Unrestricted Alternative Research Articles

A Generalization of the Savage–Dickey Density Ratio for Testing Equality and Order Constrained Hypotheses

ABSTRACT The Savage–Dickey density ratio is a specific expression of the Bayes factor when testing a precise (equality constrained) hypothesis against an unrestricted alternative. The expression greatly simplifies the computation of the Bayes factor at the cost of assuming a specific form of the prior under the precise hypothesis as a function of the unrestricted prior. A generalization was proposed by Verdinelli and Wasserman such that the priors can be freely specified under both hypotheses while keeping the computational advantage. This article presents an extension of this generalization when the hypothesis has equality as well as order constraints on the parameters of interest. The methodology is used for a constrained multivariate t-test using the JZS Bayes factor and a constrained hypothesis test under the multinomial model.

Nonparametric multi-samples test for simple stochastic ordering against unrestricted alternative

Statistical inferences under simple stochastic ordering for two continuous populations has a long and rich history. But further study for multiple populations is still expected. In this article, we consider samples test for simple stochastic ordering against unrestricted alternative. A test statistic is constructed with the empirical distribution functions and the isotonic regression estimates of distribution functions, and the null asymptotic distribution of the proposed test statistic is given. A bootstrap procedure is suggested for the critical values, and some simulation results are presented to illustrate the proposed test method.

The advantages of using excess returns to model the term structure

We advocate the use of excess returns rather than yields or log prices in analysing the risk neutral dynamics of the term structure. We show that under standard assumptions, excess returns are affine in the risk neutral innovations in the factors. This framework has several important advantages. First, it allows for an easy estimation of models that are more flexible than the AR(1). Indeed, we estimate models with more general dynamics, like ARFIMA(p, d, q), almost as easily as AR(1). Second, within our framework the dimension of the unrestricted model is the same for the AR(1) as it is for the richer models, and does not expand in line with the state vector as it does in a yield or log price framework. This makes it appropriate to test all of these risk neutral dynamic specifications against the same OLS unrestricted alternative. Our results for the US Treasury bond market show that the unrestricted model is preferred to the AR(1) by the Bayesian Information Criterion, but the opposite conclusion is reached for more flexible models. A final advantage of the excess returns framework is that the pricing errors are much lower than for the equivalent log price system.

Detecting Two-Locus Gene-Gene Effects Using Monotonisation of the Penetrance Matrix

As more genetic loci are genotyped simultaneously and as the interest in effects of combinations of loci increases, the need for more powerful analysis methods is increased. In the present paper we present a method aimed at increasing the power of likelihood ratio tests for case-control studies investigating possible two-locus effects. The method is based on the notion that the expected effect pattern of one locus, as well as the expected pattern of a penetrance matrix representing the effect of two loci, is a monotone one. By using an algorithm for making the estimated penetrance matrix monotone, the alternative hypothesis is restricted to monotone penetrance matrices only. The evaluation of the likelihood ratio tests for several underlying monotone models shows that the power is substantially increased by using a monotone alternative as compared to when an unrestricted alternative is used.

Testing for ordered failure rates under general progressive censoring

For exponentially distributed failure times under general progressive censoring schemes, testing procedures for ordered failure rates are proposed using the likelihood ratio principle. Constrained maximum likelihood estimators of the failure rates are found. The asymptotic distributions of the test statistics are shown to be mixtures of chi-square distributions. When testing the equality of the failure rates, a simulation study shows that the proposed test with restricted alternative has improved power over the usual chi-square statistic with an unrestricted alternative. The proposed methods are illustrated using data of survival times of patients with squamous carcinoma of the oropharynx.

Testing for increasing convex order in several populations

Increasing convex order is one of important stochastic orderings. It is very often used in queueing theory, reliability, operations research and economics. This paper is devoted to studying the likelihood ratio test for increasing convex order in several populations against an unrestricted alternative. We derive the null asympotic distribution of the likelihood ratio test statistic, which is precisely the chi-bar-squared distribution. The methodology for computing critical values for the test is also discussed. The test is applied to an example involving data for survival time for carcinoma of the oropharynx.

Disparity based goodness-of-fit tests for and against order restrictions for multinomial models

We consider disparity based test statistics to test the equality of a multinomial probability vector to a given probability vector against an isotonic/antitonic order restriction. The problem of testing the isotonic restriction as a null hypothesis against unrestricted alternatives is also considered. In both cases the asymptotic distributions of these test statistics are found to be of the chi-bar squared type similar to the popular likelihood ratio test statistics for these cases. However our numerical studies demonstrate that depending on the situation, several of the disparity test statistics are more powerful than the latter.

Intraday trade in dealership markets

We develop and test a structural asymmetric information transaction model to characterize the price impact of information when markets are thin. Since orders are accepted individually, the model allows for transaction costs and brokerage fees. Equilibrium demands mixed entry strategies on the part of potentially informed traders. Estimation of the structural parameters is performed using a maximum likelihood procedure on NYSE data. The structural model is rejected primarily because the nonlinear restrictions do not allow for sufficient correlation between price movements and pricing errors. This leads to unreasonably low estimates of the probability of informed trade relative to an unrestricted alternative. The price impact of information is found to be positive and significant, but economically small. This is because although the amount of private information is substantial, the quality of the information signals is poor, particularly in the middle of the trading day. Informed agents do not trade small quantities, which suggests that their ability to divide orders is limited by transaction costs.

Tests against inequality constraints in semiparametric models

We consider the problem of testing that a subvector of a finite dimensional parameter is zero against that it satisfies some inequality constraints in a semiparametric model. To our knowledge, this problem has not been studied in the literature in a general context. Huang (Ann. Statist. 24 (1996) 540) developed maximum likelihood estimation in a class of semiparametric models, and Choi et al. (Ann. Statist. 24 (1996) 84) developed efficient tests against unrestricted alternatives in a large class of semiparametric models which includes the adaptable ones. In this paper, we introduce local tests as a general approach to developing tests against inequality constraints, and apply them in different semiparametric models. In particular, we apply our local tests to develop adaptive tests and efficient tests in the classes of semiparametric models studied by Huang (1996) and Choi et al. (1996), thus extending their results to inequality constrained framework.

Tests of parameters of several gamma distributions with inequality restrictions

Gamma distribution is one of the most used methods of modeling lifetime data. However, testing homogeneity of parameters of m ≥ 3 gamma distributions against order restrictions is almost non-existent in the current literature. We propose two methods to this end: one uses quadratic forms involving ratios of cumulants as test statistic and the other is a stepwise procedure which uses Fisher's method of combining p-values when shape parameters are equal but unknown. Both procedures allow use of arbitrary sample sizes of m populations. Test of the inequality restrictions as a null hypothesis against unrestricted alternatives is also considered. A Monte Carlo study of power at various alternatives shows that both methods are competitive when they are applicable.

A Unified Approach to Likelihood Inference on Stochastic Orderings in a Nonparametric Context

Abstract For data in a two-way contingency table with ordered margins, we consider various hypotheses of stochastic orders among the conditional distributions considered by rows and show that each is equivalent to requiring that an invertible transformation of the vectors of conditional row probabilities satisfies an appropriate set of linear inequalities. This leads to the construction of a general algorithm for maximum likelihood estimation under multinomial sampling and provides a simple framework for deriving the asymptotic distribution of log-likelihood ratio tests. The usual stochastic ordering and the so called uniform and likelihood ratio orderings are considered as special cases. In particular, for each of these three orderings we determine the transformation required to apply the estimation algorithm; we then consider testing the hypothesis that the rows are identically distributed against the alternative that they are stochastically ordered, as well as testing each stochastic order against an unrestricted alternative. We show that in all cases the test statistics are asymptotically distributed as a mixture of chi-squared distributions, with weights determined by the information matrix. By exploiting the special structure of this matrix in these three cases, we find tight upper and lower bounds to the distribution of all test statistics. These bounding distributions are free of nuisance parameters and relatively easy to compute. Two examples are presented to illustrate the methodology and the required computations needed to apply these techniques.

A Unified Approach to Likelihood Inference on Stochastic Orderings in a Nonparametric Context

For data in a two-way contingency table with ordered margins, we consider various hypotheses of stochastic orders among the conditional distributions considered by rows and show that each is equivalent to requiring that an invertible transformation of the vectors of conditional row probabilities satisfies an appropriate set of linear inequalities. This leads to the construction of a general algorithm for maximum likelihood estimation under multinomial sampling and provides a simple framework for deriving the asymptotic distribution of log-likelihood ratio tests. The usual stochastic ordering and the so called uniform and likelihood ratio orderings are considered as special cases. In particular, for each of these three orderings we determine the transformation required to apply the estimation algorithm; we then consider testing the hypothesis that the rows are identically distributed against the alternative that they are stochastically ordered, as well as testing each stochastic order against an unrestricted alternative. We show that in all cases the test statistics are asymptotically distributed as a mixture of chi-squared distributions, with weights determined by the information matrix. By exploiting the special structure of this matrix in these three cases, we find tight upper and lower bounds to the distribution of all test statistics. These bounding distributions are free of nuisance parameters and relatively easy to compute. Two examples are presented to illustrate the methodology and the required computations needed to apply these techniques.

A Hotelling′s T2-Type Statistic for Testing against One-Sided Hypotheses

Suppose that Y is distributed as multivariate normal with unknown covariance matrix and that N independent observations are available on Y. An important special case of the problem studied in this paper is that of testing the null hypothesis that the mean of Y is zero against the alternative that it lies in the positive orthant. We propose a statistic T 2 for this testing problem; this closely resembles the well-known Hotelling′s statistic for testing against the unrestricted alternative, and it is also related to some other statistics in the literature. It turns out that our T 2 and the likelihood ratio test (LRT) statistic are equivalent asymptotically but not for finite samples. Some simulations and a comparison of the critical regions of T 2 and LRT in some special cases show that neither of the two can dominate the other uniformly over the parameter space in terms of power. A comparison of the critical regions of T 2 and LRT leads us to conjecture that T 2 is likely to be more (respectively, less) powerful than the LRT when the mean of the multivariate distribution is close to (respectively, away from) the boundary of the parameter space. Computation of the p-value for T 2 is almost just as straightforward as it is for LRT.

Tests of overidentification and predeterminedness in simultaneous equation models

A set of simultaneous equations that is a part of a larger complete system is considered. One hypothesis concerning this set of equations is that it is overidentified as a block by a block of zero restrictions. Another hypothesis is that a subset of the current endogenous variables is uncorrelated with the disturbances in the set of simultaneous equations (‘econometric predeterminedness’). Likelihood ratio, Lagrange multiplier, and Wald tests are derived for the overidentification hypothesis against unrestricted alternatives, for the predeterminedness hypothesis against unrestricted alternatives, and for the predeterminedness hypothesis against overidentified alternatives. The tests are derived on the basis of normality but are asymptotically valid under very general conditions. This study generalizes and unifies many test procedures proposed previously. Some new tests are developed.

Distribution-free tests in randomized blocks

Classes of distribution-free tests are proposed for testing homogeneity against order restricted as well as unrestricted alternatives in randomized block designs with multiple observations per cell. Allowing for different interblock scoring schemes, these tests are constructed based on the method of within block rankings. Asymptotic distributions (cell sizes tending to infinity) of these tests are derived under the assumption of homogeneity. The Pitman asymptotic relative efficiencies relative to the least squares statistics are studied. It is shown that when blocks are governed by different distributions, adaptive choice of scores within each block results in asymptotically more efficient tests as compared with methods that ignore such information. Monte Carlo simulations of selected designs indicate that the method of within block rankings is more power robust with respect to differing block distributions.