Testing for multivariate heteroscedasticity

In this article, we propose a testing technique for multivariate heteroscedasticity, which is expressed as a test of linear restrictions in a multivariate regression model. Four test statistics with known asymptotical null distributions are suggested, namely the Wald, Lagrange multiplier (LM), likelihood ratio (LR) and the multivariate Rao F-test. The critical values for the statistics are determined by their asymptotic null distributions, but bootstrapped critical values are also used. The size, power and robustness of the tests are examined in a Monte Carlo experiment. Our main finding is that all the tests limit their nominal sizes asymptotically, but some of them have superior small sample properties. These are the F, LM and bootstrapped versions of Wald and LR tests.     

Bootstrap LR tests of stationarity,common trends and cointegration

This paper considers the likelihood ratio (LR) tests of stationarity, common trends and cointegration for multivariate time series. As the distribution of these tests is not known, a bootstrap version is proposed via a state- space representation. The bootstrap samples are obtained from the Kalman filter innovations under the null hypothesis. Monte Carlo simulations for the Gaussian univariate random walk plus noise model show that the bootstrap LR test achieves higher power for medium-sized deviations from the null hypothesis than a locally optimal and one-sided Lagrange Multiplier (LM) test that has a known asymptotic distribution. The power gains of the bootstrap LR test are significantly larger for testing the hypothesis of common trends and cointegration in multivariate time series, as the alternative asymptotic procedure – obtained as an extension of the LM test of stationarity – does not possess properties of optimality. Finally, it is shown that the (pseudo-)LR tests maintain good size and power properties also for the non-Gaussian series. An empirical illustration is provided.     

ASYMPTOTIC TESTS FOR THE CONTINUOUS TIME MARKOV BRANCHING PROCESS WITH IMMIGRATION

This paper presents limit distributions for the modified score and the likelihood-ratio (LR) statistic for testing a composite hypothesis involving the split intensity and mean of the offspring distribution of the supercritical continuous time Markov branching process allowing immigration (CBPI). The immigration intensity and mean are treated as nuisance parameters.     

Tests for cointegration rank and choice of the alternative

This paper discusses likelihood-ratio (LR) tests on the cointegrating (CI) rank which consider any possible dimension of the CI rank under the alternative. The trace test and lambda-max test are obtained as special cases. Limit quantiles for all the tests in the class are derived. It is found that any of these tests can be used to construct an estimator of the CI rank, with no differences in asymptotic properties when the alternative is fixed. The properties of the class of tests are investigated by local asymptotic analysis, a simulation study and an empirical illustration. It is found that all the tests in the class have comparable power, which deteriorates substantially as the number of random walks increases. Tests constructed for a specific class of alternatives present minor power gains for alternatives in the class, and require the alternative to be far from the null. No test in this class is found to be asymptotically (in-)admissible. Some of the new tests in the class can also be arranged to give a constrained estimator of the CI rank, that restricts the minimum number of common trends. The power gains that these tests can obtain by constraining the minimum number of common trends appears to be limited and outweighted by the risk of inconsistency induced by the constrains. As a consequence, no value of the CI rank should be left untested, unless it can be excluded beyond any reasonable doubt.     

On Small Samples Testing for Frailty Through Homogeneity Test

We derive a test in order to examine the need of modeling survival data using frailty models based on the likelihood ratio (LR) test for homogeneity. Test is developed for both complete and censored samples from a family of baseline distributions that satisfy a closure property. Approach motivated by I-divergence distance is used in order to determine “credible” regions for all parameters of baseline distribution for which homogeneity hypothesis is not rejected. Proposed test outperforms the usual asymptotic LR test both in very small samples with known frailty and for all small sample sizes under misspecified frailty.     

Poisson approximation io the distribution of empiy cells in randomized occupancy

Let M_o denote the number of empty cells when n balls are dropped independently and at random in m cells such that each ball stays in its cell with probability p and falls through with probability 1-p. A Poisson limit is known for M_o when (n/m)→∞. We find a corresponding approximation to the distribution of M_o when m is large but finite, Ihe method is elementary, and yields the rate of convergence to the limit law. Ihe results are new for the classical case (p = 1) as well.     

Posterior likelihood ratios for evaluation of forensic trace evidence given a two-level model on the data

In forensic science, in order to determine whether sets of traces are from the same source or not, it is widely advocated to evaluate evidential value of similarity of the traces by likelihood ratios (LRs). If traces are expressed by measurements following a two-level model with random effects and known variances, closed LR formulas are available given normality, or kernel density distributions, on the effects. For the known variances estimators are used though, which leads to uncertainty on the resulting LRs which is hard to quantify. The above is analyzed in an approach in which both effects and variances are random, following standard prior distributions on univariate data, leading to posterior LRs. For non-informative and conjugate priors, closed LR formulas are obtained that are interesting in structure and generalize a known result given fixed variance. A semi-conjugate prior on the model seems usable in many applications. It is described how to obtain credible intervals using Monte Carlo Markov Chain and regular simulation, and an example is described for comparison of XTC tablets based on MDMA content. In this way, uncertainty on LR estimation is expressed more clearly which makes the evidential value more transparent in a judicial context.     

ON THE NUMBER OF RECORDS NEAR THE MAXIMUM

Recent work has considered properties of the number of observations X_j, independently drawn from a discrete law, which equal the sample maximum X_(n) The natural analogue for continuous laws is the number K_n(a) of observations in the interval (X_(n)–a, X_(n)], where a > 0. This paper derives general expressions for the law, first moment, and probability generating function of K_n(a), mentioning examples where evaluations can be given. It seeks limit laws for n→ and finds a central limit result when a is fixed and the population law has a finite right extremity. Whenever the population law is attracted to an extremal law, a limit theorem can be found by letting a depend on n in an appropriate manner; thus the limit law is geometric when the extremal law is the Gumbel type. With these results, the paper obtains limit laws for ‘top end’ spacings X_(n) - X_(n-j) with j fixed.     

面板数据单位根似然比检验研究

本文采用似然比类检验统计量进行面板单位根检验（简称为LR检验）研究,在局部备择假设成立的条件下,推导了其在无确定项、仅含截距项以及存在线性时间趋势项三种模型下所对应的渐近分布与局部渐近势函数。Monte Carlo模拟结果显示,当面板数据中含确定项（截距项或时间趋势项）时,LR检验水平比LLC和IPS检验水平更接近于给定的显著性检验水平;此外,当面板数据中包含发散个体时,经水平修正后的LR检验势要远远高于经水平修正后的LLC与IPS检验势,其中,经水平修正后的LLC与IPS检验势接近于零。     

Empirical Study of Six Tests for Equality of Populations with Zero-Inflated Continuous Distributions

We evaluated the properties of six statistical methods for testing equality among populations with zero-inflated continuous distributions. These tests are based on likelihood ratio (LR), Wald, central limit theorem (CLT), modified CLT (MCLT), parametric jackknife (PJ), and nonparametric jackknife (NPJ) statistics. We investigated their statistical properties using simulated data from mixed distributions with an unknown portion of non zero observations that have an underlying gamma, exponential, or log-normal density function and the remaining portion that are excessive zeros. The 6 statistical tests are compared in terms of their empirical Type I errors and powers estimated through 10,000 repeated simulated samples for carefully selected configurations of parameters. The LR, Wald, and PJ tests are preferred tests since their empirical Type I errors were close to the preset nominal 0.05 level and each demonstrated good power for rejecting null hypotheses when the sample sizes are at least 125 in each group. The NPJ test had unacceptable empirical Type I errors because it rejected far too often while the CLT and MCLT tests had low testing powers in some cases. Therefore, these three tests are not recommended for general use but the LR, Wald, and PJ tests all performed well in large sample applications.     

Extremal limit theorems for observations separated by random power law waiting times

This paper develops extreme value theory for random observations separated by random waiting times whose exceedence probability falls off like a power law. In the case where the waiting times between observations have an infinite mean, a limit theorem is established, where the limit is comprised of an extremal process whose time index is randomized according to the non-Markovian hitting time process for a stable subordinator. The resulting limit distributions are shown to be solutions of fractional differential equations, where the order of the fractional time derivative coincides with the power law index of the waiting time. The probability that the limit process remains below a threshold is also computed. For waiting times with finite mean but infinite variance, a two-scale argument yields a fundamentally different limit process. The resulting limit is an extremal process whose time index is randomized according to the first passage time of a positively skewed stable Lévy motion with positive drift. This two-scale limit provides a second-order correction to the usual limit behavior.     

Sparse discriminant analysis based on estimation of posterior probabilities

ABSTRACT

Fisher's linear discriminant analysis (FLDA) is known as a method to find a discriminative feature space for multi-class classification. As a theory of extending FLDA to an ultimate nonlinear form, optimal nonlinear discriminant analysis (ONDA) has been proposed. ONDA indicates that the best theoretical nonlinear map for maximizing the Fisher's discriminant criterion is formulated by using the Bayesian a posterior probabilities. In addition, the theory proves that FLDA is equivalent to ONDA when the Bayesian a posterior probabilities are approximated by linear regression (LR). Due to some limitations of the linear model, there is room to modify FLDA by using stronger approximation/estimation methods. For the purpose of probability estimation, multi-nominal logistic regression (MLR) is more suitable than LR. Along this line, in this paper, we develop a nonlinear discriminant analysis (NDA) in which the posterior probabilities in ONDA are estimated by MLR. In addition, in this paper, we develop a way to introduce sparseness into discriminant analysis. By applying L1 or L2 regularization to LR or MLR, we can incorporate sparseness in FLDA and our NDA to increase generalization performance. The performance of these methods is evaluated by benchmark experiments using last_exam17 standard datasets and a face classification experiment.     

Testing additivity in two-way classifications with no replications:the locally best invariant test

Row x column interaction is frequently assumed to be negligible in two-way classifications having one observation per cell. Absence of interaction allows the researcher to estimate experimental error and to proceed with making inferences about row and column effects. If additivity is suspect, it is conventional to test it against a structured alternative. If the structured alternative missspecifies the existing nonadditivity, then the power of the test is low, even if the magnitude of the existing nonadditivity is large. The locally best invariant (LBI) test of additivity is less subject to model misspecification because a particular structural alternative need not be hypothesized. This paper illustrates the LBI test of additivity and compares its power to that of the Johnson-Graybill likelihood ratio (LR) test. The LBI test performs as well as the LR test under a Johnson-Graybill alternative and performs better than the LR test under more general alternatives.     

Comparisons of decision tree methods using water data

This article demonstrates the application of classification trees (decision trees), logistic regression (LR), and linear discriminant function (LDR) to classify data of water quality (i.e., whether the water is fit for drinking on not fit for drinking). The data on water quality were obtained from Pakistan Council of Research in Water Resources (PCRWR) for two cities of Pakistan—one representing industrial environment (Sialkot) and the other one representing non-industrial environment (Narowal). To classify data on water quality, three statistical tools were employed—the Decision Tree methodology using Gini Index, LR, and LDA—using R software library. The results obtained by the said three techniques were compared using misclassification rates (a model with minimum value of misclassification rate is better). It was witnessed that LR performed well than the other two techniques while the Decision trees and LDA performed equally well. But for illustration purposes decision trees technique is comparatively easy to draw and interpret.     

Testing equality of mean vectors in a one-way MANOVA with monotone missing data

In this study, testing the equality of mean vectors in a one-way multivariate analysis of variance (MANOVA) is considered when each dataset has a monotone pattern of missing observations. The likelihood ratio test (LRT) statistic in a one-way MANOVA with monotone missing data is given. Furthermore, the modified test (MT) statistic based on likelihood ratio (LR) and the modified LRT (MLRT) statistic with monotone missing data are proposed using the decomposition of the LR and an asymptotic expansion for each decomposed LR. The accuracy of the approximation for the Chi-square distribution is investigated using a Monte Carlo simulation. Finally, an example is given to illustrate the methods.     

Tests for a family of random-effects covariance structures in a multivariate growth curve model

In this paper we propose test statistics for a general hypothesis concerning the adequacy of multivariate random-effects covariance structures in a multivariate growth curve model with differing numbers of random effects (Lange, N., N.M. Laird, J. Amer. Statist. Assoc. 84 (1989) 241–247). Since the exact likelihood ratio (LR) statistic for the hypothesis is complicated, it is suggested to use a modified LR statistic. An asymptotic expansion of the null distribution of the statistic is obtained. The exact LR statistic is also discussed.     

A monte carlo study of the wald,lm, and lr tests in a heteroscedastic linear model

In this paper, we examine by Monte Carlo experiments the small sample properties of the W (Wald), LM (Lagrange Multiplier) and LR (Likelihood Ratio) tests for equality between sets of coefficients in two linear regressions under heteroscedasticity. The small sample properties of the size-corrected W, LM and LR tests proposed by Rothenberg (1984) are also examined and it is shown that the performances of the size-corrected W and LM tests are very good. Further, we examine the two-stage test which consists of a test for homoscedasticity followed by the Chow (1960) test if homoscedasticity is indicated or one of the W, LM or LR tests if heteroscedasticity should be assumed. It is shown that the pretest does not reduce much the bias in the size when the sizecorrected citical values are used in the W, LM and LR tests.     

Likelihood Ratio Tests for the Non Equivalence of Means

We derive likelihood ratio (LR) tests for the null hypothesis of equivalence that the normal means fall into a practical indifference zone. The LR test can easily be constructed and applied to k ≥ 2 treatments. Simulation results indicate that the LR test might be slightly anticonservative statistically, but when the sample sizes are large, it always produces the nominal level for mean configurations under the null hypothesis. More powerful than the studentized range test, the LR test is a straightforward application that requires only current existing statistical tables, with no complicated computations.     

Inference of Seasonal Long‐memory Time Series with Measurement Error

We consider the Whittle likelihood estimation of seasonal autoregressive fractionally integrated moving‐average models in the presence of an additional measurement error and show that the spectral maximum Whittle likelihood estimator is asymptotically normal. We illustrate by simulation that ignoring measurement errors may result in incorrect inference. Hence, it is pertinent to test for the presence of measurement errors, which we do by developing a likelihood ratio (LR) test within the framework of Whittle likelihood. We derive the non‐standard asymptotic null distribution of this LR test and the limiting distribution of LR test under a sequence of local alternatives. Because in practice, we do not know the order of the seasonal autoregressive fractionally integrated moving‐average model, we consider three modifications of the LR test that takes model uncertainty into account. We study the finite sample properties of the size and the power of the LR test and its modifications. The efficacy of the proposed approach is illustrated by a real‐life example.