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We provide a statistical review of the "optimal fingerprinting" approach presented in Allen and Tett (Clim Dyn 15:419-434, 1999) in light of the severe criticism of McKitrick (Checking for model consistency in optimal fingerprinting: a comment. Clim Dyn 58:405–411, 2022). Our review finds that the "optimal fingerprinting" approach would survive much of McKitrick (2022)'s criticism by enforcing two conditions related to the conduct of the null simulation of the climate model, and the accuracy of the null setting climate model. The conditions we proposed are simpler and easier to verify than those in McKitrick (2022). We provide additional remarks on the residual consistency test in Allen and Tett (1999), showing that it is operational for checking the agreement between the residual covariance matrices of the null simulation and the physical internal variation under certain conditions. We further provide the reason why the Feasible Generalized Least Square method, much advocated by McKitrick (2022), is not regarded as operational by geophysicists. [ABSTRACT FROM AUTHOR]

Allen and Tett (1999, herein AT99) introduced a Generalized Least Squares (GLS) regression methodology for decomposing patterns of climate change for attribution purposes and proposed the "Residual Consistency Test" (RCT) to check the GLS specification. Their methodology has been widely used and highly influential ever since, in part because subsequent authors have relied upon their claim that their GLS model satisfies the conditions of the Gauss-Markov (GM) Theorem, thereby yielding unbiased and efficient estimators. But AT99 stated the GM Theorem incorrectly, omitting a critical condition altogether, their GLS method cannot satisfy the GM conditions, and their variance estimator is inconsistent by construction. Additionally, they did not formally state the null hypothesis of the RCT nor identify which of the GM conditions it tests, nor did they prove its distribution and critical values, rendering it uninformative as a specification test. The continuing influence of AT99 two decades later means these issues should be corrected. I identify 6 conditions needing to be shown for the AT99 method to be valid. [ABSTRACT FROM AUTHOR]

This article provides the best linear unbiased estimators (â and...) and the best linear invariant estimators (âBLIE and... BLIE) for location a and scale b parameters of Burr type III distribution based on dual generalized order statistics. [ABSTRACT FROM AUTHOR]

The Dirichlet process has been used extensively in Bayesian non parametric modeling, and has proven to be very useful. In particular, mixed models with Dirichlet process random effects have been used in modeling many types of data and can often outperform their normal random effect counterparts. Here we examine the linearmixed model with Dirichlet process random effects from a classical view, and derive the best linear unbiased estimator (BLUE) of the fixed effects. We are also able to calculate the resulting covariancematrix and find that the covariance is directly related to the precision parameter of the Dirichlet process, giving a new interpretation of this parameter. We also characterize the relationship between the BLUE and the ordinary least-squares (OLS) estimator and show how confidence intervals can be approximated. [ABSTRACT FROM AUTHOR]

Observations that are realizations of some continuous process are frequently found in science, engineering, economics, and other fields. In this paper, we consider linear models with possible random effects and where the responses are random functions in a suitable Sobolev space. In particular, the processes cannot be observed directly. By using smoothing procedures on the original data, both the response curves and their derivatives can be reconstructed, both as an ensemble and separately. From these reconstructed functions, one representative sample is obtained to estimate the vector of functional parameters. A simulation study shows the benefits of this approach over the common method of using information either on curves or derivatives. The main theoretical result is a strong functional version of the Gauss–Markov theorem. This ensures that the proposed functional estimator is more efficient than the best linear unbiased estimator (BLUE) based only on curves or derivatives. [ABSTRACT FROM AUTHOR]

The 1997 Biometrics paper by Mike Kenward and James Roger has become a citation classic (more than 1260 citations by End June 2013 according to Google Scholar) and the solution that they proposed to deal with the problem of significance tests of fixed effects in REML models is now incorporated in many software packages and accepted by all biostatisticians as the method of choice. Nevertheless, it does not solve all problems, since there is more to analysis than just significance and since the problems that models with more than one variance pose arise in many contexts. In this paper, I discuss some problems and applications and make some tentative suggestions as to how they may be tackled. My excuse for raising problems I do not solve is that it may inspire James and Mike to complete what they started. [ABSTRACT FROM AUTHOR]

This paper deals with an application of regression analysis to the regulation of the blood-sugar under diabetes mellitus. Section 2 gives a description of Gram-Schmidt orthogonalization, while Section 3 discusses the difference between Gauss-Markov estimation and Least Squares Estimation. Section 4 is devoted to the statistical analysis of the blood-sugar during the night. The response change of blood-sugar is explained by three variables: time, food and physical activity ("Bewegung"). At the beginning of the section it is shown that the proposed method was very successful in 2007. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a method of estimating the regression and variance parameters in the multiple linear regression Berkson model for a continuous-time stochastic process with uncorrelated increments. Under minimal conditions, we establish (i) the Gauss–Markov theorem and the quadratic mean—as well as the strong consistency of the proposed estimate of the regression parameter and (ii) the weak consistency of the proposed estimate of the variance parameter. [Copyright &y& Elsevier]

Abstract: The minimum variance linear unbiased estimators (MVLUE), the best linear invariant estimators (BLIE) and the maximum likelihood estimators (MLE) based on n-selected generalized order statistics are presented for the parameters of the Burr XII distribution. [Copyright &y& Elsevier]

In this paper, first we make a maximal extension of the well-known Gauss–Markov Theorem (GMT) in its linear framework. In particular, the maximal class of distributions of error term for which the GMT holds is derived. Second, we establish a nonlinear version of the maximal GMT and describe some interesting families of distributions of error term for which the nonlinear GMT holds. [Copyright &y& Elsevier]

This article gives a nonlinear version of the Gauss-Markov theorem. It is shown that in a linear regression model y = X beta + u, the lower bound for the risk matrix E[beta - beta][beta - beta]' of a nonlinear estimator of beta belonging to a certain class is the covariance matrix of the Gauss-Markov estimator, provided the distribution of error term u belongs to the class of elliptically symmetric distributions with second moments. [ABSTRACT FROM AUTHOR]