Nearly optimal truncated group sequential test on binomial proportions

In this article, we systematically study the optimal truncated group sequential test on binomial proportions. Through analysis of the cost structure, average test cost is introduced as a new optimality criterion. According to the new criterion, the optimal tests on different design parameters including the boundaries, success discriminant value, stage sample vector, stage size, and the maximum sample size are defined. Since the computation time in finding optimal designs by exhaustive search is intolerably long, group sequential sample space sorting method and procedures are developed to find the near-optimal ones. In comparison with the international standard ISO2859-1, the truncated group sequential designs proposed in this article can reduce the average test costs around 20%.     

A generalization of anderson's modified sequential probability ratio test

A generalization of Anderson's sequential probability ratio test procedure is proposed in which the continuation region is bounded by a pair of converging lines up to a certain stage of the experiment and later by another pair of converging lines until the procedure is truncated at a predetermined stage of the experiment. The OC and the ASN functions have been derived. For certain parameter values the proposed procedure attains lower average sample numbers than that attainable by any other known procedure.     

A new nonlinear unit root test with Fourier function

Abstract

Traditional unit root tests display a tendency to be nonstationary in the case of structural breaks and nonlinearity. To eliminate this problem this paper proposes a new flexible Fourier form nonlinear unit root test. This test eliminates this problem to add structural breaks and nonlinearity together to the test procedure. In this test procedure, structural breaks are modeled by means of a Fourier function and nonlinear adjustment is modeled by means of an exponential smooth threshold autoregressive (ESTAR) model. The simulation results indicate that the proposed unit root test is more powerful than the Kruse and KSS tests.     

Exponentiality test based on Renyi distance between equilibrium distributions

ABSTRACT

On the basis of Csiszar's φ-divergence discrimination information, we propose a measure of discrepancy between equilibriums associated with two distributions. Proving that a distribution can be characterized by associated equilibrium distribution, a Renyi distance of the equilibrium distributions is constructed that made us to propose an EDF-based goodness-of-fit test for exponential distribution. For comparing the performance of the proposed test, some well-known EDF-based tests and some entropy-based tests are considered. Based on the simulation results, the proposed test has better powers than those of competing entropy-based tests for the alternatives with decreasing hazard rate function. The use of the proposed test is evaluated in an illustrative example.     

A combined p-value test for multiple hypothesis testing

Tests that combine p-values, such as Fisher's product test, are popular to test the global null hypothesis H₀ that each of n component null hypotheses, H₁,…,H_n, is true versus the alternative that at least one of H₁,…,H_n is false, since they are more powerful than classical multiple tests such as the Bonferroni test and the Simes tests. Recent modifications of Fisher's product test, popular in the analysis of large scale genetic studies include the truncated product method (TPM) of Zaykin et al. (2002), the rank truncated product (RTP) test of Dudbridge and Koeleman (2003) and more recently, a permutation based test—the adaptive rank truncated product (ARTP) method of Yu et al. (2009). The TPM and RTP methods require users' specification of a truncation point. The ARTP method improves the performance of the RTP method by optimizing selection of the truncation point over a set of pre-specified candidate points. In this paper we extend the ARTP by proposing to use all the possible truncation points {1,…,n} as the candidate truncation points. Furthermore, we derive the theoretical probability distribution of the test statistic under the global null hypothesis H₀. Simulations are conducted to compare the performance of the proposed test with the Bonferroni test, the Simes test, the RTP test, and Fisher's product test. The simulation results show that the proposed test has higher power than the Bonferroni test and the Simes test, as well as the RTP method. It is also significantly more powerful than Fisher's product test when the number of truly false hypotheses is small relative to the total number of hypotheses, and has comparable power to Fisher's product test otherwise.     

A note on the most powerful invariant test of Rayleigh against exponential distribution

Abstract

The problem of testing Rayleigh distribution against exponentiality, based on a random sample of observations is considered. This problem arises in survival analysis, when testing a linearly increasing hazard function against a constant hazard function. It is shown that for this problem the most powerful invariant test is equivalent to the “ratio of maximized likelihoods” (RML) test. However, since the two families are separate, the RML test statistic does not have the usual asymptotic chi-square distribution. Normal and saddlepoint approximations to the distribution of the RML test statistic are derived. Simulations show that saddlepoint approximation is more accurate than the normal approximation, especially for tail probabilities that are the main values of interest in hypothesis testing.     

An adjusted cumulative Kullback-Leibler information with application to test of exponentiality

Abstract

In order to discriminate between two probability distributions extensions of Kullback–Leibler (KL) information have been proposed in the literature. In recent years, an extension called cumulative Kullback–Leibler (CKL) information is considered by authors which is closely related to equilibrium distributions. In this paper, we propose an adjusted version of CKL based on equilibrium distributions. Some properties of the proposed measure of divergence are investigated. A test of exponentiality based on the adjusted measure, is proposed. The empirical power of the presented test is calculated and compared with some existing standard tests of exponentiality. The results show that our proposed test, for some important alternative distributions, has better performance than some of the existing tests.     

Data-driven smooth test for a location-scale family

A combination of a smooth test statistic and (an approximate) Schwarz's selection rule has been proposed by Inglot, T., Kallenberg, W. C. M. and Ledwina, T. ((1997). Data-driven smooth tests for composite hypotheses. Ann. Statist. 25, 1222–1250) as a solution of a standard goodness-of-fit problem when nuisance parameters are present. In the present paper we modify the above solution in the sense that we propose another analogue of Schwarz's rule and rederive properties of it and the resulting test statistic. To avoid technicalities we restrict our attention to location-scale family and method of moments estimators of its parameters. In a parallel paper [Janic-Wróblewska, A. (2004). Data-driven smooth tests for the extreme value distribution. Statistics, in press] we illustrate an application of our solution and advantages of modification when testing of fit to extreme value distribution.     

Combining the t test and Wilcoxon's rank-sum test

In the two-sample location-shift problem, Student's t test or Wilcoxon's rank-sum test are commonly applied. The latter test can be more powerful for non-normal data. Here, we propose to combine the two tests within a maximum test. We show that the constructed maximum test controls the type I error rate and has good power characteristics for a variety of distributions; its power is close to that of the more powerful of the two tests. Thus, irrespective of the distribution, the maximum test stabilizes the power. To carry out the maximum test is a more powerful strategy than selecting one of the single tests. The proposed test is applied to data of a clinical trial.     

Asymptotic power comparison of T2-type test and likelihood ratio test for a mean vector based on two-step monotone missing data

Abstract

In a 2-step monotone missing dataset drawn from a multivariate normal population, T²-type test statistic (similar to Hotelling’s T² test statistic) and likelihood ratio (LR) are often used for the test for a mean vector. In complete data, Hotelling’s T² test and LR test are equivalent, however T²-type test and LR test are not equivalent in the 2-step monotone missing dataset. Then we interest which statistic is reasonable with relation to power. In this paper, we derive asymptotic power function of both statistics under a local alternative and obtain an explicit form for difference in asymptotic power function. Furthermore, under several parameter settings, we compare LR and T²-type test numerically by using difference in empirical power and in asymptotic power function. Summarizing obtained results, we recommend applying LR test for testing a mean vector.     

A test for the complete independence of high-dimensional random vectors

ABSTRACT

This paper discusses the problem of testing the complete independence of random variables when the dimension of observations can be much larger than the sample size. It is reported that two typical tests based on, respectively, the biggest off-diagonal entry and the largest eigenvalue of the sample correlation matrix lose their control of type I error in such high-dimensional scenarios, and exhibit distinct behaviours in type II error under different types of alternative hypothesis. Given these facts, we propose a permutation test procedure by synthesizing these two extreme statistics. Simulation results show that for finite dimension and sample size the proposed test outperforms the existing methods in various cases.     

The limit test statistic distribution of the maximum value test for right-censored data

ABSTRACT

In this paper, the maximum value test is proposed and considered for two-sample problem solving with lifetime data. This test is a distribution-free test under non-censoring and is a not distribution-free test under censoring. The formula of the limit distribution of the proposed maximal value test is represented in the general case. The distribution of the test statistic has been studied experimentally. Also, we propose the estimate of a p-value calculation of the maximum value test instead of the Monte-Carlo simulation. This test is useful and applicable in case of choosing among the logrank test, the Cox–Mantel test, the Q test and Generalized Wilcoxon tests, for instance, the Gehan's Generalized Wilcoxon test and the Peto and Peto's Generalized Wilcoxon test.     

A two- sample partially sequential probability ratio test

A two-sample partially sequential probability ratio test (PSPRT) is considered for the two-sample location problem with one sample fixed and the other sequential. Observations are assumed to come from two normal poptilatlons with equal and known variances. Asymptotically in the fixed-sample size the PSPRT is a truncated Wald one sample sequential probability test. Brownian motion approximations for boundary-crossing probabilities and expected sequential sample size are obtained. These calculations are compared to values obtained by Monte Carlo simulation.     

An adaptive test for the one-way layout

An adaptive test is proposed for the one-way layout. This test procedure uses the order statistics of the combined data to obtain estimates of percentiles, which are used to select an appropriate set of rank scores for the one-way test statistic. This test is designed to have reasonably high power over a range of distributions. The adaptive procedure proposed for a one-way layout is a generalization of an existing two-sample adaptive test procedure. In this Monte Carlo study, the power and significance level of the F-test, the Kruskal-Wallis test, the normal scores test, and the adaptive test were evaluated for the one-way layout. All tests maintained their significance level for data sets having at least 24 observations. The simulation results show that the adaptive test is more powerful than the other tests for skewed distributions if the total number of observations equals or exceeds 24. For data sets having at least 60 observations the adaptive test is also more powerful than the F-test for some symmetric distributions.     

Hypothesis testing in smoothing spline models

Nonparametric regression models are often used to check or suggest a parametric model. Several methods have been proposed to test the hypothesis of a parametric regression function against an alternative smoothing spline model. Some tests such as the locally most powerful (LMP) test by Cox et al. (Cox, D., Koh, E., Wahba, G. and Yandell, B. (1988). Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models. Ann. Stat., 16, 113–119.), the generalized maximum likelihood (GML) ratio test and the generalized cross validation (GCV) test by Wahba (Wahba, G. (1990). Spline models for observational data. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM.) were developed from the corresponding Bayesian models. Their frequentist properties have not been studied. We conduct simulations to evaluate and compare finite sample performances. Simulation results show that the performances of these tests depend on the shape of the true function. The LMP and GML tests are more powerful for low frequency functions while the GCV test is more powerful for high frequency functions. For all test statistics, distributions under the null hypothesis are complicated. Computationally intensive Monte Carlo methods can be used to calculate null distributions. We also propose approximations to these null distributions and evaluate their performances by simulations.     

A test of normality using nonparametrlic residuals

In this paper, we develop a test of the normality assumption of the errors using the residuals from a nonparametric kernel regression. Contrary to the existing tests based on the residuals from a parametric regression, our test is thus robust to misspecification of the regression function. The test statistic proposed here is a Bera-Jarque type test of skewness and kurtosis. We show that the test statistic has the usual x ²(2) limit distribution under the null hypothesis. In contrast to the results of Rilstone (1992), we provide a set of primitive assumptions that allow weakly dependent observations and data dependent bandwidth parameters. We also establish consistency property of the test. Monte Carlo experiments show that our test has reasonably good size and power performance in small samples and perfornu better than some of the alternative tests in various situations.     

On empirical likelihood test for predictability

Predicting asset prices is a critical issue in statistics and finance. In this article, by incorporating the recent advances in nonparametric approaches, we propose the empirical likelihood test for the predictability for the direction of price changes. Under some regularity conditions, the test statistic has an asymptotic χ² distribution under the null hypothesis that the direction of price change cannot be predicted. This test procedure is easy to implement and presents better finite sample performances than other popular causality tests, as reported in some Monte Carlo experiments.

1. Hightlights

2. We propose a non parametric likelihood test for predictability.

3. The test involves no user-chosen parameter or estimation of covariance matrix.

4. The test is simple to implement and has standard asymptotics.

5. The test has significantly better sizes than several popular tests with satisfactory power.

     

A note on a family of criteria for evaluating test statistics

ABSTRACT

In noting that the usual criteria for choosing an optimal test, Uniform Power and Local Power are at opposite ends of a spectrum of dominance criteria, a complete “Power Dominance” family of criteria for classifying and choosing optimal tests on the basis of their power characteristics is identified, wherein successive orders of dominance attach increasing weight to power close to the null hypothesis. Indices of the extent to which a preferred test has superior power characteristics over other members in its class, and an index of the proximity of a test to the envelope function of alternative tests are also provided. The ideas are exemplified using various optimal test statistics for Normal and Laplace population distributions.     

The modified permutation entropy-based independence test of time series

Abstract

In time series, it is essential to check the independence of data by means of a proper method or an appropriate statistical test before any further analysis. Therefore, among different independence tests, a powerful and productive test has been introduced by Matilla-García and Marín via m-dimensional vectorial process, in which the value of the process at time t includes m-histories of the primary process. However, this method causes a dependency for the vectors even when the independence assumption of random variables is considered. Considering this dependency, a modified test is obtained in this article through presenting a new asymptotic distribution based on weighted chi-square random variables. Also, some other alterations to the test have been made via bootstrap method and by controlling the overlap. Compared with the primary test, it is obtained that not only the modified test is more accurate but also, it possesses higher power.