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.     

Asymptotic expansions for the distributions of statistics based on a correlation matrix

An asymptotic expansion is given for the distribution of the α-th largest latent root of a correlation matrix, when the observations are from a multivariate normal distribution. An asymptotic expansion for the distribution of a test statistic based on a correlation matrix, which is useful in dimensionality reduction in principal component analysis, is also given. These expansions hold when the corresponding latent root of the population correlation matrix is simple. The approach here is based on a perturbation method.     

Power of largest root on canonical correlation

We consider the testing hypothesis that two random vectors of p and q components are independent in canonical correlation analysis. In this paper we investigate the powers of the test based on the largest root criterion. As the exact distribution are expressed by the zonal polynomials, the computation is possible only for p=2, and also it is necessary to calculate using quadruplex precision because we lose the significance by subtraction. So in Table I we obtain the percentage points of the largest root criterion for the computation of the quadruplex precision. Then we calculate the power when p=2 and q = 3 to 11 (2). The results show that for the fixed n–q the power becomes smaller when q increases, and for the fixed p1 of the alternative hypothesis (p1, P2) the power does not become significantly large when P2 increases. We can also find the sample size required for the power agnist some alternative hypothesis to be about 0.9. the numerical results may be useful to find the quality of approximation by using formula of the asyptotic distribution.     

Stepwise estimators for three-phase sampling of categorical variables

Three-phase sampling can be a very effective design for the estimation of regional and national forest cover type frequencies. Simultaneous estimation of frequencies and sampling variances require estimation of a large number of parameters; often so many that consistency and robustness of results becomes an issue. A new stepwise estimation model, in which bias in phase one and two is corrected sequentially instead of simultaneously, requires fewer parameters. Simulated three-phase sampling tested the new model with 144 settings of sample sizes, the number of classes and classification accuracy. Relative mean absolute deviations and root mean square errors were, in most cases, about 8% lower with the stepwise method than with a simultaneous approach. Differences were a function of design parameters. Average expected relative root mean square errors, derived from the assumption of a Dirichlet distribution of cover-type frequencies, tracked the empirical root mean square errors obtained from repeated sampling with ±10%. Resampling results indicate that the relative bias of the most frequent cover types was slightly inflated by the stepwise method. For the least common cover type, the simultaneous method produced the largest relative bias.     

On Comparisons of Exact Powers of Bivariate GMANOVA Tests

Exact powers of four classical tests in a GMANOVA model are compared numerically when the order of the error sum of square matrix is 2. The four tests are likelihood ratio (=LR), Pillai's V, Hotelling's T ², and Roy's largest root tests. It turns out that for small sizes, there are a few cases in which Rothenberg's condition for the relative magnitude of asymptotic powers of three standard tests does not hold.     

检验式设定错误对时间序列单位根检验小样本性质的影响

首先对单位根检验的两类常见的数据生成系统进行比较，然后利用蒙特卡洛实验研究了时间序列单位根检验式的设定问题。研究发现在利用DF检验和DF-GLS检验进行时间序列的单位根检验时，检验式设定错误直接影响着检验结果，尤其在推断时间序列是趋势平稳过程还是有时间趋势项的随机游走过程或有二阶时间趋势多项式的随机游走过程时，检验式的错误设定很容易将趋势平稳过程误判为非平稳过程。     

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.     

A Unified Approach to the Characterization of Equivalence Classes of DAGs, Chain Graphs with no Flags and Chain Graphs

Abstract.  A Markov property associates a set of conditional independencies to a graph. Two alternative Markov properties are available for chain graphs (CGs), the Lauritzen–Wermuth–Frydenberg (LWF) and the Andersson–Madigan– Perlman (AMP) Markov properties, which are different in general but coincide for the subclass of CGs with no flags . Markov equivalence induces a partition of the class of CGs into equivalence classes and every equivalence class contains a, possibly empty, subclass of CGs with no flags itself containing a, possibly empty, subclass of directed acyclic graphs (DAGs). LWF-Markov equivalence classes of CGs can be naturally characterized by means of the so-called largest CGs , whereas a graphical characterization of equivalence classes of DAGs is provided by the essential graphs . In this paper, we show the existence of largest CGs with no flags that provide a natural characterization of equivalence classes of CGs of this kind, with respect to both the LWF- and the AMP-Markov properties. We propose a procedure for the construction of the largest CGs, the largest CGs with no flags and the essential graphs, thereby providing a unified approach to the problem. As by-products we obtain a characterization of graphs that are largest CGs with no flags and an alternative characterization of graphs which are largest CGs. Furthermore, a known characterization of the essential graphs is shown to be a special case of our more general framework. The three graphical characterizations have a common structure: they use two versions of a locally verifiable graphical rule. Moreover, in case of DAGs, an immediate comparison of three characterizing graphs is possible.     

Testing for the maximum mean in a mixture of normals

We consider the test of the null hypothesis that the largest mean in a mixture of an unknown number of normal components is less than or equal to a given threshold. This test is motivated by the problem of assessing whether the Soviet Union has been operating in compliance with the Nuclear Test Ban Treaty. In our analysis, the number of normal components is determined using Akaike's Information Criterion while the hypothesis test itself is based on asymptotic results given by Behboodian for a mixture of two normal components. A bootstrap approach is also considered for estimating the standard error of the largest estimated mean. The performance of the testa are examined through the use of simulation.     

A simple root selection method for univariate finite normal mixture models

It is well known that there exist multiple roots of the likelihood equations for finite normal mixture models. Selecting a consistent root for finite normal mixture models has long been a challenging problem. Simply using the root with the largest likelihood will not work because of the spurious roots. In addition, the likelihood of normal mixture models with unequal variance is unbounded and thus its maximum likelihood estimate (MLE) is not well defined. In this paper, we propose a simple root selection method for univariate normal mixture models by incorporating the idea of goodness of fit test. Our new method inherits both the consistency properties of distance estimators and the efficiency of the MLE. The new method is simple to use and its computation can be easily done using existing R packages for mixture models. In addition, the proposed root selection method is very general and can be also applied to other univariate mixture models. We demonstrate the effectiveness of the proposed method and compare it with some other existing methods through simulation studies and a real data application.     

Some complex variable transformations and exact power comparisons of two-sided tests of equality of two Hermitian covariance matrices

Power studies of tests of equality of covariance matrices of two p-variate complex normal populations σ₁ = σ₂ against two-sided alternatives have been made based on the following five criteria: (1) Roy's largest root, (2) Hotelling's trace, (4) Wilks' criterion and (5) Roy's largest and smallest roots. Some theorems on transformations and Jacobians in the two-sample complex Gaussian case have been proved in order to obtain a general theorem for establishing the local unbiasedness conditions connecting the two critical values for tests (1)–(5). Extensive unbiased power tabulations have been made for p=2, for various values of n₁, n₂, λ₁ and λ₂ where n₁ is the df of the SP matrix from the ith sample and λ₁ is the ith latent root of σ₁σ^-1₂ (i=1, 2). Equal tail areas approach has also been used further to compute powers of tests (1)–(4) for p=2 for studying the bias and facilitating comparisons with powers in the unbiased case. The inferences have been found similar to those in the real case. (Chu and Pillai, Ann. Inst. Statist. Math. 31.     

Least significant spacing for ‘one versus the rest’ normal populations

Consider sample means from k(≥2) normal populations where the variances and sample sizes are equal. The problem is to find the ‘least significant difference’ or ‘spacing’ (LSS) between the two largest means, so that if an observed spacing is larger we have confidence 1 - α that the population with largest sample mean also has the largest population mean.

When the variance is known it is shown that the maximum LSS occurs when k = 2, provided a < .2723. In other words, for any value of k we may use the usual (one-tailed) least significant difference to demonstrate that one population has a population mean greater than (or equal to) the rest.

When the variance is estimated bounds are obtained for the confidence which indicate that this last result is approximately correct.     

On estimating the mean of the selected population with unknown variance

This paper compares four estimators of the mean of the selected population from two normal populations with unknown means and common but unknown variance. The selection procedure is that the population yielding the largest sample mean is selected. The four estimators considered are invariant under both location and scale transformations. The bias and mean square errors of the four estimators are computed and compared. The conclusions are close to those reported by Dahiya ‘1974’, even for small sample sizes     

Panel data unit roots tests: The role of serial correlation and the time dimension

We investigate the influence of residual serial correlation and of the time dimension on statistical inference for a unit root in dynamic longitudinal data, known as panel data in econometrics. To this end, we introduce two test statistics based on method of moments estimators. The first is based on the generalized method of moments estimators, while the second is based on the instrumental variables estimator. Analytical results for the Instrumental Variables (IV) based test in a simplified setting show that (i) large time dimension panel unit root tests will suffer from serious size distortions in finite samples, even for samples that would normally be considered large in practice, and (ii) negative serial correlation in the error terms of the panel reduces the power of the unit root tests, possibly up to a point where the test becomes biased. However, near the unit root the test is shown to have power against a wide range of alternatives. These findings are confirmed in a more general set-up through a series of Monte Carlo experiments.     

Bias-Corrected maximum likelihood estimation of the parameters of the weighted Lindley distribution

The two-parameter weighted Lindley distribution is useful for modeling survival data, whereas its maximum likelihood estimators (MLEs) are biased in finite samples. This motivates us to construct nearly unbiased estimators for the unknown parameters. We adopt a “corrective” approach to derive modified MLEs that are bias-free to second order. We also consider an alternative bias-correction mechanism based on Efron’s bootstrap resampling. Monte Carlo simulations are conducted to compare the performance between the proposed and two previous methods in the literature. The numerical evidence shows that the bias-corrected estimators are extremely accurate even for very small sample sizes and are superior than the previous estimators in terms of biases and root mean squared errors. Finally, applications to two real datasets are presented for illustrative purposes.     

Comparison of Maximum Likelihood with Conditional Pairwise Likelihood Estimation of Person Parameters in the Rasch Model

This paper is concerned with person parameter estimation in the binary Rasch model. The loss of efficiency of a pseudo, quasi, or composite likelihood approach investigated. By means of a Monte Carlo study, two quasi likelihood estimators are compared to two well-established maximum likelihood approaches, one of which being a weighted likelihood procedure. The results show that the observed values of the root mean squared error are practically equivalent for the compared estimators in the case of a sufficiently large number of items.     

Simultaneous upper confidence bounds for distances from the best two-parameter exponentlal distribution

Consider k independent exponential distributions possibly with different location parameters and a common scale parameter. If the best population is defined to be the one having the largest mean or equivalently having the largest location parameter, we then derive a set of simultaneous upper confidence bounds for all distances of the means from the largest one. These bounds not only can serve as confidence intervals for all distances from the largest parameter but they also can be used to identify the best population. Relationships to ranking and selection procedures are pointed out. Cases in which scale parameters are known or unknown and samples are complete or type II censored are considered. Tables to implement this procedure are given.     

TWO STAGE SELECTION PROBLEM FOR NORMAL POPULATIONS WITH UNEQUAL VARIANCES

The paper discusses the Dudewicz-Dalal modification of a Stein-type two sample procedure for the goal of selecting the population with the largest mean from k normal populations with unknown variances. Largest k values are obtained such that a procedure based on sample means is preferred to the Dudewicz-Dalal procedure. The more general goal of choosing those populations with the t largest means is also considered.     

Estimating the variance of the sample median

The small-sample bias and root mean squared error of several distribution-free estimators of the variance of the sample median are examined. A new estimator is proposed that is easy to compute and tends to have the smallest bias and root mean squared error.     

Least Squares Programs-A Look at the Square Root Procedure

Various algorithms are in use on computers to solve least squares problems. Apparently very few programs use the square root procedure. In this paper, results from programs based on the square root procedure are compared with results based on some other algorithms. Remarkably good results are obtained using the square root procedure.