Summary This paper reviews the wide variety of approaches to both univariate and multivariate calibration. In particular, in univariate calibration, it considers the classical, inverse, Bayesian and non-parametric approaches together with the approaches via tolerance regions. In multivariate calibration, it considers the ridge regression, multiple linear regression, partial least squares regression and principal components regression approaches together with the Bayesian and profile likelihood approaches. An extensive bibliography is provided.

Statistical Calibration: A Review

Preface Introduction One Sample A General Theory for Testing and Confidence Intervals Two Sample Problems One-Way Analysis of Variance Multiple Comparison Methods Simple Linear and Polynomial Regression The Analysis of Count Data Basic Experimental Designs Analysis of Covariance Factorial Treatment Structures Split Plots, Repeated Measures, Random Effects, and Subsampling Multiple Regression: Matrix Formation Unbalanced Multifactor Analysis of Variance Confounding and Fractional Replication in 2n Factorial Systems Nonlinear Regression Appendix A: Matrices Appendix B: Tables References Author Index Subject Index

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Analysis of Variance, Design, and Regression: Applied Statistical Methods

The concept of predictive likelihood is reviewed and studied. The emphasis is on comparing and clarifying some of the more important predictive likelihoods suggested in the literature. A unified modification and simplification of the sufficiency-based predictive likelihoods is sug- gested. Other predictive likelihoods discussed include the profile predictive likelihood and various modifications of it.

Predictive Likelihood: A Review

The Schwarz information criterion (SIC, BTC, SBC) is one of the most widely known and used tools in statistical model selection. The criterion was derived by Schwarz (1978) to serve as an asymptotic approximation to a transformation of the Bayesian posterior probability of a candidate model. Although the original derivation assumes that the observed data is independent, identically distributed, and arising from a probability distribution in the regular exponential family, SIC has traditionally been used in a much larger scope of model selection problems. To better justify the widespread applicability of SIC, we derive the criterion in a very general framework: one which does not assume any specific form for the likelihood function, but only requires that it satisfies certain non-restrictive regularity conditions.

Generalizing the derivation of the schwarz information criterion

: Since the publication of the seminal note, Kuilback and Leiber (1951), there has been continual endeavor in statistics and related fields to explicate the existing statistical methods and to develop new methods based on the logarithmic information of Shannon (1948). During the last four decades numerous information theoretic regression methods have been developed. Kullback and Rosenblatt (1957) pioneered the information theoretic approach to regression by explicating the usual regression qyantities such as sums of squares and F ratios in terms of information functions. We have now information theoretic methods for model and predictive density derivation, parameter estimation and testing, model selection, collinearity analysis, and influential observation detection which can be used in sampling theory and Bayesian regression analyses. The purpose of this paper is to integrate the existing entropy-based methods in a single work, to explore their interrelationships, to elaborate on information theoretic interpretations of the existing entropy-based diagnostics and to present information theoretic interpretations for traditional diagnostics.

Information theoretic regression methods

Let X 1, …, X m and Y 1, …, Y n be two independent random samples from normal populations N(μ1, σ1 2) and N(μ2, σ2 2), respectively. The statistical problem is to test H 0: μ1 = μ2 and σ1 2 = σ2 2 against H a : μ1 ≠ μ2 or σ1 2 ≠ σ2 2. Using Fisher's method of combining two independent test statistics, we suggest a test and prove that it is asymptotically optimal in the sense of Bahadur efficiency.

A Test of Equality of Two Normal Population Means and Variances

A prediction density function g* for the normal linear model is derived. This function is shown to dominate three well-known prediction densities by first constructing a specified class of densities that includes these three and then proving that g* is the optimal member of this class in the sense of minimizing a criterion based on the Kullback—Leibler divergence. g* coincides with a Bayesian prediction density assuming diffuse prior.

An Optimal Prediction Function for the Normal Linear Model

An alternative technique to current methods for constructing a prediction function for the normal linear regression model is proposed based on the concept of maximum likelihood. The form of this prediction function is evaluated and normalized to produce a multivariate Student's t-density. Consistency properties are established under regularity conditions, and an empirical comparison, based on the Kullback-Leibler information divergence, is made with some other prediction functions.

A maximum likelihood prediction function for the linear model with consistency results

Let x1,…, xr be the first r (1 ≤ r ≤ n) order statistics of a random sample of size n from an exponential distribution with probability density where σ > 0 and −∞ < β < + ∞ both are unknown parameters. The statistical problem is to test against H0 : θ e Ω0 H1 : θ e(Ω − Ω0) on the basis of x1,…, xr, where θ = (β, σ), Ω = {θ | 0 < σ ≤ σ0, β ≥ β0}, and Ω0 = {θ | σ = σ0, β = β0} Using Fisher’s method of combining two independent test statistics, we suggest a test and then prove that the test is asymptotically optimal in the sense of Bahadur efficiency.

An asymptotically efficient test for the location parameter and the scale parameter of an exponential distribution

We present a class of asymptotically distribution free tests for the equality of selected quantiles of two continuous distributions F and G based on independent random samples summarized by their empirical distribution functions, denoted [Fcirc] and Ĝ. Our test statistics are based on [Fcirc] evaluated at a quantile of the distribution function H defined as the weighted average Ĥ = b[Fcirc] + (l - b)Ĝ, 0≤b<1. The choice b = 0 yields the control quantile test studied by Chakraborti [4] and others. Inversion of the test statistic defined by b = 0.5 leads to a measure of the separation between F and G. We investigate and compare the performance of some of these tests and a normal theory test proposed by Perng et al. [12] via simulation.

S.K. Perng

Papers

A Test of Equality of Two Normal Population Means and Variances

An Optimal Prediction Function for the Normal Linear Model

A maximum likelihood prediction function for the linear model with consistency results

An asymptotically efficient test for the location parameter and the scale parameter of an exponential distribution

An asymptotically distribution freetest for assessing the separationbetween two distributions