Exact power comparison of three criteria to assess the independence between two sets of variates

Powers of the three criteria are evaluated for testing the hypothesis of the independence between a -set and a q-set of variates in a (p + q) -variate normal population. They are: (1) the likelihood ratio type criterion, Wt (2) the largest root criterion, r1, and (3) criterion of the sum of roots, V. For p= 2, Pillai and Jayachandran, and others have studied for the restricted range of the alternative hypothesis. Recently the power of the largest root was investigated in detail by Sugiyama and %%. In this paper, their power functions are compared in a wide range of the alternative hypotheses. The powers of rl and V are locally optimum, but the W shows a large power in a wide range.     

数据初始值对DF类IPS检验势的影响研究

 本文研究了DF类面板数据单位根IPS检验势受时序数据初始值的影响,推导了DF类面板单位根IPS检验统计量在局部备择假设下的极限分布和局部渐近势函数,发现了DF类面板数据单位根IPS检验统计量局部渐近势在异质性局部备择假设下是初始条件的单调递增函数;小样本Monte Carlo模拟分析结果表明,若假设初始条件为零,DF类IPS统计量的检验势将被低估。     

On Testing the Mean Equivalence of Treatments from Correlated Normal Populations

Let there be k equally correlated treatment populations under consideration, a Studentized range test is proposed to test the hypothesis of average mean equivalence against the alternative hypothesis of inequivalence. The maximum level and minimum power at some least favorable configurations of means are used to calculate the critical value and the required sample size simultaneously when testing a null against an alternative hypothesis. The range test is applied to a real world problem to find out if the stress levels among children at four time periods due to a newly built nearby airport are equivalent.     

Expected Power for the False Discovery Rate with Independence

The Benjamini-Hochberg procedure is widely used in multiple comparisons. Previous power results for this procedure have been based on simulations. This article produces theoretical expressions for expected power. To derive them, we make assumptions about the number of hypotheses being tested, which null hypotheses are true, which are false, and the distributions of the test statistics under each null and alternative. We use these assumptions to derive bounds for multiple dimensional rejection regions. With these bounds and a permanent based representation of the joint density function of the largest p-values, we use the law of total probability to derive the distribution of the total number of rejections. We derive the joint distribution of the total number of rejections and the number of rejections when the null hypothesis is true. We give an analytic expression for the expected power for a false discovery rate procedure that assumes the hypotheses are independent.     

TESTING FOR PERIODIC STATIONARITY

This paper proposes a test for the null hypothesis of periodic stationarity against the alternative hypothesis of periodic integration. We derive the limiting distribution of the test statistic and its characteristic function, which are the same as those of the test developed in Kwiatkowski, Phillips, Schmidt and Shin.[15] We find that some parameters, which we must assume under the alternative, have an important effect on the limiting power, so we should choose such parameters carefully. A Monte Carlo simulation reveals that the test has reasonable power but may be affected by the lag truncation parameter that is used for the correction of nuisance parameters.     

Expected Power for the False Discovery Rate with Independence

The Benjamini–Hochberg procedure is widely used in multiple comparisons. Previous power results for this procedure have been based on simulations. This article produces theoretical expressions for expected power. To derive them, we make assumptions about the number of hypotheses being tested, which null hypotheses are true, which are false, and the distributions of the test statistics under each null and alternative. We use these assumptions to derive bounds for multiple dimensional rejection regions. With these bounds and a permanent based representation of the joint density function of the largest p-values, we use the law of total probability to derive the distribution of the total number of rejections. We derive the joint distribution of the total number of rejections and the number of rejections when the null hypothesis is true. We give an analytic expression for the expected power for a false discovery rate procedure that assumes the hypotheses are independent.     

Testing stationarity and trend stationarity against the unit root hypothesis

In this paper we propose a family of relativel simple nonparametrics tests for a unit root in a univariate time series. Almost all the tests proposed in the literature test the unit root hypothesis against the alternative that the time series involved is stationarity or trend stationary. In this paper we take the (trend) stationarity hypothesis as the null and the unit root hypothesis as the alternative. The order differnce with most of the tests proposed in the literature is that in all four cases the asymptotic null distribution is of a well-known type, namely standard Cauchy. In the first instance we propose four Cauchy tests of the stationarity hypothesis against the unit root hypothesis. Under H1 these four test statistics involved, divided by the sample size n, converge weakly to a non-central Cauchy distribution, to one, and to the product of two normal variates, respectively. Hence, the absolute values of these test statistics converge in probability to infinity 9at order n). The tests involved are therefore consistent against the unit root hypothesis. Moreover, the small sample performance of these test are compared by Monte Carlo simulations. Furthermore, we propose two additional Cauchy tests of the trend stationarity hypothesis against the alternative of a unit root with drift.     

The asymptotic distributions of maximum likelihood ratio test and maximally selected χ2-test in binomial observations

We considered binomial distributed random variables whose parameters are unknown and some of those parameters need to be estimated. We studied the maximum likelihood ratio test and the maximally selected χ²-test to detect if there is a change in the distributions among the random variables. Their limit distributions under the null hypothesis and their asymptotic distributions under the alternative hypothesis were obtained when the number of the observations is fixed. We discussed the properties of the limit distribution and found an efficient way to calculate the probability of multivariate normal random variables. Finally, those results for both tests have been applied to examples of Lindisfarne's data, the Talipes Data. Our conclusions are consistent with other researchers' findings.     

Which Wald statistic? Choosing a parameterization of the Wald statistic to maximize power in k-sample generalized estimating equations

The Wald statistic is known to vary under reparameterization. This raises the question: which parameterization should be chosen, in order to optimize power of the Wald statistic? We specifically consider k-sample tests of generalized linear models (GLMs) and generalized estimating equations (GEEs) in which the alternative hypothesis contains only two parameters. An example is presented in which such an alternative hypothesis is of interest. Amongst a general class of parameterizations, we find the parameterization that maximizes power via analysis of the non-centrality parameter, and show how the effect on power of reparameterization depends on sampling design and the differences in variance across samples. There is no single parameterization with optimal power across all alternatives. The Wald statistic commonly used under the canonical parameterization is optimal in some instances but it performs very poorly in others. We demonstrate results by example and by simulation, and describe their implications for likelihood ratio statistics and score statistics. We conclude that due to poor power properties, the routine use of score statistics and Wald statistics under the canonical parameterization for GEEs is a questionable practice.     

Testing a Regression Model When We Have Smooth Alternatives in Mind

Goodness-of-fit tests based on residual sums of squares are standard procedures used when fitting regression models. Often we have a smooth alternative in mind, a qualitative feature that the χ²-test does not take into account. We show that the power of detecting a smooth alternative increases when we smooth the current model as well. The proposed test is shown to be able to detect any continuous local alternative tending to zero slower than n ^−1/2. Theoretical results also address minimax non-parametric hypothesis testing in Sobolev spaces. A simulation study is presented, and the procedure is applied to expenditure curve estimation.     

Testing for Unit Roots in Dynamic Panels in the Presence of a Deterministic Trend: Re-examining the Unit Root Hypothesis for Real Stock Prices and Dividends

In this paper, we suggest a similar unit root test statistic for dynamic panel data with fixed effects. The test is based on the LM, or score, principle and is derived under the assumption that the time dimension of the panel is fixed, which is typical in many panel data studies. It is shown that the limiting distribution of the test statistic is standard normal. The similarity of the test with respect to both the initial conditions of the panel and the fixed effects is achieved by allowing for a trend in the model using a parameterisation that has the same interpretation under both the null and alternative hypotheses. This parameterisation can be expected to increase the power of the test statistic. Simulation evidence suggests that the proposed test has empirical size that is very close to the nominal level and considerably more power than other panel unit root tests that assume that the time dimension of the panel is large. As an application of the test, we re-examine the stationarity of real stock prices and dividends using disaggregated panel data over a relatively short period of time. Our results suggest that while real stock prices contain a unit root, real dividends are trend stationary.     

Testing for a Unit Root in a Time Series With a Changing Mean

This study considers testing for a unit root in a time series characterized by a structural change in its mean level. My approach follows the “intervention analysis” of Box and Tiao (1975) in the sense that I consider the change as being exogenous and as occurring at a known date. Standard unit-root tests are shown to be biased toward nonrejection of the hypothesis of a unit root when the full sample is used. Since tests using split sample regressions usually have low power, I design test statistics that allow the presence of a change in the mean of the series under both the null and alternative hypotheses. The limiting distribution of the statistics is derived and tabulated under the null hypothesis of a unit root. My analysis is illustrated by considering the behavior of various univariate time series for which the unit-root hypothesis has been advanced in the literature. This study complements that of Perron (1989), which considered time series with trends.     

The Likelihood Ratio Test with the Box–Cox Transformation for the Normal Mixture Problem: Power and Sample Size Study

Abstract

Through simulation and regression, we study the alternative distribution of the likelihood ratio test in which the null hypothesis postulates that the data are from a normal distribution after a restricted Box–Cox transformation and the alternative hypothesis postulates that they are from a mixture of two normals after a restricted (possibly different) Box–Cox transformation. The number of observations in the sample is called N. The standardized distance between components (after transformation) is D = (μ₂ ? μ₁)/σ, where μ₁ and μ₂ are the component means and σ² is their common variance. One component contains the fraction π of observed, and the other 1 ? π. The simulation results demonstrate a dependence of power on the mixing proportion, with power decreasing as the mixing proportion differs from 0.5. The alternative distribution appears to be a non-central chi-squared with approximately 2.48 + 10N ^?0.75 degrees of freedom and non-centrality parameter 0.174N(D ? 1.4)² × [π(1 ? π)]. At least 900 observations are needed to have power 95% for a 5% test when D = 2. For fixed values of D, power, and significance level, substantially more observations are necessary when π ≥ 0.90 or π ≤ 0.10. We give the estimated powers for the alternatives studied and a table of sample sizes needed for 50%, 80%, 90%, and 95% power.     

STAR模型中的递归退势单位根检验研究

在STAR模型框架下,考虑时间序列具有线性确定性趋势成分,本文建立了一个递归退势单位根检验统计量,推导了其渐近分布;并在考虑初始条件情形下,对递归退势、OLS和GLS退势单位根检验统计量的有限样本性质进行了细致的比较研究。若忽略初始条件的影响,GLS退势和递归退势单位根检验统计量的检验势都显著高于OLS退势。随着初始条件的增大,GLS退势单位根检验统计量的检验势下降得比较厉害,递归退势单位根检验统计量的检验势较为稳定,且在样本量较大情形下更具优势。     

Sample size requirements to test the equality of raters' precision

Exact and approximate methods are developed to calculate the required number of subjects n in a repeatability study, where repeatability is measured by the precision of measurements made by a rater. The exact method is based on power calculations under the non-null distribution of the multiple coefficient of determination, which requires intensive numerical computation. The approximate method is based on predictions from families of non-linear curves fitted by the method of least squares.     

From statistical power to statistical assurance: It's time for a paradigm change in clinical trial design

A well-designed clinical trial requires an appropriate sample size with adequate statistical power to address trial objectives. The statistical power is traditionally defined as the probability of rejecting the null hypothesis with a pre-specified true clinical treatment effect. This power is a conditional probability conditioned on the true but actually unknown effect. In practice, however, this true effect is never a fixed value. Thus, we discuss a newly proposed alternative to this conventional statistical power: statistical assurance, defined as the unconditional probability of rejecting the null hypothesis. This kind of assurance can then be obtained as an expected power where the expectation is based on the prior probability distribution of the unknown treatment effect, which leads to the Bayesian paradigm. In this article, we outline the transition from conventional statistical power to the newly developed assurance and discuss the computations of assurance using Monte Carlo simulation-based approach.     

TESTING THE LARGEST OF A SET OF CORRELATION COEFFICIENTS

A previous paper which studied the distribution of the smallest distance between N independent random points on the surface of a sphere is generalised to higher dimensions in order to study the distribution of the largest sample correlation coefficient between a set of independent normally distributed variables. Inclusion-exclusion arguments provide accurate bounds for the tail of this distribution, and by another argument more exact bounds are also found, one of which is an improvement on the result in the previous paper. Bounds are also found for the power of the test against the alternative hypothesis that one only of the population correlation coefficients is non-zero. The test is also shown to be the likelihood ratio test against the latter alternative.     

Precedence-type test based on Kaplan–Meier estimator of cumulative distribution function

In this paper, we introduce a precedence-type test based on Kaplan–Meier estimator of cumulative distribution function (CDF) for testing the hypothesis that two distribution functions are equal against a stochastically ordered hypothesis. This test is an alternative to the precedence life-test proposed first by Nelson (1963). After deriving the null distribution of the test statistic, we present its exact power function under the Lehmann alternative, and compare the exact power as well as simulated power (under location-shift) of the proposed test with other precedence-type tests. Next, we extend this test to the case of progressively Type-II censored data. Critical values for some combination of sample sizes and progressive censoring schemes are presented. We then examine the power properties of this test procedure and compare them to those of the weighted precedence and weighted maximal precedence tests under a location-shift alternative by means of Monte Carlo simulations. Finally, we present two examples to illustrate all the test procedures discussed here, and then make some concluding remarks.     

An approximation to the distribution of the ratio of two general quadratic forms with application

Based on mixed cumulants up to order six, this paper provides a four moment approximation to the distribution of a ratio of two general quadratic forms in normal variables. The approximation is applied to calculate the percentile points of modified F-test statistics for testing treatment effects when standard F-ratio test is misleading because of dependence among observations. For the special case, when data is generated by an AR(1) process, the approximation is evaluated by a simulation study. For the general SARMA (p,q)(P,Q)_s process, a modified F-test statistic Is given, and its distribution for the (0,1)(0,l)₁₂ process, is approximated by the moment approximation technique.     

Testing for a Unit Root in ARIMA Processes

A unit root has important long-run implications for many time series in economics and finance. This paper develops a unit-root test of an ARIMA(p-1, 1, q) with drift null process against a trend-stationary ARMA(p, q) alternative process, where the order of the time series is assumed known through previous statistical testing or relevant theory. This test uses a point-optimal test statistic, but it estimates the null and alternative variance-covariance matrices that are used in the test statistic. Consequently, this test approximates a point-optimal test. Simulations show that its small-sample size is close to the nominal test level for a variety of unit-root processes, that it has a robust power curve against a variety of stationary alternatives, that its combined small-sample size and power properties are highly competitive with previous unit-root tests, and that it is robust to conditional heteroskedasticity. An application to post-Second World War real per capita gross domestic product is provided.