The sizes of the three popular asymptotic tests for testing homogeneity of two binomial proportions

In statistical hypothesis testing it is important to ensure that the type I error rate is preserved under the nominal level. This paper addresses the sizes and the type I errors rates of the three popular asymptotic tests for testing homogeneity of two binomial proportions: the chi-square test with and without continuity correction, the likelihood ratio test. Although it has been recognized that, based on limited simulation studies, the sizes of the tests are inflated in small samples, it has been thought that the sizes are well preserved under the nominal level when the sample size is sufficiently large. But, Loh [1989. Bounds on the size of the χ² test of independence in a contingency table. Ann. Statist. 17, 1709-1722], and Loh and Yu [1993. Bounds on the size of the likelihood ratio test of independence in a contingency table. J. Multivariate Anal. 45, 291-304] showed theoretically that the sizes are always greater than or equal to the nominal level when the sample size is infinite. In this paper, we confirm their results by computing the large-sample lower bounds of the sizes numerically. Applying complete enumeration which does not have any error, we confirm again the results by computing the sizes precisely on computer in moderate sample sizes. When the sample sizes are unbalanced, the peaks of the type I error rates occur at the extremes of the nuisance parameter. But, the type I error rates of the three tests are close to the nominal level in most values of the nuisance parameter except the extremes. We also find that, when the sample sizes are severely unbalanced and the value of the nuisance parameter is very small, the size of the chi-square test with continuity correction can exceed the nominal level excessively (for instance, the size could be at least 0.877 at 5% nominal level in some cases).     

Empirical likelihood based diagnostics for heteroscedasticity in partial linear models

In this paper, we propose a diagnostic technique for checking heteroscedasticity based on empirical likelihood for the partial linear models. We construct an empirical likelihood ratio test for heteroscedasticity. Also, under mild conditions, a nonparametric version of Wilk’s theorem is derived, which says that our proposed test has an asymptotic chi-square distribution. Simulation results reveal that the finite sample performance of our proposed test is satisfactory in both size and power. An empirical likelihood bootstrap simulation is also conducted to overcome the size distortion in small sample sizes.     

Backoff hierarchical class n-gram language models: effectiveness to model unseen events in speech recognition

In this paper, we introduce the backoff hierarchical class n-gram language models to better estimate the likelihood of unseen n-gram events. This multi-level class hierarchy language modeling approach generalizes the well-known backoff n-gram language modeling technique. It uses a class hierarchy to define word contexts. Each node in the hierarchy is a class that contains all the words of its descendant nodes. The closer a node to the root, the more general the class (and context) is. We investigate the effectiveness of the approach to model unseen events in speech recognition. Our results illustrate that the proposed technique outperforms backoff n-gram language models. We also study the effect of the vocabulary size and the depth of the class hierarchy on the performance of the approach. Results are presented on Wall Street Journal (WSJ) corpus using two vocabulary set: 5000 words and 20,000 words. Experiments with 5000 word vocabulary, which contain a small numbers of unseen events in the test set, show up to 10% improvement of the unseen event perplexity when using the hierarchical class n-gram language models. With a vocabulary of 20,000 words, characterized by a larger number of unseen events, the perplexity of unseen events decreases by 26%, while the word error rate (WER) decreases by 12% when using the hierarchical approach. Our results suggest that the largest gains in performance are obtained when the test set contains a large number of unseen events.     

On Birnbaum–Saunders inference

The Birnbaum–Saunders distribution, also known as the fatigue-life distribution, is frequently used in reliability studies. We obtain adjustments to the Birnbaum–Saunders profile likelihood function. The modified versions of the likelihood function were obtained for both the shape and scale parameters, i.e., we take the shape parameter to be of interest and the scale parameter to be of nuisance, and then consider the situation in which the interest lies in performing inference on the scale parameter with the shape parameter entering the modeling in nuisance fashion. Modified profile maximum likelihood estimators are obtained by maximizing the corresponding adjusted likelihood functions. We present numerical evidence on the finite sample behavior of the different estimators and associated likelihood ratio tests. The results favor the adjusted estimators and tests we propose. A novel aspect of the profile likelihood adjustments obtained in this paper is that they yield improved point estimators and tests. The two profile likelihood adjustments work well when inference is made on the shape parameter, and one of them displays superior behavior when it comes to performing hypothesis testing inference on the scale parameter. Two empirical applications are briefly presented.     

Maximum likelihood ratio test for the stability of sequence of Gaussian random processes

Based on the Karhunen-Loeve expansion, the maximum likelihood ratio test for the stability of sequence of Gaussian random processes is investigated. The likelihood function is based on the first p scores of eigenfunctions in the Karhunen-Loeve expansion for Gaussian random processes. Though the scores are unobservable, we show that the effect of the difference between scores and their estimators is negligible as the sample size tends to infinity. The asymptotic distribution is proved to be the Gumbel extreme value distribution. Under the alternative the test is shown to be consistent. For different choices of p, simulation results show that the test behaves quite well in finite samples. The test procedure is also applied to the annual temperature data of central England. The results show that the temperatures have risen in the last twenty years, however there is no evidence to show that the autocovariance functions of the temperatures have changed among the range of the observations.     

One-sided multiple comparisons for treatment means with a control mean

We study comparisons of several treatments with a common control when it is believed a priori that the treatment means, μ_i, are at least as large as the control mean, μ₀. In this setting, which is called a tree ordering, we study multiple comparisons that determine whether μ_i>μ₀ or μ_i=μ₀ for each treatment. The classical procedure by Dunnett (1955) and the step-down and step-up techniques by [Dunnett and Tamhane, 1991] and [Dunnett and Tamhane, 1992] are well known. The results in Marcus and Talpaz (1992) provide multiple comparisons based on the maximum likelihood estimates restricted by the tree ordering. We also study two-stage procedures that consist of the likelihood ratio test of homogeneity with the alternative constrained by the tree ordering followed by two-sample t comparisons with possibly different critical values for the two-sample comparisons. Marcus et al. (1976) discuss the use of closed tests in such situations and propose using a closed version of the restricted likelihood ratio test. We describe step-down versions of the Marcus-Talpaz, the two-stage, and the likelihood ratio procedures, as well as a closed version of the Marcus-Talpaz multiple comparison procedure. Using Monte Carlo techniques, we study the familywise errors and powers of these procedures and make some recommendations concerning techniques that perform well for all tree ordered mean vectors.     

Improved inference on a scalar fixed effect of interest in nonlinear mixed-effects models

Likelihood-based inference on a scalar fixed effect of interest in nonlinear mixed-effects models usually relies on first-order approximations. If the sample size is small, tests and confidence intervals derived from first-order solutions can be inaccurate. An improved test statistic based on a modification of the signed likelihood ratio statistic is presented which was recently suggested by Skovgaard [1996. An explicit large-deviation approximation to one-parameter tests. Bernoulli 2, 145-165]. The finite sample behaviour of this statistic is investigated through a set of simulation studies. The results show that its finite-sample null distribution is better approximated by the standard normal than it is for its first-order counterpart. The R code used to run the simulations is freely available.     

Finite-Sample Adjustments for Homogeneity and Symmetry Tests in Systems of Demand Equations: A Monte Carlo Evaluation

This paper gives simulation results comparing the finite-sample performance of three commonly used homogeneity and symmetry asymptotic tests, and some size-corrected tests that can be used when the sample size is small. The results suggest that such finite-sample corrections can be effective in bringing the empirical sizes of the tests closer to their nominal levels. They also suggest that the likelihood ratio test is, in general, more reliable than the Wald and Lagrange multiplier tests. Finally, it is shown that size-corrections of homogeneity tests tend to introduce reductions in power, which can be very large when bootstrap corrections are used. An application to a well known data set is also presented.     

Truncated sequential probability ratio test

It is well known that in the testing of a simple hypothesis H versus a simple alternative K, the sequential probability ratio test (SPRT) has the smallest average sample number (ASN) under H and K. Compared to the corresponding best fixed sample size (FSS) test, the saving in the average number of samples under H or K in the SPRT is significant. However, when the parameter values of the sample distribution lie between those hypothesized under H and K, the ASN for the SPRT can become much larger than the sample size of the corresponding FSS test, especially for small probabilities of error. It is shown here that a properly truncated SPRT can eliminate this undesirable feature. For small probabilities of error, truncating the SPRT at the sample size needed for the corresponding FSS test serves as a remedy, while the test is essentially unaffected when the samples are distributed according to H or K.     

ANOVA extensions for mixed discrete and continuous data

This paper is concerned with ANOVA-like tests in the context of mixed discrete and continuous data. The likelihood ratio approach is used to obtain a location test in the mixed data setting after specifying a general location model for the joint distribution of the mixed discrete and continuous variables. The approach allows the problem to be treated from a multivariate perspective to simultaneously test both the discrete and continuous parameters of the model, thus avoiding the problem of multiple significance testing. Moreover, associations among variables are accounted for, resulting in improved power performance of the test. Unlike existing distance-based alternatives which rely on asymptotic theory, the likelihood ratio test is exact. In addition, it can be viewed as an extension to the mixed data setting of the classical multivariate ANOVA. We compare its performance against those of currently available tests via Monte Carlo simulations. Two real-data examples are presented to illustrate the methodology.     

Improved testing inference in mixed linear models

Mixed linear models are commonly used in repeated measures studies. They account for the dependence amongst observations obtained from the same experimental unit. Often, the number of observations is small, and it is thus important to use inference strategies that incorporate small sample corrections. In this paper, we develop modified versions of the likelihood ratio test for fixed effects inference in mixed linear models. In particular, we derive a Bartlett correction to such a test, and also to a test obtained from a modified profile likelihood function. Our results generalize those in [Zucker, D.M., Lieberman, O., Manor, O., 2000. Improved small sample inference in the mixed linear model: Bartlett correction and adjusted likelihood. Journal of the Royal Statistical Society B, 62, 827-838] by allowing the parameter of interest to be vector-valued. Additionally, our Bartlett corrections allow for random effects nonlinear covariance matrix structure. We report simulation results which show that the proposed tests display superior finite sample behavior relative to the standard likelihood ratio test. An application is also presented and discussed.     

On using cache conscious clustering for improving OODBMS performance

The two main techniques of improving I/O performance of Object Oriented Database Management Systems (OODBMS) are clustering and buffer replacement. Clustering is the placement of objects accessed near to each other in time into the same page. Buffer replacement involves the selection of a page to be evicted, when the buffer is full. The page evicted ideally should be the page needed least in the future. These two techniques both influence the likelihood of a requested object being memory resident. We believe an effective way of reducing disk I/O is to take advantage of the synergy that exists between clustering and buffer replacement. Hence, we design a framework, whereby clustering algorithms incorporating buffer replacement cache behaviour can be conveniently employed for enhancing the I/O performance of OODBMS. We call this new type of clustering algorithm, Cache Conscious Clustering (C3). In this paper, we present the C3 framework, and a C3 algorithm that we have developed, namely C3-GP. We have tested C3-GP against three well known clustering algorithms. The results show that C3-GP out performs them by up to 40% when using popular buffer replacement algorithms such as LRU and CLOCK. C3-GP offers the same performance as the best existing clustering algorithm when the buffer size compared to the database size is very small.     

Voice activity detection based on statistical models and machine learning approaches

The voice activity detectors (VADs) based on statistical models have shown impressive performances especially when fairly precise statistical models are employed. Moreover, the accuracy of the VAD utilizing statistical models can be significantly improved when machine-learning techniques are adopted to provide prior knowledge for speech characteristics. In the first part of this paper, we introduce a more accurate and flexible statistical model, the generalized gamma distribution (GΓD) as a new model in the VAD based on the likelihood ratio test. In practice, parameter estimation algorithm based on maximum likelihood principle is also presented. Experimental results show that the VAD algorithm implemented based on GΓD outperform those adopting the conventional Laplacian and Gamma distributions. In the second part of this paper, we introduce machine learning techniques such as a minimum classification error (MCE) and support vector machine (SVM) to exploit automatically prior knowledge obtained from the speech database, which can enhance the performance of the VAD. Firstly, we present a discriminative weight training method based on the MCE criterion. In this approach, the VAD decision rule becomes the geometric mean of optimally weighted likelihood ratios. Secondly, the SVM-based approach is introduced to assist the VAD based on statistical models. In this algorithm, the SVM efficiently classifies the input signal into two classes which are voice active and voice inactive regions with nonlinear boundary. Experimental results show that these training-based approaches can effectively enhance the performance of the VAD.     

Size and power properties of some tests in the Birnbaum-Saunders regression model

The Birnbaum-Saunders distribution has been used quite effectively to model times to failure for materials subject to fatigue and for modeling lifetime data. In this paper we obtain asymptotic expansions, up to order n^−1/2 and under a sequence of Pitman alternatives, for the non-null distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the Birnbaum-Saunders regression model. The asymptotic distributions of all four statistics are obtained for testing a subset of regression parameters and for testing the shape parameter. Monte Carlo simulation is presented in order to compare the finite-sample performance of these tests. We also present two empirical applications.     

Testing for jumps in the EGARCH process

This paper considers testing for jumps in the exponential GARCH (EGARCH) models with Gaussian and Student-t innovations. The Wald and log likelihood ratio tests contain a nuisance parameter unidentified under the null hypothesis of no jumps, and hence are unavailable for this problem, because jump probability and variance of jumps in the test statistic cannot be estimated under the null hypothesis of no jumps. It is shown that the nuisance parameter is cancelled out in the Lagrange multiplier (LM) test statistic, and hence that the test is nuisance parameter-free. The one-sided test is also proposed using the nonnegative constraint on jump variance. The actual size and power of the tests are examined in a Monte Carlo experiment. The test is applied to daily returns of S&P 500 as an illustrative example.     

An extension of likelihood-ratio-test for testing linear hypotheses in the baseline-category logit model

A new family of test statistics for testing linear hypotheses in baseline-category logit models is introduced and its asymptotic distribution is obtained. The new family is a natural extension of the classical likelihood ratio test. A simulation study is carried out to find new test statistics that offer an attractive alternative to the classical likelihood ratio test in terms of both exact size and exact power.     

Bootstrapping the likelihood ratio cointegration test in error correction models with unknown lag order

The finite-sample size and power properties of bootstrapped likelihood ratio system cointegration tests are investigated via Monte Carlo simulations when the true lag order of the data generating process is unknown. Recursive bootstrap schemes are employed which differ in the way in which the lag order is chosen. The order is estimated by minimizing different information criteria and by combining the corresponding order estimates. It is found that, in comparison to the standard asymptotic likelihood ratio test based on an estimated lag order, bootstrapping can lead to improvements in small samples even when the true lag order is unknown, while the power loss is moderate.     

A Cascade Visual Front End for Speaker Independent Automatic Speechreading

We propose a three-stage pixel-based visual front end for automatic speechreading (lipreading) that results in significantly improved recognition performance of spoken words or phonemes. The proposed algorithm is a cascade of three transforms applied on a three-dimensional video region-of-interest that contains the speaker's mouth area. The first stage is a typical image compression transform that achieves a high-energy, reduced-dimensionality representation of the video data. The second stage is a linear discriminant analysis-based data projection, which is applied on a concatenation of a small amount of consecutive image transformed video data. The third stage is a data rotation by means of a maximum likelihood linear transform that optimizes the likelihood of the observed data under the assumption of their class-conditional multivariate normal distribution with diagonal covariance. We applied the algorithm to visual-only 52-class phonetic and 27-class visemic classification on a 162-subject, 8-hour long, large vocabulary, continuous speech audio-visual database. We demonstrated significant classification accuracy gains by each added stage of the proposed algorithm which, when combined, can achieve up to 27% improvement. Overall, we achieved a 60% (49%) visual-only frame-level visemic classification accuracy with (without) use of test set viseme boundaries. In addition, we report improved audio-visual phonetic classification over the use of a single-stage image transform visual front end. Finally, we discuss preliminary speech recognition results.     

Axis estimation and grouping of rotationally symmetric object segments

This paper proposes a new method for estimating the symmetric axis of a pottery from its small fragment using surface geometry. Also, it provides a scheme for grouping such fragments into shape categories using distribution of surface curvature. For automatic assembly of pot from broken sherds, axis estimation is an important task and when a fragment is small, it is difficult to estimate axis orientation since it looks like a patch of a sphere and conventional methods mostly fail. But the proposed method provides fast and robust axis estimation by using multiple constraints. The computational cost is also too lowered. To estimate the symmetric axis, the proposed algorithm uses three constraints: (1) The curvature is constant on a circumference C_H. (2) The curvature is invariant in any scale. (3) Also the principal curvatures does not vary on C_H. C_H is a planar circle which is one of all the possible circumferences of a pottery or sherd. A hypothesis test for axis is performed using maximum likelihood. The variance of curvature, multi-scale curvature and principal curvatures is computed in the likelihood function. We also show that the principal curvatures can be used for grouping of sherds. The grouping of sherds will reduce the computation significantly by omitting impossible configurations in broken pottery assembly process.     

Generalized likelihood ratio test for varying-coefficient models with different smoothing variables

Varying-coefficient models are popular multivariate nonparametric fitting techniques. When all coefficient functions in a varying-coefficient model share the same smoothing variable, inference tools available include the F-test, the sieve empirical likelihood ratio test and the generalized likelihood ratio (GLR) test. However, when the coefficient functions have different smoothing variables, these tools cannot be used directly to make inferences on the model because of the differences in the process of estimating the functions. In this paper, the GLR test is extended to models of the latter case by the efficient estimators of these coefficient functions. Under the null hypothesis the new proposed GLR test follows the χ²-distribution asymptotically with scale constant and degree of freedom independent of the nuisance parameters, known as Wilks phenomenon. Further, we have derived its asymptotic power which is shown to achieve the optimal rate of convergence for nonparametric hypothesis testing. A simulation study is conducted to evaluate the test procedure empirically.