Generalized manova double linear hypothesis with double linear restrictions

For the generalized MANOVA model of Potthoff and Roy [7], Gleser and Olkin [3] give a likelihood ratio test criterion for testing double linear parametric functions of the regression parameters. Their theory is extended in this paper to the testing of double linear parametric functions with double linear restrictions on the parameters. The theory is presented in terms of the original variates unlike Gleser and Olkin [3] who resort to canonical transformations of the original variates.     

Estimating and testing generalized linear models under inequality restrictions

We consider maximum likelihood estimation and likelihood ratio tests under inequality restrictions on the parameters. A special case are order restrictions, which may appear for example in connection with effects of an ordinal qualitative covariate. Our estimation approach is based on the principle of sequential quadratic programming, where the restricted estimate is computed iteratively and a quadratic optimization problem under inequality restrictions is solved in each iteration. Testing for inequality restrictions is based on the likelihood ratio principle. Under certain regularity assumptions the likelihood ratio test statistic is asymptotically distributed like a mixture of χ², where the weights are a function of the restrictions and the information matrix. A major problem in theory is that in general there is no unique least favourable point. We present some empirical findings on finite-sample behaviour of tests and apply the methods to examples from credit scoring and dentistry.     

A note on the manova model with double linear restrictions

In this note we show how one may construct goodness-of-fit tests to test hypotheses for the restricted MANOVA and GMANOVA models using the multivariate seemingly unrelated regression (MSUR) model.     

Recursive improvement of estimates in a Gauss-Markov model with linear restrictions

The minimum-dispersion linear unbiased estimator of a set of estimable functions in a general Gauss-Markov model with double linear restrictions is considered. The attention is focused on developing a recursive formula in which an initial estimator, obtained from the unrestricted model, is corrected with respect to the restrictions successively incorporated into the model. The established formula generalizes known results developed for the simple Gauss-Markov model.     

On f-tests and linear hypothesis in a general linear model

A simple method of setting linear hypotheses testable by F-tests in a general linear model when the covariance matrix has a general form and is completely unknown, is provided. With some additional conditions imposed on the covariance matrix, there exist the UMP invariant tests of certain linear hypotheses. We derive them to compare the powers with those of F-tests obtained under no restrictions on the covariance matrix. The results are illustrated in a multiple regression model with some examples.     

Manova with covariates and an application in profile analysis

Replacing one of the two marginal distributions in a bivariate normal by a family of symmetrical distributions, we obtain a new family of symmetric bivariate distributions. We use the Tiku - Suresh (1990) method to estimate the parameters of this new bivariate family. We define a Hotelling - type statistic to test the mean vector and evaluate the asymptotic power of this statistic relative to the Hotelling T² statistic. We show that the former is considerably more powerful.     

Estimation in a linear regression model with stochastic linear restrictions: a new two-parameter-weighted mixed estimator

The present paper considers the weighted mixed regression estimation of the coefficient vector in a linear regression model with stochastic linear restrictions binding the regression coefficients. We introduce a new two-parameter-weighted mixed estimator (TPWME) by unifying the weighted mixed estimator of Schaffrin and Toutenburg [1] and the two-parameter estimator (TPE) of Özkale and Kaç?ranlar [2]. This new estimator is a general estimator which includes the weighted mixed estimator, the TPE and the restricted two-parameter estimator (RTPE) proposed by Özkale and Kaç?ranlar [2] as special cases. Furthermore, we compare the TPWME with the weighted mixed estimator and the TPE with respect to the matrix mean square error criterion. A numerical example and a Monte Carlo simulation experiment are presented by using different estimators of the biasing parameters to illustrate some of the theoretical results.     

Regularised Manova for High‐Dimensional Data

The traditional and readily available multivariate analysis of variance (MANOVA) tests such as Wilks' Lambda and the Pillai–Bartlett trace start to suffer from low power as the number of variables approaches the sample size. Moreover, when the number of variables exceeds the number of available observations, these statistics are not available for use. Ridge regularisation of the covariance matrix has been proposed to allow the use of MANOVA in high‐dimensional situations and to increase its power when the sample size approaches the number of variables. In this paper two forms of ridge regression are compared to each other and to a novel approach based on lasso regularisation, as well as to more traditional approaches based on principal components and the Moore‐Penrose generalised inverse. The performance of the different methods is explored via an extensive simulation study. All the regularised methods perform well; the best method varies across the different scenarios, with margins of victory being relatively modest. We examine a data set of soil compaction profiles at various positions relative to a ridgetop, and illustrate how our results can be used to inform the selection of a regularisation method.     

Testing the disturbance variance after a pre-test for a linear hypothesis on coefficients in a linear regression

In this paper, we examine the sampling performance of a two-stage test which consists of a pre-test for a linear hypothesis on regression coeffiecients followed by a main-test for a disturbance variance in a linear regression. It is shown that the actual size of the two-stage test can be well-controlled around the normal size if the suggested sizes presented in this paper are used in the pre-test. It is also shown that the two-stage test when the suggested sizes are used in the preferable to the usual test for the disturbance variable which incorporates no pre-test in terms of the power.     

Distribution of the biased hypothesis sum of squares in linear models with missing observations

In a linear model with missing observations, one can substitute algebraic quantities and then minimize the error sum of squares for the augmented model. This gives the correct error sum of squares. But this method does not produce the correct hypothesis sum of squares for testing a linear hypothesis about the parameters. The sum of squares obtained is biased but practitioners still use it. The distribution of this biased sum of squares is derived in this paper and the consequences of using this biased sum of squares on the type I and II errors is examined.     

Performance analysis of the preliminary test estimator with series of stochastic restrictions

In this paper, the problem of estimation of the regression coefficients in a multiple regression model is considered under the multicollinearity situation when there are series of stochastic linear restrictions available on the regression parameter vector. We have considered the preliminary test ridge regression estimators (PTRREs) based on the Wald, likelihood ratio, and lagrangian multiplier tests. Tables for the maximum and minimum guaranteed efficiency of the PTRREs are obtained, which allow us to determine the optimum choice of the level of significance corresponding to the optimum estimator. Some numerical results support the findings.     

Statistical inferences for partially linear single-index models with error-prone linear covariates

We consider statistical inference for partially linear single-index models (PLSIM) when some linear covariates are not observed, but ancillary variables are available. Based on the profile least-squared estimators of the unknowns, we study the testing problems for parametric components in the proposed models. It is to see whether the generalized likelihood ratio (GLR) tests proposed by Fan et al. (2001) are applicable to testing for the parametric components. We show that under the null hypothesis the proposed GLR statistics follow asymptotically the χ²-distributions with the scale constants and the degrees of freedom being independent of the nuisance parameters or functions, which is called the Wilks phenomenon. Simulated experiments are conducted to illustrate our proposed methodology.     

Influence on tests with focus on linear models

To assess the influence of single observations on the parameter estimates, case-deletion diagnostics are commonly used in linear regression models; one example is Cook's distance. For nested parametric models we consider a deletion diagnostic for evaluating the influence of a single observation on the likelihood ratio (LR) test. In order to have a common scale as reference, the asymptotic distribution of the diagnostic is derived and the values of the diagnostic are converted to percentiles. We focus on linear models and general linear models, and in these cases explicit results are derived. The performance of the diagnostic is explored in two small bench mark examples from linear regression and in a larger linear mixed model example.     

Local power properties of some asymptotic tests in symmetric linear regression models

In this paper we obtain asymptotic expansions, up to order n^−1/2 and under a sequence of Pitman alternatives, for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the class of symmetric linear regression models. This is a wide class of models which encompasses the t model and several other symmetric distributions with longer-than normal tails. The asymptotic distributions of all four statistics are obtained for testing a subset of regression parameters. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, Monte Carlo simulations are presented. An empirical application to a real data set is considered for illustrative purposes.     

Asymptotic hypothesis test to compare likelihood ratios of multiple diagnostic tests in unpaired designs

The accuracy of a binary diagnostic test is usually measured in terms of its sensitivity and its specificity. Other measures of the performance of a diagnostic test are the positive and negative likelihood ratios, which quantify the increase in knowledge about the presence of the disease through the application of a diagnostic test, and which depend on the sensitivity and specificity of the diagnostic test. In this article, we construct an asymptotic hypothesis test to simultaneously compare the positive and negative likelihood ratios of two or more diagnostic tests in unpaired designs. The hypothesis test is based on the logarithmic transformation of the likelihood ratios and on the chi-square distribution. Simulation experiments have been carried out to study the type I error and the power of the constructed hypothesis test when comparing two and three binary diagnostic tests. The method has been extended to the case of multiple multi-level diagnostic tests.     

Test of equality of normal means in the absence of independent estimator of variance

Let X_l,…,X_n be normally and independently distributed with means θ_l,…,θ_nand a cornmorl variance. Thus there are n observations and n+i unknwon parameters. A test of the null hypothesis that, the θ_i's are all zero and the alternative that the vector (θ_l,…,θ_n) lies in a convex cone with its vertex a.t the origin is connsidered in this paper. It is shown that under a mild condition the likelihood ratio test is possible. The ordinary one sided t - test belongs to the class of tests considered in this paper. The hypothesis of equality of means against the simple order alternative can be tested in certain cases .