On the use of curve fitting to model the error of the Cornish-Fisher expansion of the pearson type-VI distribution

The Cornish-Fisher expansion of the Pearson type VI distribution is known to be reasonably accurate when both degrees of freedom are relatively large (say greater than or equal to 5). However, when either or both degrees of freedom are less than 5, the accuracy of the computed percentage point begins to suffer; in some cases severely. To correct for this, the error surface in the degrees of freedom plane is modeled by least squares curve fitting for selected levels of tail probability (.025, .05, and .10) which can be used to adjust the percentage point obtained from the usual Cornish-Fisher expansion. This adjustment procedure produces a computing algorithm that computes percentage points of the Pearson type VI distribution at the above probability levels, accurate to at least + 1 in 3 digits in approximately 11 milliseconds per subroutine call on an IBM 370/145. This adjusted routine is valid for both degrees of freedom greater than or equal to 1.     

Geometry and Degrees of Freedom of Linearly Constrained Generalized Lasso

The least squares fit in a linear regression is always unique even when the design matrix has rank deficiency. In this paper, we extend this classic result to linearly constrained generalized lasso. It is shown that under a mild condition, the fit can be represented as a projection onto a polytope and, hence, is unique no matter whether design matrix X has full column rank or not. Furthermore, a formula for the degrees of freedom is derived to characterize the effective number of parameters. It directly yields an unbiased estimate of degrees of freedom, which can be incorporated in an information criterion for model selection.     

A decomposition of the Bradley–Blackwood paired-samples omnibus test

We show that the Bradley–Blackwood simultaneous test for equal means and equal variances in paired-samples additively decomposes into separate tests of these hypotheses. The test of equal variances in the decomposition is the standard Pitman–Morgan procedure. The test of equal means in the decomposition is based on a t-ratio with (n ? 2) degrees of freedom and has the additional restriction that the variances are equal.     

从统计学史解析“自由度”概念

自由度是统计学中一个十分重要而又长期没有被圆满解释的概念。对此,从统计学史角度,对皮尔逊、费歇尔有关自由度问题争论原始文献细致考察,彻底澄清了自由度概念的内涵及与其相关的统计思想,弥补了Fienberg、Stigler与陈希孺已有解释的缺陷。研究表明：皮尔逊关于卡方检验中无论总体分布已知还是其来自于样本推断统计量都具有同一分布的错误判断,导致卡方检验的准确性出现偏差,这种偏差虽被同时代少数几个统计学家察觉但他们却无法解释其根源。费歇尔提出自由度概念并结合n维几何、假设检验与最大似然方法的论证不仅修正了皮尔逊的错误,也完善了从样本统计量估计总体参数的数理逻辑。     

A Decomposition of Pearson–Fisher and Dzhaparidze–Nikulin Statistics and Some Ideas for a More Powerful Test Construction

An explicit decomposition on asymptotically independent distributed as chi-squared with one degree of freedom components of the Pearson–Fisher and Dzhaparidze–Nikulin tests is presented. The decomposition is formally the same for both tests and is valid for any partitioning of a sample space. Vector-valued tests, components of which can be not only different scalar tests based on the same sample, but also scalar tests based on components or groups of components of the same statistic are considered. Numerical examples illustrating the idea are presented.     

The Effect of Estimation of Unknown Degrees of Freedom on Estimation of Remaining Parameters in Spanos’ Conditional t Heteroskedastic Model

This article investigates the effect of estimation of unknown degrees of freedom on efficient estimation of remaining parameters in Spanos’ conditional t heteroskedastic model. We compare by simulation three maximum likelihood estimators (MLEs) of the remaining parameters in the model: the MLE of the remaining parameters when all the parameters are estimated by the MLE, when the degrees of freedom is estimated by a method of moments estimator, and when the degrees of freedom is erroneously specified. The latter two methods are found to perform poorly compared to the former method for the inference of variance parameters in the model. Thus, efficient estimation of degrees of freedom by the MLE is important to estimate efficiently the remaining variance parameters.     

A note on the power of the f-test for testing linear hypothesis with elliptic t errors

The power of the classical .F-test for testing the regression coefficient of a general linear model with elliptic t error variable depends on the degrees of freedom of the t- distribution. In this note it is shown that the power of the F-test based on t-distribution is greater than the normal based test at smaller level of significance.     

ON LEVENE'S TESTS OF VARIANCE HOMOGENEITY

Levene's tests of homogeneity of treatment variances in completely randomised and randomised complete block experiments are examined. These tests are essentially standard analysis of variance F-tests performed on functions of the absolute values of residuals. It is found that in order to achieve (i) equality of component mean squares under the null hypothesis, and (ii) nominal significance levels, the various standard degrees of freedom need to be modified.     

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 Tests for Group Variation With a Small to Moderate Number of Groups

This paper considers a family of penalized likelihood score tests for group variation. The tests can be indexed by a measure of degrees of freedom. At one extreme, with degrees of freedom one less than the number of groups, is the usual score test for a fixed effects alternative using indicator variables for the groups, while at the other extreme, in the limit as the degrees of freedom 0, is a test closely related to a score test based on a random effects alternative. Asymptotic power comparisons are made for the tests in the family. As would be expected, different members of the family are more efficient for different alternatives. Generally the tests with smaller degrees of freedom appear to have better power than the standard test for alternatives focusing on differences among the larger groups, and lower power for alternatives focusing on differences among the smaller groups. Simulations indicate the asymptotic approximation to the distribution performs better for the tests with small degrees of freedom.     

Robustness of the student t based M-estimator

This paper considers the maximum likelihood type (M) estimator based on Student's t distribution for the location/scale model. The Student t M-estimator is generally thought to be robust to outliers. This paper shows that this is only true if the degrees of freedom parameter is kept fixed. By contrast, if the degrees of freedom parameter is also estimated from the data, the influence functions for the scale and degrees of freedom parameter become unbounded. Moreover, the influence function of the location parameter remains bounded, but its change-of-variance function is unboi~nded. The intuitioil behind these results is explained in the paper. The rates at which both the influence functions and the change-of-variance function diverge to infinity, are very slow. Tliis implies that outliers have to be extremely large in order to become detrimental to the performance of the Student t based M-estimator with estimated degrees of freedom. The theoretical results are illustrated in a a simulation experiment using several related competing estimators and several distributions for the error process.     

Approximating the difference of two t-variables for all degrees of freedom using truncated variables

A scaled t‐distribution is used to approximate the distribution of a linear combination of two independent t‐variables for any number of degrees of freedom, and in particular for low degrees of freedom where moments do not exist. The approximation is the method‐of‐moments solution to the analogous problem with truncated t‐variables. The approximation exists for all degrees of freedom, is very accurate for more than two degrees of freedom, and performs as well as other approximations of this form when they exist.     

Kruskal–Wallis, Multiple Comparisons and Efron Dice

The Kruskal–Wallis test is a rank–based one way ANOVA. Its test statistic is shown here to be a quadratic form among the Mann–Whitney or Kendall tau concordance measures between pairs of treatments. But the full set of such concordance measures has more degrees of freedom than the Kruskal–Wallis test uses, and the independent surplus is attributable to circularity, or non–transitive effects. The meaning of circularity is well illustrated by Efron dice. The cases of k = 3, 4 treatments are analysed thoroughly in this paper, which also shows how the full sum of squares among all concordance measures can be decomposed into uncorrelated transitive and non–transitive circularity effects. A multiple comparisons procedure based on patterns of transitive orderings among treatments is implemented. The testing of circularities involves non–standard asymptotic distributions. The asymptotic theory is deferred, but Monte Carlo permutation tests are easy to implement.     

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     

Approximate F-tests of multiple degree of freedom hypotheses in generalized least squares analyses of unbalanced split-plot experiments

Approximate t-tests of single degree of freedom hypotheses in generalized least squares analyses (GLS) of mixed linear models using restricted maximum likelihood (REML) estimates of variance components have been previously developed by Giesbrecht and Burns (GB), and by Jeske and Harville (JH), using method of moment approximations for the degrees of freedom (df) for the tstatistics. This paper proposes approximate Fstatistics for tests of multiple df hypotheses using one-moment and two-moment approximations which may be viewed as extensions of the GB and JH methods. The paper focuses specifically on tests of hypotheses concerning the main-plot treatment factor in split-plot experiments with missing data. Simulation results indicate usually satisfactory control of Type I error rates.     

Alternative transformations to eliminate fixed effects

In a panel data model with fixed individual effects, a number of alternative transformations are available to eliminate these effects such that the slope parameters can be estimated from ordinary least squares on transformed data. In this note we show that each transformation leads to algebraically the same estimator if the transformed data are used efficiently (i.e. if GLS is applied). If OLS is used, however, differences may occur and the routinely computed variances, even after degrees of freedom correction, are incorrect. In addition, it may matter whether “redundant” observations are used or not.     

A method for choosing the smoothing parameter in a semi-parametric model for detecting change-points in blood flow

In a smoothing spline model with unknown change-points, the choice of the smoothing parameter strongly influences the estimation of the change-point locations and the function at the change-points. In a tumor biology example, where change-points in blood flow in response to treatment were of interest, choosing the smoothing parameter based on minimizing generalized cross-validation (GCV) gave unsatisfactory estimates of the change-points. We propose a new method, aGCV, that re-weights the residual sum of squares and generalized degrees of freedom terms from GCV. The weight is chosen to maximize the decrease in the generalized degrees of freedom as a function of the weight value, while simultaneously minimizing aGCV as a function of the smoothing parameter and the change-points. Compared with GCV, simulation studies suggest that the aGCV method yields improved estimates of the change-point and the value of the function at the change-point.     

Detection of non-Gaussianity

We develop two tests sensitive to various departures from composite goodness-of-fit hypothesis of normality. The tests are based on the sums of squares of some components naturally arising in decomposition of the Shapiro–Wilk-type statistic. Each component itself has diagnostic properties. The numbers of squared components in sums are determined via some novel selection rules based on the data. The new solutions prove to be effective tools in detecting a broad spectrum of sources of non-Gaussianity. We also discuss two variants of the new tests adjusted to verification of simple goodness-of-fit hypothesis of normality. These variants also compare well to popular competitors.     

Evaluating the Effective Degrees of Freedom of the Delete-a-Group Jackknife

The delete-a-group jackknife is sometimes used when estimating the variances of statistics based on a large sample. We investigate heavily poststratified estimators for a population mean and a simple regression coefficient, where both full-sample and domain estimates are of interest. The delete-a-group (DAG) jackknife employing 30, 60, and 100 replicates is found to be highly unstable, even for large sample sizes. The empirical degrees of freedom of these DAG jackknives are usually much less than their nominal degrees of freedom. This analysis calls into question whether coverage intervals derived from replication-based variance estimators can be trusted for highly calibrated estimates.     

Estimation of the parameters of a general student’s t distribution

We have observations for a t distribution with unknown mean, variance, and degrees of freedom, each of which we wish to estimate. The major problem lies in the estimate of the degrees of freedom. We show that a relatively efficient yet very simple estimator is a given function of the ratio of percentile estimates. We derive the appropriate estimator, provide equations for transformation and standard errors, contrast this with other estimators, and give examples.