Some Elementary Theory of Permutation Tests

In literature, permutation tests are mostly derived by means of heuristic arguments (Edgington and Onghena, 2007 Edgington, E.S., Onghena, P. (2007). Randomization Tests (4th ed.). Boca Raton: Chapman & Hall/CRC.[Crossref] , [Google Scholar]; Good, 2005 Good, P. (2005). Permutation, Parametric, and Bootstrap Tests of Hypotheses (3rd ed.). New York: Springer-Verlag. [Google Scholar]). In this paper, we derive them within the sufficiency and conditionality principles of inference. Most important of their properties are exposed without formal proofs that can be found in the book by Pesarin and Salmaso (2010).     

Bayesian estimation for time-series regressions improved with exact likelihoods

We propose an estimation procedure for time-series regression models under the Bayesian inference framework. With the exact method of Wise [Wise, J. (1955). The autocorrelation function and spectral density function. Biometrika, 42, 151–159], an exact likelihood function can be obtained instead of the likelihood conditional on initial observations. The constraints on the parameter space arising from the stationarity conditions are handled by a reparametrization, which was not taken into consideration by Chib [Chib, S. (1993). Bayes regression with autoregressive errors: A Gibbs sampling approach. J. Econometrics, 58, 275–294] or Chib and Greenberg [Chib, S. and Greenberg, E. (1994). Bayes inference in regression model with ARMA(p, q) errors. J. Econometrics, 64, 183–206]. Simulation studies show that our method leads to better inferential results than their results.     

线性GMDH参数模型的无偏估计研究

 多元线性回归分析中,参数无偏性是参数估计方法的一个重要指标。本文对线性GMDH参数模型建立多元线性模型进行了研究,得到以下结论：一,在满足经典线性回归模型的假设条件下,其参数估计量具有无偏的性质;二,在满足其它假设条件下,可以在样本量少于待估参数的情况下建模,估计的参数也是无偏的;三,用参数GMHD方法建模时,它对完全多重共线性是免疫的。     

Approximate is Better than “Exact” for Interval Estimation of Binomial Proportions

For interval estimation of a proportion, coverage probabilities tend to be too large for “exact” confidence intervals based on inverting the binomial test and too small for the interval based on inverting the Wald large-sample normal test (i.e., sample proportion ± z-score × estimated standard error). Wilson's suggestion of inverting the related score test with null rather than estimated standard error yields coverage probabilities close to nominal confidence levels, even for very small sample sizes. The 95% score interval has similar behavior as the adjusted Wald interval obtained after adding two “successes” and two “failures” to the sample. In elementary courses, with the score and adjusted Wald methods it is unnecessary to provide students with awkward sample size guidelines.     

Distribution of the Λ-Criterion for Sphericity Test and Its Connection to a Generalized Dirichlet Model

It is shown that the exact null distribution of the likelihood ratio criterion for sphericity test in the p-variate normal case and the marginal distribution of the first component of a (p ? 1)-variate generalized Dirichlet model with a given set of parameters are identical. The exact distribution of the likelihood ratio criterion so obtained has a general format for every p. A novel idea is introduced here through which the complicated exact null distribution of the sphericity test criterion in multivariate statistical analysis is converted into an easily tractable marginal density in a generalized Dirichlet model. It provides a direct and easiest method of computation of p-values. The computation of p-values and a table of critical points corresponding to p = 3 and 4 are also presented.     

Exact Bootstrap Variances of the Area Under ROC Curve

The area under the Receiver Operating Characteristic (ROC) curve (AUC) and related summary indices are widely used for assessment of accuracy of an individual and comparison of performances of several diagnostic systems in many areas including studies of human perception, decision making, and the regulatory approval process for new diagnostic technologies. Many investigators have suggested implementing the bootstrap approach to estimate variability of AUC-based indices. Corresponding bootstrap quantities are typically estimated by sampling a bootstrap distribution. Such a process, frequently termed Monte Carlo bootstrap, is often computationally burdensome and imposes an additional sampling error on the resulting estimates. In this article, we demonstrate that the exact or ideal (sampling error free) bootstrap variances of the nonparametric estimator of AUC can be computed directly, i.e., avoiding resampling of the original data, and we develop easy-to-use formulas to compute them. We derive the formulas for the variances of the AUC corresponding to a single given or random reader, and to the average over several given or randomly selected readers. The derived formulas provide an algorithm for computing the ideal bootstrap variances exactly and hence improve many bootstrap methods proposed earlier for analyzing AUCs by eliminating the sampling error and sometimes burdensome computations associated with a Monte Carlo (MC) approximation. In addition, the availability of closed-form solutions provides the potential for an analytical assessment of the properties of bootstrap variance estimators. Applications of the proposed method are shown on two experimentally ascertained datasets that illustrate settings commonly encountered in diagnostic imaging. In the context of the two examples we also demonstrate the magnitude of the effect of the sampling error of the MC estimators on the resulting inferences.     

A Test of Independence in Two-Way Contingency Tables Based on Maximal Correlation

Maximal correlation has several desirable properties as a measure of dependence, including the fact that it vanishes if and only if the variables are independent. Except for a few special cases, it is hard to evaluate maximal correlation explicitly. We focus on two-dimensional contingency tables and discuss a procedure for estimating maximal correlation, which we use for constructing a test of independence. We compare the maximal correlation test with other tests of independence by Monte Carlo simulations. When the underlying continuous variables are dependent but uncorrelated, we point out some cases for which the new test is more powerful.     

Formulas for the Exact LMS and LQS Estimators

The author presents the derivation of formulas for the calculation of critical values of the median function or the general version of it, namely, the quantile functions. In statistics, these functions are used to detect outliers in the data set and to make predictions that are resistant to outliers. Therefore, these formulas can also be used as estimators for these regressions. The fact that these formulas are able to calculate the global optimum gives the exact least median squares or the exact least quantile of squares estimators. The author provides the theoretical background for deriving these estimator formulas and derives the estimator formulas for regression models up to three parameters. In addition, the author provides guides for the derivation of formulas for other models, illustrates the use of these formulas, and emphasizes their properties that are useful for future works. One important conclusion is that each regression model has its own set of formulas.