Aligned rank statistics for repeated measurement models with orthonormal design, employing a Chernoff–Savage approach

In this paper, aligned rank statistics are considered for testing hypotheses regarding the location in repeated measurement designs, where the design matrix for each set of measurements is orthonormal. Such a design may, for instance, be used when testing for linearity. It turns out that the centered design matrix is not of full rank, and therefore it does not quite satisfy the usual conditions in the literature. The number of degrees of freedom of the limiting chi-square distribution of the test statistic under the null hypothesis, however, is not affected, unless rather special hypotheses are tested. An independent derivation of this limiting distribution is given, using the Chernoff–Savage approach. In passing, it is observed that independence of the choice of aligner, which in the location problem is well-known to be due to cancellation, may in scale problems occur as a result of the type of score function suitable for scale tests. A possible extension to multivariate data is briefly indicated.     

Secondary design considerations for minimum bias estimation

Several authors have suggested the method of minimum bias estimation for estimating response surfaces. The minimum bias estimation procedure achieves minimum average squared bias of the fitted model without depending on the values of the unknown parameters of the true surface. The only requirement is that the design satisfies a simple estimability condition. Subject to providing minimum average squared bias, the minimum bias estimator also provides minimum average variance of ?(x) where ?(x) is the estimate of the response at the point x.

To support the estimation of the parameters in the fitted model, very little has been suggested in the way of experimental designs except to say that a full rank matrix X of independent variables should be used. This paper presents a closer look at the estimability conditions that are required for minimum bias estimation, and from the form of the matrix X, a formula is derived which measures the amount of design flexibility available. The design flexibility is termed “the degrees of freedom” of the X matrix and it is shown how the degrees of freedom can be used to decide if other design optimality criteria might be considered along with minimum bias estimation. Several examples are provided.     

Fourier smoother and additive models

Suppose the observations (t_i,y_i), i = 1,… n, follow the model where g_j are unknown functions. The estimation of the additive components can be done by approximating g_j, with a function made up of the sum of a linear fit and a truncated Fourier series of cosines and minimizing a penalized least-squares loss function over the coefficients. This finite-dimensional basis approximation, when fitting an additive model with r predictors, has the advantage of reducing the computations drastically, since it does not require the use of the backfitting algorithm. The cross-validation (CV) [or generalized cross-validation (GCV)] for the additive fit is calculated in a further 0(n) operations. A search path in the r-dimensional space of degrees of freedom is proposed along which the CV (GCV) continuously decreases. The path ends when an increase in the degrees of freedom of any of the predictors yields an increase in CV (GCV). This procedure is illustrated on a meteorological data set.     

Predicting pickle harvests using a parametric feedforward neural network

Feedforward networks have demonstrated their ability to model non-linear data. Despite this success, their use as a statistical analysis tool has been limited by the persistent assumption that these networks can only be implemented as non-parametric models. In fact, a feedforward network can be used for parametric modeling, with the result that many of the common parametric testing procedures can be applied to the nonlinear network. In this paper, a feedforward network for predicting the biological growth rate of pickles is developed. Using this network, the parametric nature of the network is demonstrated. Once trained, the network model is tested using standard parametric methods. In order to facilitate this testing, it is first necessary to develop a method for calculating the degrees of freedom for the neural network, and the residual covariance matrix. It is shown that the degrees of freedom is determined by the number of parameters that actually contribute to an output. With this information, the covariance matrix can be created by adapting the error matrix. Using these results, the trained network is tested using a simple F-statistic.     

Some remarks about the gasser-sroka-jennen-steinmetz variance estimator

This article proposes some simplifications of the residual variance estimator of Gasset, Sroka, and Jeneen-Steinmetz (GSJ, 1986) which is often used in conjunction with non parametric regression. The GSJ estimator is a quadratic form of the data, which depends on the relative spacings of the design points. When the errors are independent, identically distributed Gaussian variables, and the true regression curve is flat, the estimate is distributed as a weighted sum of x² variables. By matching the first two moments, the distribution can be approximated by a x² with degrees of freedom determined by the coefficients of the. quadratic form. Computation of the estimated degrees of freedom requires computing the trace of the square of an n x n matrix, where n is the number of design points. In this article, (n-2)/3 is shown to be a conservative estimate of the approximate degrees of freedom, and (n-2)/2 is shown to be conservative for many designs. In addition, a simplified version of the estimator is shown to be asymptotically equivalent, under many conditions.     

Singular gauss-markov models

For the general linear model Y = X$sZ + e in which e has a singular dispersion matrix $sG²A, $sG > 0, where A is n x n and singular, Mitra [2] considers the problem of testing F$sZ, where F is a known q x q matrix and claims that the sum of squares (SS) due to hypothesis is not distributed (as a x² variate with degrees of freedom (d. f.) equal to the rank of F) independent of the SS due to error, when a generalized inverse of A is chosen as (A + X'X)^–. This claim does not hold if a pseudo-inverse of A is taken to be (A + X'X)⁺ where A⁺ denotes the unique Moore-Penrose inverse (MPI) of A.     

Testing Goodness of Fit for Parametric Families of Copulas—Application to Financial Data

ABSTRACT

This article suggests a chi-square test of fit for parametric families of bivariate copulas. The marginal distribution functions are assumed to be unknown and are estimated by their empirical counterparts. Therefore, the standard asymptotic theory of the test is not applicable, but we derive a rule for the determination of the appropriate degrees of freedom in the asymptotic chi-square distribution. The behavior of the test under H ₀ and for selected alternatives is investigated by Monte Carlo simulation. The test is applied to investigate the dependence structure of daily German asset returns. It turns out that the Gauss copula is inappropriate to describe the dependencies in the data. A t _ν-copula with low degrees of freedom performs better.     

Nonnull distribution of Wilks' statistic for Manova in the complex case

In this paper, the exact distribution of Wilks' likelihood ratio criterion, A, for MANOVA, in the complex case when the alternate hypothesis is of unit rank (i.e. the linear case) has been derived and the explicit expressions for the same for p = 2 and 3 (where p is the number of variates) and general f₁ (the error degrees of freedom) and f₂ (the hypothesis degrees of freedom), are given. For an unrestricted number of variables, a general form of the density and the distribution of A in this case, is also given. It has been shown that the total integral of the series obtained by taking a few terms only, rapidly approaches the theoretical value one as more terms are taken into account, and some percentage points have also been computed.     

Goodness‐of‐fit testing based on a weighted bootstrap: A fast large‐sample alternative to the parametric bootstrap

The process comparing the empirical cumulative distribution function of the sample with a parametric estimate of the cumulative distribution function is known as the empirical process with estimated parameters and has been extensively employed in the literature for goodness‐of‐fit testing. The simplest way to carry out such goodness‐of‐fit tests, especially in a multivariate setting, is to use a parametric bootstrap. Although very easy to implement, the parametric bootstrap can become very computationally expensive as the sample size, the number of parameters, or the dimension of the data increase. An alternative resampling technique based on a fast weighted bootstrap is proposed in this paper, and is studied both theoretically and empirically. The outcome of this work is a generic and computationally efficient multiplier goodness‐of‐fit procedure that can be used as a large‐sample alternative to the parametric bootstrap. In order to approximately determine how large the sample size needs to be for the parametric and weighted bootstraps to have roughly equivalent powers, extensive Monte Carlo experiments are carried out in dimension one, two and three, and for models containing up to nine parameters. The computational gains resulting from the use of the proposed multiplier goodness‐of‐fit procedure are illustrated on trivariate financial data. A by‐product of this work is a fast large‐sample goodness‐of‐fit procedure for the bivariate and trivariate t distribution whose degrees of freedom are fixed. The Canadian Journal of Statistics 40: 480–500; 2012 © 2012 Statistical Society of Canada     

Designs for discrimination between binary response models

We propose a test statistic for discrimination between alternative univariate binary response models which is asymptotically equivalent to the likelihood ratio statistic and Pearson's goodness of fit statistic. We propose an optimal design procedure. Under certain conditions we prove that the maximum value of the power can be obtained when the degrees of freedom of the test statistic is one. Several mathematical properties of the incomplete gamma function ratio and the non-central chi-squared distribution are required in the discussion and these are established.     

Uncorrelated residuals and an exact test for two variance components in experimental design

The error contrasts from an experimental design can be constructed from uncorrelated residuals normally associated with the linear model. In this paper uncorrelated residuals are defined for the linear model that has a design matrix which is less than full rank, typical of many experimental design representations. It transpires in this setting, that for certain choices of uncorrelated residuals, corresponding to recursive type residuals, there is a natural partition of information when two variance components are known to be present. Under an assumtion of normality of errors this leads to construction of appropriate F-tests for testing heteroscedasticity. The test, which can be optimal, is applied to two well known data sets to illustrate its usefullness.     

Simple approximations to the behrens—fisher distribution

Dayal and Dickey (1977) have published in this journal a rather efficient numerical integration procedure for the product of k Student t-densities, and point out the evaluation of Behrens-Fisher (BF) densities as an important special case. The present note adds to this three simple normal approximations to Behrens-Fisher tail probabilities, that will save computer time for someone using the Dayal-Dickey results, and even allow evaluation on a desk calculator for moderately large degrees of freedom.

A direct normal approximation (method U) will be too coarse unless both degrees of freedom are large. A combination of the Peizer-Pratt (1968) approximation to the t-distribution and the Patil (1965) t-approximation to the BF distribution turns out to be very accurate. For very small degrees of freedom it may still be refined by an adhoc correction presented below. Other approximations and expansions turn out to be less satisfactory than the present trio. It facilitates a quick evaluation of BF probabilities and quantiles on a small computer or even a pocket calculator.     

Sample size calculations for clinical studies allowing for uncertainty about the variance

One of the most important steps in the design of a pharmaceutical clinical trial is the estimation of the sample size. For a superiority trial the sample size formula (to achieve a stated power) would be based on a given clinically meaningful difference and a value for the population variance. The formula is typically used as though this population variance is known whereas in reality it is unknown and is replaced by an estimate with its associated uncertainty. The variance estimate would be derived from an earlier similarly designed study (or an overall estimate from several previous studies) and its precision would depend on its degrees of freedom. This paper provides a solution for the calculation of sample sizes that allows for the imprecision in the estimate of the sample variance and shows how traditional formulae give sample sizes that are too small since they do not allow for this uncertainty with the deficiency being more acute with fewer degrees of freedom. It is recommended that the methodology described in this paper should be used when the sample variance has less than 200 degrees of freedom.     

On the wald,lagrangian multiplier and likelihood ratio tests when the information matrix is singular

Summary Modified formulas for the Wald and Lagrangian multiplier statistics are introduced and considered together with the likelihood ratio statistics for testing a typical null hypothesisH ₀ stated in terms of equality constraints. It is demonstrated, subject to known standard regularity conditions, that each of these statistics and the known Wald statistic has the asymptotic chi-square distribution with degrees of freedom equal to the number of equality constraints specified byH ₀ whether the information matrix is singular or nonsingular. The results of this paper include a generalization of the results of Sively (1959) concerning the equivalence of the Wald, Lagrange multiplier and likelihood ratio tests to the case of singular information matrices.     

A Retrievable Recipe for Inverse t

Although the percentage points of the Student-t distribution have been widely tabulated, a simple approximation is given and derived in this article. The approximation can be re-derived easily, since it is based on the percentage points from the Gaussian distribution, and can thus be used for applications requiring non-integer degrees of freedom (e.g., Welch's two-sample t test) and for arbitrary significance levels (e.g., for Bonferroni multiple comparison procedures). Comparisons between this approximation and others suggested in the literature indicate three-digit accuracy for even small degrees of freedom and tail areas.     

Restricted maximum likelihood estimation of joint mean‐covariance models

The class of joint mean‐covariance models uses the modified Cholesky decomposition of the within subject covariance matrix in order to arrive to an unconstrained, statistically meaningful reparameterisation. The new parameterisation of the covariance matrix has two sets of parameters that separately describe the variances and correlations. Thus, with the mean or regression parameters, these models have three sets of distinct parameters. In order to alleviate the problem of inefficient estimation and downward bias in the variance estimates, inherent in the maximum likelihood estimation procedure, the usual REML estimation procedure adjusts for the degrees of freedom lost due to the estimation of the mean parameters. Because of the parameterisation of the joint mean covariance models, it is possible to adapt the usual REML procedure in order to estimate the variance (correlation) parameters by taking into account the degrees of freedom lost by the estimation of both the mean and correlation (variance) parameters. To this end, here we propose adjustments to the estimation procedures based on the modified and adjusted profile likelihoods. The methods are illustrated by an application to a real data set and simulation studies. The Canadian Journal of Statistics 40: 225–242; 2012 © 2012 Statistical Society of Canada     

SIMULATIONS AND DERIVED APPROXIMATIONS FOR THE MEANS AND STANDARD DEVIATIONS OF THE CHARACTERISTIC ROOTS OF A WISHART MATRIX

Monte Carlo simulations were done to estimate the means and standard deviations of the characteristic roots of a Wishart matrix which can be used in computing tests of hypotheses concerning multiplicative terms in balanced linear-bilinear (multiplicative) models for an m × n table of data. In this report we extend the previous results (Mandel, 1971; Cornelius, 1980) to r ≤ 199, c ≤ 149 or r ≤ 149, c ≤ 199, where r and c are row and column degrees of freedom, respectively, of the two-way array of residuals (with total degrees of freedom rc) after fitting the linear effects. For 187 combinations of r and c at intervals over this domain, we used 5000 simulations to estimate expectations and standard deviations of the Wishart roots. Using weighted linear regression variable selection techniques, symmetric functions of r and c were obtained for approximating the simulated means and standard deviations. Use of these approximating functions will avoid the need for reference to tables for input to computer programs which require these values for tests of significance of sequentially fitted terms in the analyses of balanced linear-bilinear models.     

On the use of corrections for overdispersion

In studying fluctuations in the size of a blackgrouse ( Tetrao tetrix ) population, an autoregressive model using climatic conditions appears to follow the change quite well. However, the deviance of the model is considerably larger than its number of degrees of freedom. A widely used statistical rule of thumb holds that overdispersion is present in such situations, but model selection based on a direct likelihood approach can produce opposing results. Two further examples, of binomial and of Poisson data, have models with deviances that are almost twice the degrees of freedom and yet various overdispersion models do not fit better than the standard model for independent data. This can arise because the rule of thumb only considers a point estimate of dispersion, without regard for any measure of its precision. A reasonable criterion for detecting overdispersion is that the deviance be at least twice the number of degrees of freedom, the familiar Akaike information criterion, but the actual presence of overdispersion should then be checked by some appropriate modelling procedure.     

Evaluation of probabilities for the noncentral distributions and the difference of two T-variables with a desk calculator

It is demonstrated that integrals of the noncentral chi-square, noncentral F and noncentral T distributions can be evaluated on desk calculators. The same procedure can be used to compute probabilities for the distribution of the difference of two T-variables with equal degrees of freedom. The proposed method of computation can be used with any computer which yields probabilities for the chi-square and F distributions.     

A Note on Comparing the Power of Test Statistics at Low Significance Levels

It is an obvious fact that the power of a test statistic is dependent upon the significance (alpha) level at which the test is performed. It is perhaps a less obvious fact that the relative performance of two statistics in terms of power is also a function of the alpha level. Through numerous personal discussions, we have noted that even some competent statisticians have the mistaken intuition that relative power comparisons at traditional levels such as α=0.05 will be roughly similar to relative power comparisons at very low levels, such as the level α=510^?8, which is commonly used in genome-wide association studies. In this brief note, we demonstrate that this notion is in fact quite wrong, especially with respect to comparing tests with differing degrees of freedom. In fact, at very low alpha levels the cost of additional degrees of freedom is often comparatively low. Thus we recommend that statisticians exercise caution when interpreting the results of power comparison studies which use alpha levels that will not be used in practice.