Testing hypotheses about the common mean of two normal populations

Two simple tests which allow for unequal sample sizes are considered for testing hypothesis for the common mean of two normal populations. The first test is an exact test of size a based on two available t-statistics based on single samples made exact through random allocation of α among the two available t-tests. The test statistic of the second test is a weighted average of two available t-statistics with random weights. It is shown that the first test is more efficient than the available two t-tests with respect to Bahadur asymptotic relative efficiency. It is also shown that the null distribution of the test statistic in the second test, which is similar to the one based on the normalized Graybill-Deal test statistic, converges to a standard normal distribution. Finally, we compare the small sample properties of these tests, those given in Zhou and Mat hew (1993), and some tests given in Cohen and Sackrowitz (1984) in a simulation study. In this study, we find that the second test performs better than the tests given in Zhou and Mathew (1993) and is comparable to the ones given in Cohen and Sackrowitz (1984) with respect to power..     

Approximations of the critical region of the fbietkan statistic

The Friedman (1937) test for the randomized complete block design is used to test the hypothesis of no treatment effect among k treatments with b blocks. Difficulty in determination of the size of the critical region for this hypothesis is com¬pounded by the facts that (1) the most recent extension of exact tables for the distribution of the test statistic by Odeh (1977) go up only to the case with k6 and b6, and (2) the usual chi-square approximation is grossly inaccurate for most commonly used combinations of (k,b). The purpose of this paper 2 is to compare two new approximations with the usual x² and F large sample approximations. This work represents an extension to the two-way layout of work done earlier by the authors for the one-way Kruskal-Wallis test statistic.     

Likelihood ratio test for testing equality of location parameters of two exponential distributions from doubly censored samples

The Likelihood Ratio (LR) test for testing equality of two exponential distributions with common unknown scale parameter is obtained. Samples are assumed to be drawn under a type II doubly censored sampling scheme. Effects of left and right censoring on the power of the test are studied. Further, the performance of the LR test is compared with the Tiku(1981) test.     

A robust modification of the ordered-heterogeneity test

An ordered heterogeneity (OH) test is a test for a trend that combines a non-directional heterogeneity test with the rank-order information specified under the alternative. We propose two modifications of the OH test procedure: (1) to use the mean ranks of the groups rather than the sample means to determine the observed ordering of the groups, and (2) to use the maximum correlation out of the 2^k???1 – 1 possibilities under the alternative rather than the single ordering (1, 2, … , k), where k is the number of independent groups. A simulation study indicates that these two changes increase the power of the ordered heterogeneity test when, as common in practice, the underlying distribution may deviate from a normal distribution and the trend pattern is a priori unknown. In contrast to the original OH test, the modified OH test can detect all possible patterns under the alternative with a relatively high power.     

Granger Causality Test in the Presence of Spillover Effects

In this article, we investigate the effect of spillover (i.e., causality in variance) on the reliability of Granger causality test based on ordinary least square estimates. We studied eight different versions of the test both, with and without Whites heteroskedasticity consistent covariance matrix (HCCME). The properties of the tests are investigated by means of a Monte Carlo experiment where 21 different data generating processes (DGP) are used and a number of factors that might affect the test are varied. The result shows that the best choice to test for Granger causality under the presence of spillover is the Lagrange Multiplier test with HCCME.     

A comparison of some conditional and unconditional exact tests for 2x2 contingency tables

In 1935, R.A. Fisher published his well-known “exact” test for 2x2 contingency tables. This test is based on the conditional distribution of a cell entry when the rows and columns marginal totals are held fixed. Tocher (1950) and Lehmann (1959) showed that Fisher s test, when supplemented by randomization, is uniformly most powerful among all the unbiased tests UMPU). However, since all the practical tests for 2x2 tables are nonrandomized - and therefore biased the UMPU test is not necessarily more powerful than other tests of the same or lower size. Inthis work, the two-sided Fisher exact test and the UMPU test are compared with six nonrandomized unconditional exact tests with respect to their power. In both the two-binomial and double dichotomy models, the UMPU test is often less powerful than some of the unconditional tests of the same (or even lower) size. Thus, the assertion that the Tocher-Lehmann modification of Fisher's conditional test is the optimal test for 2x2 tables is unjustified.     

Testing for superadditivity of the mean value function of a non-homogeneous poisson process

Consider a non-homogeneous Poisson process, N(t), with mean value function Λ(t) and intensity function λ(t). A conditional test of the hypothesis that the process is homogeneous, versus alternatives for which Λ(t) is superadditive, was proposed by Hollander and Proschan (1974). A new test for superadditivity of Λ(t), which is based on a linear combination of the occurrence times of the process N{t) is suggested in this paper. Though this test has the same Pitman efficiency as the Hollander-Proschan test, it is shown by Monte-Carlo simulation that our test has more power for many important alternatives. Tables for the exact null distribution of the test statistic have been given.     

On the power of durbin-watson statistic against fractionally integrated processes

This paper provides the theoretical explanation and Monte Carlo experiments of using a modified version of Durbin-Watson ( D W ) statistic to test an 1 ( 1 ) process against I ( d ) alternatives, that is, integrated process of order d, where d is a fractional number. We provide the exact order of magnitude of the modified D W test when the data generating process is an I ( d ) process with d E (0. 1.5). Moreover, the consistency of the modified DW statistic as a unit root test against I ( d ) alternatives with d E ( 0 , l ) U ( 1 , 1.5) is proved in this paper. In addition to the theoretical analysis, Monte Carlo experiments show that the performance of the modified D W statistic reveals that it can be used as a unit root test against I ( d ) alternatives.     

Locally optimal test for simultaneously testing the mean and variance of a normal distribution

A generalization of the locally most powerful unbiased (LMPU) test for the single parameter case to the k-parameter case was proposed by SenGupta and Vermeire (1986). In particular we defined a locally most mean power unbiased (LMMPU) test based on the mean curvature of the power hypersurface. Compared to the type C tests of Neyman and Pearson and the type D tests (Isaacson, 1951), LMMPU tests possess better theoretical properties and enjoy ease of construction of critical regions. In this paper we present an interesting example of a two-parameter univariate normal population for which Isaacson (1951, p. 233) was unsuccessful in finding a type D test. For the case of one observation, we prove that no Type D region exists but the LMMPU test is obtained - it is an example of a test with singular Hessian matrix for its power but is nevertheless a strictly locally unbiased (LU) test.     

A new distribution-free k-sample test: Analysis of kernel density functionals

A novel distribution-free k-sample test of differences in location shifts based on the analysis of kernel density functional estimation is introduced and studied. The proposed test parallels one-way analysis of variance and the Kruskal–Wallis (KW) test aiming at testing locations of unknown distributions. In contrast to the rank (score)-transformed non-parametric approach, such as the KW test, the proposed F-test uses the measurement responses along with well-known kernel density estimation (KDE) to estimate the locations and construct the test statistic. A practical optimal bandwidth selection procedure is also provided. Our simulation studies and real data example indicate that the proposed analysis of kernel density functional estimate (ANDFE) test is superior to existing competitors for fat-tailed or heavy-tailed distributions when the k groups differ mainly in location rather than shape, especially with unbalanced data. ANDFE is also highly recommended when it is unclear whether test groups differ mainly in shape or location. The Canadian Journal of Statistics 48: 167–186; 2020 © 2019 Statistical Society of Canada     

Behrens-fisher problem under the assumption of homogeneous coefficients of variation

The powers of the likelihood ratio (LR) test and an “asymptotically (in some sense) optimum” invariant test are examined and compared by simulation techniques with those of several other relevant tests for the problem of testing the equality of two univariate normal population means under the assumption of heterogeneous variances but homogeneous coefficients of variation. It is seen that the LR test is highly satisfactory for all values of the coefficient of variation and the “asymptotically optimum” invariant test, which is computationally much simpler than the LR test, is a reasonably good competitor for cases where the value of the coefficient of variation is greater than or equal to 3. Also, a     

Multiple Monte Carlo testing,with applications in spatial point processes

The rank envelope test (Myllymäki et al. in J R Stat Soc B, doi: 10.1111/rssb.12172, 2016) is proposed as a solution to the multiple testing problem for Monte Carlo tests. Three different situations are recognized: (1) a few univariate Monte Carlo tests, (2) a Monte Carlo test with a function as the test statistic, (3) several Monte Carlo tests with functions as test statistics. The rank test has correct (global) type I error in each case and it is accompanied with a p-value and with a graphical interpretation which determines subtests and distances of the used test function(s) which lead to the rejection at the prescribed significance level of the test. Examples of null hypotheses from point process and random set statistics are used to demonstrate the strength of the rank envelope test. The examples include goodness-of-fit test with several test functions, goodness-of-fit test for a group of point patterns, test of dependence of components in a multi-type point pattern, and test of the Boolean assumption for random closed sets. A power comparison to the classical multiple testing procedures is given.     

A Simple Test For The Independence of Sets of Variables

A test for the mutual independence of subvectors of the p-dimensional random vector X , distributed as N( 0, S? ), is described. The test is based on the maximum likelihood estimates (MLEs) of the off-(block) diagonal elements of S?. It is shown that the resulting test statistic is much easier to compute than the likelihood ratio (LR) test statistic while retaining the same asymptotic power properties in view of the general properties of tests based on the MLEs (ML test) and the likelihood ratio (LR test).     

A New Goodness-of-Fit Test Based on the Laplace Statistic for a Large Class of NHPP Models

ABSTRACT

Nonhomogeneous Poisson processes (NHPP) provide many models for hardware and software reliability analysis. In order to get an appropriate NHPP model, goodness-of-Fit (GOF for short) tests have to be carried out. For the power-law processes, lots of GOF tests have been developed. For other NHPP models, only the Conditional Probability Integral Transformation (CPIT) test has been proposed. However, the CPIT test is less powerful and cannot be applied to some NHPP models. This article proposes a general GOF test based on the Laplace statistic for a large class of NHPP models with intensity functions of the form αλ(t, β). The simulation results show that this test is more powerful than CPIT test.     

Empirical likelihood test for equality of two distributions using distance of characteristic functions

In this paper, we investigate the problem of testing for the equality of two distributions. We employ a two-sample Jackknife Empirical Likelihood (JEL) approach to construct a test statistic whose limiting distribution is Chi-square distribution with degree of freedom 1, no matter what the data dimension (fixed) is. A variety of synthetic data experiments demonstrate that our JEL test statistic performs very well, with a very neat asymptotic distribution under the null hypothesis. Furthermore, we apply the test procedure to a real dataset to obtain competitive results.     

Exact tests for equality of two proportions: fisher v. bayes

Yates (1984) using theoretical and philosophical arguments claims to have proved that the Fisher exact test for comparing the proportions of two binomial experiments is the best exact test. The present article uses objective and practical arguments to confront the Fisher exact test with a Bayes exact test. Using simulated samples we claim to have proved here the inferiority of the Fisher exact test in relation to a Bayes exact test. The comparison is based on the quality concept of Dawid (1982).     

On the choice of precedence tests

The best precedence test (BPT) is derived for testing the hypothesis that the lifetimes of two types of items on test have the same distribution. The test has maximum power in the class of the Lehmann type of alternatives F - 1 - (1-G) , A > 1, where F and G are probability distributions of the lifetimes of two types of items on test. This class includes exponential distributions, the Weibull distribution differing only in scale and distributions with proportional hazard rates. Exact power of the BPT is compared with other nonparametrie and parametric tests. The test may terminate before all the lifetimes of the items on test are recorded. In comparing with competing tests of equal size, the power functions are similar but a considerable number of items can be saved and the time on test can be reduced by using the BPT     

Ratio tests of a unit root

Based on the maximal invariant principle, we derive two ratio tests (locally best invariant test and point optimal test) for a unit root and compare them with previously proposed ratio tests. We also show that our ratio tests tend to have better powers than the Dickey-Fuller test and the modified Dickey-Fuller test.     

A residual-based test of the null of cointegration in panel data

This paper proposes a residual-based Lagrange Multiplier (LM) test for the null of cointegration in panel data. The test is analogous to the locally best unbiased invariant (LBUI) for a moving average (MA) unit root. The asymptotic distribution of the test is derived under the null. Monte Carlo simulations are performed to study the size and power properties of the proposed test.

overall, the empirical sizes of the LM-FM and LM-DOLs are close to the true size even in small samples. The power is quite good for the panels where T ≥ 50, and decent with panels for fewer observation in T. In our fixed sample of N = 50 and T = 50, the presence of a moving average and correlation between the LM-DOLS test seems to be better at correcting these effects, although in some cases the LM-FM test is more powerful.

Although much of the non-stationary time series econometrics has been criticized for having more to do with the specific properties of the data set rather than underlying economic models, the recent development of the cointegration literature has allowed for a concrete bridge between economic long run theory and time series methods. Our test now allows for the testing of the null of cointegration in a panel setting and should be of considerable interest to economists in a wide variety of fields.     

Some comparative studies on testing parallelism of several straight lines under heteroscedastic variances

This paper examines the robustness of the Welch test, the James test as well as Tan's ANOVA test (to be referred as F_β test) for testing parallelism in k straight lines under heteroscedasticity and nonnormality. Results of Monte Carlo studies demonstrate the robustness of all tests with respect to departure from normality. Further, there is hardly any difference between these methods with respect to both power and size of the test.