Bivariate Sign Test for One-Sample Bivariate Location Model Using Ranked Set Sample

In this article, we introduce a bivariate sign test for the one-sample bivariate location model using a bivariate ranked set sample (BVRSS). We show that the proposed test is asymptotically more efficient than its counterpart sign test based on a bivariate simple random sample (BVSRS). The asymptotic null distribution and the non centrality parameter are derived. The asymptotic distribution of the vector of sample median as an estimator of the locations of the bivariate model is introduced. Theoretical and numerical comparisons of the asymptotic efficiency of the BVRSS sign test with respect to the BVSRS sign test are also given.     

Sign test based on k-tuple general ranked set samples

ABSTRACT

The sign test based on the k-tuple ranked set samples is discussed here. We first derive the distribution of the k-tuple ranked set sample sign test statistic, and then the asymptotic distribution is also obtained. We then compare its performance with its counterparts based on simple random sample and classical ranked set sample. The asymptotic relative efficiency and the power are then derived. Finally, the effect of imperfect ranking on the procedure is assessed.     

On small sample properties of the generalized signed rank and generalized sign tests

The generalized signed rank (GSR) and generalized sign (GS) tests were recently proposed for matched pair studies with censored observations (Woolson and Lechenbruch, 1980). The results provided in that paper were asymptotic, and no indicatin of small sample behavior was given. In this paper we report on simulation studied of these statistics for a variety of distributions. We find that the GSR is more powerful than the GS, and that censoring does not affect power greatly. In the original paper, we assumed each member of the pair has the same censoring time. We consider a variant of this in which each member of the pair has a censoring time chosen from a uniform distribution, and the minimum of these times is selected as the censoring time for the pair. It is found that the power of the test is slightly reduced because the number of doubly censored pairs is increased.     

On the application of improved symplectic integrators in Hamiltonian Monte Carlo

We explore the construction of new symplectic numerical integration schemes to be used in Hamiltonian Monte Carlo and study their efficiency. Integration schemes from Blanes et al., and a new scheme are considered as candidates to the commonly used leapfrog method. All integration schemes are tested within the framework of the No-U-Turn sampler (NUTS), both for a logistic regression model and a student t-model. The results show that the leapfrog method is inferior to all the new methods both in terms of asymptotic expected acceptance probability for a model problem and the efficient sample size per computing time for the realistic models.     

Nonparametric two-sample tests for dispersion differences based on placements

Two different two-sample tests for dispersion differences based on placement statistics are proposed. The means and variances of the test statistics are derived, and asymptotic normality is established for both. Variants of the proposed tests based on reversing the X and Y labels in the test statistic calculations are shown to have different small-sample properties; for both pairs of tests, one member of the pair will be resolving, the other nonresolving. The proposed tests are similar in spirit to the dispersion tests of both Mood and Hollander; comparative simulation results for these four tests are given. For small sample sizes, the powers of the proposed tests are approximately equal to the powers of the tests of both Mood and Hollander for samples from the normal, Cauchy and exponential distributions. The one-sample limiting distributions are also provided, yielding useful approximations to the exact tests when one sample is much larger than the other. A bootstrap test may alternatively be performed. The proposed test statistics may be used with lightly censored data by substituting Kaplan-Meier estimates for the empirical distribution functions.     

Optimal rank set sampling estimates for a population proportion

This paper examines two different classes of estimates for a population proportion based on an unbalanced rank set sample. Specifically, the two classes correspond to the maximum likelihood estimator (MLE) and a weighted average (WA) estimate. Both estimators are asymptotically normal, so standard inference procedures can still be implemented. Furthermore, these results can be used to develop optimal allocation schemes for both estimators. The performances of the optimal estimators are studied in terms of both finite sample and asymptotic relative efficiency. In general, the MLE is more efficient than the WA estimate. Lastly, the practicality of the optimal sampling plans is addressed and illustrated via an example.     

Adaptive Test for Testing the Difference in Survival Distributions

An adaptive test is proposed for the problem of testing the difference in survival distributions when the shape of the hazard ratio is unknown, hence the efficient test is unknown. The proposed adaptive test selects a test statistic from a finite set of the weighted logrank statistics T on the basis of the estimates of the efficiencies of the tests in T for given data. The efficiency estimator uses the length of the test based nonparametric confidence interval for the shift in a time transformed shift model. The suggested adaptive test is shown to be asymptotically efficient among the tests in T under the time transformed shift model and conditions commonly used in survival analysis. Simulations demonstrate that the adaptive test enjoys good small sample properties and in most situations is more powerful than the test using the maximum of the tests in T.     

Uniform robustness against nonnormality of the t and f tests

The size of the two-sample t test is generally thought to be robust against nonnormal distributions if the sample sizes are large. This belief is based on central limit theory, and asymptotic expansions of the moments of the t statistic suggest that robustness may be improved for moderate sample sizes if the variance, skewness, and kurtosis of the distributions are matched, particularly if the sample sizes are also equal.

It is shown that asymptotic arguments such as these can be misleading and that, in fact, the size of the t test can be as large as unity if the distributions are allowed to be completely arbitrary. Restricting the distributions to be identical or symmetric (but otherwise arbitrary) does not guarantee that the size can be controlled either, but controlling the tail-heaviness of the distributions does. The last result is proved more generally for the k-sample F test.     

Rank tests for the k-sample problem with restricted alternatives

An asymptotically maximin most powerful rank test among somewhere asymptotically most powerful linear rank tests with scores generating function cf> is derived for each of the simple order alternative, the simple loop alternative and the simple tree alternative in the k-sample problem. The comparisons of the tests obtained with the rank analogues of the Bartholomew's xv tests are made in terms of local asymptotic relative efficiency. It is found that our tests are better than the rank analogues of the xk tests. Furthermore, the asymptotic equivalence of the ranking by the pooled sample to the ranking in pairs are discuss¬ed and the tests which are asymptotically equivalent to ours are given.     

Nonparametric tests for right-censored data with biased sampling

Testing the equality of two survival distributions can be difficult in a prevalent cohort study when non random sampling of subjects is involved. Due to the biased sampling scheme, independent censoring assumption is often violated. Although the issues about biased inference caused by length-biased sampling have been widely recognized in statistical, epidemiological and economical literature, there is no satisfactory solution for efficient two-sample testing. We propose an asymptotic most efficient nonparametric test by properly adjusting for length-biased sampling. The test statistic is derived from a full likelihood function, and can be generalized from the two-sample test to a k-sample test. The asymptotic properties of the test statistic under the null hypothesis are derived using its asymptotic independent and identically distributed representation. We conduct extensive Monte Carlo simulations to evaluate the performance of the proposed test statistics and compare them with the conditional test and the standard logrank test for different biased sampling schemes and right-censoring mechanisms. For length-biased data, empirical studies demonstrated that the proposed test is substantially more powerful than the existing methods. For general left-truncated data, the proposed test is robust, still maintains accurate control of type I error rate, and is also more powerful than the existing methods, if the truncation patterns and right-censoring patterns are the same between the groups. We illustrate the methods using two real data examples.     

Efficiency and Validity Analyses of Two-Stage Estimation Procedures and Derived Testing Procedures in Quantitative Linear Models with AR(1) Errors

Abstract

In a quantitative linear model with errors following a stationary Gaussian, first-order autoregressive or AR(1) process, Generalized Least Squares (GLS) on raw data and Ordinary Least Squares (OLS) on prewhitened data are efficient methods of estimation of the slope parameters when the autocorrelation parameter of the error AR(1) process, ρ, is known. In practice, ρ is generally unknown. In the so-called two-stage estimation procedures, ρ is then estimated first before using the estimate of ρ to transform the data and estimate the slope parameters by OLS on the transformed data. Different estimators of ρ have been considered in previous studies. In this article, we study nine two-stage estimation procedures for their efficiency in estimating the slope parameters. Six of them (i.e., three noniterative, three iterative) are based on three estimators of ρ that have been considered previously. Two more (i.e., one noniterative, one iterative) are based on a new estimator of ρ that we propose: it is provided by the sample autocorrelation coefficient of the OLS residuals at lag 1, denoted r(1). Lastly, REstricted Maximum Likelihood (REML) represents a different type of two-stage estimation procedure whose efficiency has not been compared to the others yet. We also study the validity of the testing procedures derived from GLS and the nine two-stage estimation procedures. Efficiency and validity are analyzed in a Monte Carlo study. Three types of explanatory variable x in a simple quantitative linear model with AR(1) errors are considered in the time domain: Case 1, x is fixed; Case 2, x is purely random; and Case 3, x follows an AR(1) process with the same autocorrelation parameter value as the error AR(1) process. In a preliminary step, the number of inadmissible estimates and the efficiency of the different estimators of ρ are compared empirically, whereas their approximate expected value in finite samples and their asymptotic variance are derived theoretically. Thereafter, the efficiency of the estimation procedures and the validity of the derived testing procedures are discussed in terms of the sample size and the magnitude and sign of ρ. The noniterative two-stage estimation procedure based on the new estimator of ρ is shown to be more efficient for moderate values of ρ at small sample sizes. With the exception of small sample sizes, REML and its derived F-test perform the best overall. The asymptotic equivalence of two-stage estimation procedures, besides REML, is observed empirically. Differences related to the nature, fixed or random (uncorrelated or autocorrelated), of the explanatory variable are also discussed.     

An extended class of permutation techniques for matched pairs

A class of matched-pairs permutation techniques based on distances between each pair of observed signed values is considered. Although many commonly-used inference techniques for matched pairs are members of this class, some of the more appealing inference techniques among this class have received very little attention. Two new simple rank tests of this class jointly possess both intuitive properties and location-alternative power characteristics which appear more appealing than the corresponding characteristics of either the sign test or the Wllcoxon signed-ranks test. In particular, power comparisons based on slmula-tions indicate that these new rank tests are jointly as good or even vastly superior to the sign test or the Wilcoxon signed-ranks test for location alternatives involving five symmetric distributions. The five distributions selected for these com-parisons include the Laplace, logistic, normal, uniform and a U-shaped distribution     

An optimal sign test for one-sample bivariate location model using an alternative bivariate ranked-set sample

The aim of this paper is to find an optimal alternative bivariate ranked-set sample for one-sample location model bivariate sign test. Our numerical and theoretical results indicated that the optimal designs for the bivariate sign test are the alternative designs with quantifying order statistics with labels {((r+1)/2, (r+1)/2)}, when the set size r is odd and {(r/2+1, r/2), (r/2, r/2+1)} when the set size r is even. The asymptotic distribution and Pitman efficiencies of these designs are derived. A simulation study is conducted to investigate the power of the proposed optimal designs. Illustration using real data with the Bootstrap algorithm for P-value estimation is used.     

On the Ghoudi,Khoudraji, and Rivest test for extreme‐value dependence

Ghoudi, Khoudraji & Rivest [The Canadian Journal of Statistics 1998;26:187–197] showed how to test whether the dependence structure of a pair of continuous random variables is characterized by an extreme‐value copula. The test is based on a U‐statistic whose finite‐ and large‐sample variance are determined by the present authors. They propose estimates of this variance which they compare to the jackknife estimate of Ghoudi, Khoudraji & Rivest ( 1998 ) through simulations. They study the finite‐sample and asymptotic power of the test under various alternatives. They illustrate their approach using financial and geological data. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

More powerful sign test using median ranked set sample: Finite sample power comparison

Sign test using median ranked set samples (MRSS) is introduced and investigated. We show that, this test is more powerful than the sign tests based on simple random sample (SRS) and ranked set sample (RSS) for finite sample size. It is found that, when the set size of MRSS is odd, the null distribution of the MRSS sign test is the same as the sign test obtained by using SRS. The exact null distributions and the power functions, in case of finite sample sizes, of these tests are derived. Also, the asymptotic distribution of the MRSS sign tests are derived. Numerical comparison of the MRSS sign test power with the power of the SRS sign test and the RSS sign test is given. Illustration of the procedure, using real data set of bilirubin level in Jaundice babies who stay in neonatal intensive care is introduced.     

A new family of random graphs for testing spatial segregation

The authors discuss a graph‐based approach for testing spatial point patterns. This approach falls under the category of data‐random graphs, which have been introduced and used for statistical pattern recognition in recent years. The authors address specifically the problem of testing complete spatial randomness against spatial patterns of segregation or association between two or more classes of points on the plane. To this end, they use a particular type of parameterized random digraph called a proximity catch digraph (PCD) which is based on relative positions of the data points from various classes. The statistic employed is the relative density of the PCD, which is a U‐statistic when scaled properly. The authors derive the limiting distribution of the relative density, using the standard asymptotic theory of U‐statistics. They evaluate the finite‐sample performance of their test statistic by Monte Carlo simulations and assess its asymptotic performance via Pitman's asymptotic efficiency, thereby yielding the optimal parameters for testing. They further stress that their methodology remains valid for data in higher dimensions.     

Ranked set sample sign test for quantiles

A ranked set sample version of the sign test is proposed for testing hypotheses concerning the quantiles of a population characteristic. Both equal and unequal allocations are considered and the relative performance of different allocations is assessed in terms of Pitman's asymptotic relative efficiency. In particular, for each quantile, the allocation that maximizes the efficacy is identified and shown to not depend on the population distribution.     

The sample breakdown points of tests

Several authors have taken the worst case breakdown measures in analyzing the robustness of a test. In general, these kinds of measures give only a rough picture of breakdown robustness of a test. To overcome this limitation, a new kind of breakdown measure of a test is defined as the smallest proportion of arbitrary outliers in the sample that can distort the test decision. It is called as the sample breakdown point of a test in this paper. A distinct advantage of this new measure is that it is directly concerned with the test decision based on the present sample and with the critical region of the test. The sample breakdown points of several commonly used tests of one-sided or two-sided hypotheses are calculated and their asymptotic properties are also established. By Monte Carlo simulations and asymptotic analysis, we show that the acceptance breakdown of the t-test and the Hotelling T²-test is slightly better than that of the sample mean test. Finally, we prove that, for a one-sided hypothesis testing of location, the sign test has the maximum sample breakdown points asymptotically within a class of M-tests and score-tests.     

Estimates of regression coefficients based on the sign covariance matrix

Summary. A new estimator of the regression parameters is introduced in a multivariate multiple-regression model in which both the vector of explanatory variables and the vector of response variables are assumed to be random. The affine equivariant estimate matrix is constructed using the sign covariance matrix (SCM) where the sign concept is based on Oja's criterion function. The influence function and asymptotic theory are developed to consider robustness and limiting efficiencies of the SCM regression estimate. The estimate is shown to be consistent with a limiting multinormal distribution. The influence function, as a function of the length of the contamination vector, is shown to be linear in elliptic cases; for the least squares (LS) estimate it is quadratic. The asymptotic relative efficiencies with respect to the LS estimate are given in the multivariate normal as well as the t -distribution cases. The SCM regression estimate is highly efficient in the multivariate normal case and, for heavy-tailed distributions, it performs better than the LS estimate. Simulations are used to consider finite sample efficiencies with similar results. The theory is illustrated with an example.     

TESTING NEW BETTER THAN RENEWAL USED LIFE DISTRIBUTIONS BASED ON U-TEST

In this paper, the problem of testing exponentiality against new better (worse) than renewal used in expectation is investigated and similarly for the case of nuharmonic new better than renewal used in expectation. For each of these two aging properties, a nonparametric procedure (U-statistic) is presented. Selected critical values are tabulated for sample sizes n = 5(1)30(10)50. The Pitman asymptotic relative efficiency to the test relative to other classes are studied. A real example is given to elucidate the use of the proposed test statistics for the reliability analysis.