. For this problem the standard results, such as the asymptotic chi-squared distribution of the Wald test statistic, are generally false. In the paper we propose an estimator of the parameters of interest, x, so that a Wald-type test statistic can be used for testing H0. Such an estimator is obtained through the maximization of a modified (penalized) log-likelihood function. We show that a solution to the (penalized) likelihood equation is consistent and asymptotically normally distributed with variance-covariance matrix approximated by the Moore-Penrose pseudoinverse of the information matrix. These properties allow one to construct a Wald-type test statistic useful for inferential purposes.

Analysis of the risk difference of marginal and conditional probabilities in an incomplete correlated [formula omitted] table

Incomplete correlated 2 × 2 tables are common in infectious disease studies and 2-step treatment studies in which one of the comparative measures of interest is the rate/risk difference (RD) between the marginal probability and the conditional probability. In this paper, we consider an interval hypothesis test about the RD in an incomplete correlated 2 × 2 table. We propose two asymptotic test statistics, a Wald-type (sample-based) test statistic and a constrained maximum likelihood estimation (CMLE-based) test statistic to test the interval hypothesis. The sample sizes based on the above two tests for two types of analyses: hypothesis testing and confidence interval estimation are proposed. The true type I error rates and powers of the two tests are computed by enumerating the exact probabilities in the rejection region. Our empirical results show that the CMLE-based test performs better than the sample-based test in the sense that its significance level is much closer to the prespecified nominal level, its confidence interval estimator generally possesses shorter confidence interval width with coverage probability close to the prespecified confidence level. Two real examples are used to illustrate the proposed methods.

Analysis of clustered matched-pair data for a non-inferiority study design.

Hypothesis testing of matched-pair data has been adapted for equivalence (one- and two-sided) study designs; however, a statistical problem arises when more than one measurement is recorded for each study subject. Ignoring the correlation between the repeated measurements per subject may underestimate the standard error of the parameter estimate. A method of testing for non-inferiority or equivalence of clustered matched-pair data is not yet available. This paper proposes a Wald-type test statistic for a non-inferiority hypothesis and applies it to the comparison of two diagnostic procedures.

Statistical Properties of the Two-Stage Least Squares Estimator Under Cointegration

We derive the limiting properties of the two-stage least squares estimator of an equation in a dynamic simultaneous model when variables are nonstationary and cointegrated. The implication on hypothesis testing is also discussed. It is shown that in a structural equation approach what one needs to worry about are the classical issues of identification and estimation, not nonstationarity and cointegration. Conventional formulae for computing the asymptotic covariance of the 2SLS estimator and the Wald-type test statistics remain good approximations despite the fact that variables may be integrated.

Estimating odds-ratio: Homogeneity constraints

This paper considers the asymptotic theory of estimating odds ratios ink tables when there is uncertain prior information about homogeneity constraint on them. Using a preliminary test approach, we propose seven estimators and study their properties of asymptotic dominance under local alternatives. In the process, we propose a Wald-type test statistic for testing homogeneity and obtain its asymptotic distribution under local alternatives.

Inference on a Structural Parameter in Instrumental Variables Regression with Weak Instruments

In this paper we consider the problem of making inference on a structural parameter in instrumental variables regression when the instruments are only weakly correlated with the explanatory endogenous variables. Adopting a local-to-zero assumption as in Staiger and Stock (1994) on the coefficients of the instruments in the first stage equation, the asymptotic distributions of various test statistics are derived under a limited infomation framework. We show that Wald-type test statistics are not asymptotically pivotal, thus (1 - a)*100% confidence intervals implied by those test statistics can have zero coverage probability if standard asymptotic distribution theory is used. In contrast, likelihood-type test statistics are asymptotically pivotal when the model is just identified thus providing valid confidence intervals. Even when the model is overidentified, we show that the distributions of the likelihood-type test statistics are asymptotically bounded from above by a chi- square distribution with degrees of freedom given by the number of instruments. Hence, we can always invert the likelihood-type statistics to obtain valid, although conservative, confidence intervals. The confidence intervals obtained by using this bounding distribution are compared with those obtained by using the standard chi-square 1 asymptotic distribution and an alternative bounding distribution, a transformation of the distribution of the Wilks statistic, suggested by Dufour (1994). Using Monte Carlo methods, confidence intervals based on our chi-square bounding distribution are shown to be tighter in finite samples than those based on the Wilks bounding distribution.

Band Covariance Matrix Estimation Using Restricted Residuals: A Monte Carlo Analysis

Using Monte Carlo simulations, the authors examine the performance of Wald-type test statistics based on alternative versions of a heteroskedasticity consistent band covariance matrix estimator that is algorithmically constrained to be positive definite in finite samples. They find that the test statistic based on the originally proposed estimator tends to result in excessive type I errors. This problem can be alleviated to some extent by employing a quasi-maximum likelihood procedure. However, by simply using restricted, as opposed to the usual OLS residuals when constructing the band covariance matrix estimator, excessive type I errors can be substantially reduced, if not eliminated. Copyright 1995 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.

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.

Tests of non-nested linear regression models subject to linear restrictions

The standard methods for testing between non-nested or separate regression models require that the models have the same dependent variable. If the models are subject to non-homogeneous linear restrictions, substituting out these restrictions results in a redefinition of the dependent variable before estimation and the standard results will not apply. We derive a Wald-type test statistic to cater for this situation.