Testing for multiple upper and lower outliers in an exponential sample

Approximating a class of goodness-of-fit test statistics

In this paper, we propose an algorithm to derive the exact distributions of discordancy tests for exponential samples under the slippage alternative providing that their survival functions involve the linear combinations of independent and identically distributed exponential random variables with arbitrary real coefficients. In addition, we define the various performance measures in terms of conditional probabilities that the observed value of the test statistic exceeds the critical value given that the contaminants have the specific position numbers in the ordered sample. These make possible to calculate various performance measures of discordancy tests for the exponential samples to any desired degree of accuracy. For the purpose of illustration, we derive the distributions of the maximum likelihood ratio tests for testing single and multiple outliers in the exponential samples and then we calculate their performance measures accurately to six decimal places. Moreover, the definitions of the performance criteria are not restricted to the discordancy tests for exponential samples only, they are also equally applicable to the discordancy tests for samples from other distributions.

A new distribution free inferential procedure for comparing paired observations, called the ‘rank difference test’, is presented. The new procedure is based on ranks, and is applicable as a robust alternative to the related f test in all situations where the Wilcoxon signed ranks test is applicable. In additions it may be applied to ordinal data or operational measures which do not meet the assumptions underlying the Wilcoxon. Since the rank difference statistic, D, can take on half‐integer, as well as integer, values, it has continuity advantages over the Wilcoxon for small samples. The exact null distribution of the rank difference test statistic, D, is derived and tabulated for samples sizes from 2 to 7 pairs. The null distribution of the rank difference test statistic is shown to be asymptotically the same as that of the Wilcoxon test statistic for large sample sizes. Permutation methods are used to derive the null distribution for sample sizes from 8 to 20 based on 400000 simulations for each sample size. The practical application of the rank difference test is evaluated by comparing its power and performance with that of the Wilcoxon, the sign test and the related t test, for several classic sets of data in the literature, and for several sets of simulated data.

In this paper, a new method of sensing primary user's signal for cognitive radios in Grassmann manifold is proposed. The Grassmann covariance matrix (GCM) is formed with the help of covariance matrix of the transmitted and the received symbols. By using GCM, a new test statistic is defined, which is the modification of Binet-Cauchy metric. On the basis of this new test statistic, primary user's signal is detected. We also show that the new test statistic is a valid detector since it follows the concentration phenomena. We derive the distribution of new test statistic under null hypothesis and alternative hypothesis. Lower bound for the probability of detection of signal is also derived using separating function and distribution of new test statistic. Simulations using the real-world measurements of digital television (DTV) signal show performance gain of the proposed method in terms of signal detection over existing methods and their agreement with the derived distribution. Additionally, we extend the proposed method for the cooperative spectrum sensing and derive the distribution under both hypotheses. Experimental verification on the software defined radio is also performed and it is found that the proposed method fulfills the requirement of maximum protection of the DTV signal.

Choice of a null distribution in resampling-based multiple testing

A distinction between Fisher's implied data-generating process for Monte Carlo cycles and the more general Markov process leads to non-parametric tests for duration dependence. Tests are based on the method of moments, Tauchen's generalized method of moments (GMM) procedure, and a statistic whose null distribution probability limit is zero. Using finite-sample critical values obtained by Monte Carlo methods, our test results are remarkably consistent. The null distribution of the GMM test statistic for samples of the size considered is distinctly non-normal, so that asymptotic critical values give erroneous results. The tests are applied to UK business cycle data for 1854–1992. There is evidence for duration dependence in expansions but not in contractions.

In order to analyze multivariate data for the randomized block design, a test based on permutation procedures was introduced by Mielke and Iyer (1982). Under the null hypothesis of no difference in the treatment effects, the distribution of this test statistic assigns equal probabilities to all possible permutations. When the number of blocks is large, this approach becomes unreasonable. In order to approximate its distribution, they obtained the first three moments of this test statistic under the null hypothesis. Due to negative skewness, the distribution of this test statistic was approximated by the Pearson type III distribution. We obtain the fourth moment of this test statistic for the case of three treatments and obtain a better approximating distribution. Then, we compare the power performance of this test based on this additional information with that based on three moment results for simulated samples from several underlying populations.

Seneta & Chen (2005) tightened the familywise error rate control of Holm's procedure by sharpening its critical values using pairwise dependencies of the $p$-values. In this paper we further sharpen these critical values in the case where the distribution functions of the pairwise maxima of null $p$-values are convex, a property shown to hold in some applications of Holm's procedure. The newer critical values are uniformly larger, providing tighter familywise error rate control than the approach of Seneta & Chen (2005), significantly so under high pairwise positive dependencies. The critical values can be further improved under exchangeable null $p$-values.

Most of the existing methodologies for evaluating heterogeneity in zero-inflated Poisson (ZIP) models are often assuming that the Poisson mean is a function of nuisance parameters. However, these nuisance parameters can be misspecified when performing these methodologies, the validity and the power of the test may be affected. In this article, we primarily focus on investigating the impact of misspecification on the performance of score test for homogeneity in ZIP models. Through an intensive simulation study, we find that: 1) under misspecification, the limiting distribution of the score test statistic under the null no longer follows a distribution. A parametric bootstrap methodology is suggested to use to find the true null limiting distribution of the score test statistic; 2) the power of the test decreases as the number of covariates in the Poisson mean increases. The test with a constant Poisson mean has the highest power, even compared to the test with a well-specified mean. At last, simulation results are applied to the Wuhan Inpatient Care Insurance data which contain excess zeros.

An exhaustive unconditional permutation distribution of a test statistic is necessary in the construction of exact tests of significance. Tables of exact critical values for the Mood test are scarce. In this paper, the exact permutation distribution of the Mood test statistic is generated empirically by actually obtaining all the distinct permutations of the variates in an experiment. The tables of exact critical values for the Mood test are produced.

In this paper, we suggest a least squares procedure for the determination of the number of upper outliers in an exponential sample by minimizing sample mean squared error. Moreover, the method can reduce the masking or “swamping” effects. In addition, we have also found that the least squares procedure is easy and simple to compute than test procedure Tk, suggested by Zhang (1998) for determining the number of upper outliers, since Zhang (1998) need to use the complicated null distribution of Tk. Moreover, we give three practical examples and a simulated example to illustrate the procedures. Further, simulation studies are given to show the advantages of the proposed method. Finally, the proposed least squares procedure can also determine the number of upper outliers in other continuous univariate distributions (for example, Pareto, Gumbel, Weibull, etc.).

Reconstruction regions for unobserved middle order statistics in the exponential distribution

In this paper, we suggest a new set of regression‐based statistics for testing the seasonal unit root null hypothesis. These tests are based on combining conventional Hylleberg et al. (1990) ‐type seasonal unit root test statistics calculated from both forward and reverse estimation of the auxiliary regression equation. We derive the asymptotic distributions of the new test statistics under the seasonal unit root null hypothesis. We provide finite sample critical values appropriate for the case of quarterly data together with asymptotic critical values, the latter appropriate for any seasonal aspect. Monte Carlo simulation of the finite‐sample size and power properties of the new tests reveals that, overall, they perform rather better than extant tests of the seasonal unit root hypothesis.

Asymptotic expansions for the distributions of multivariate basic statistics and one-way MANOVA tests under nonnormality

Outliers often occur during data collection, which could impact the result seriously and lead to a large inference error; therefore, it is important to detect outliers before data analysis. Gamma distribution is a popular distribution in statistics; this paper proposes a method for detecting multiple upper outliers from gamma (m,θ). For computing the critical value of the test statistic in our method, we derive the density function for the case of a single outlier and design two algorithms based on the Monte Carlo and the kernel density estimation for the case of multiple upper outliers. A simulation study shows that the test statistic proposed in this paper outperforms some common test statistics. Finally, we propose an improved testing method to reduce the impact of the swamping effect, which is demonstrated by real data analyses.