Traditional Variance Research Articles

Simulation Designs for Quadratic Response Surface Models in the Presence of Model Misspecification

This article considers the selection of experimental designs for the estimation of second-order response surface metamodels in a simulation environment. Rather than construct designs based on the premise that the postulated model exactly represents the simulated response, as is the case in optimal design theory, we assume that the estimation process may be biased by the presence of unfitted third-order terms. We therefore seek to specify experimental plans that address the bias due to possible model misspecification as well as traditional variance considerations. The performance measure used for this “fit-protect” scenario is Box and Draper's design criterion of average mean squared error of predicted response. Four important classes of response surface experimental plans are examined: (1) central composite designs, (2) Box-Behnken plans, (3) three-level factorial experiments, and (4) small central composite designs. Each design class is studied in conjunction with three pseudorandom number assignment strategies: (1) the use of a unique set of streams at each design point; (2) the assignment of a common stream set to all experimental points: and (3) the simultaneous use of common and antithetic streams through design blocking. Comparisons of both design classes and assignment strategies are presented to assist the user in the selection of an appropriate experimental strategy.

Variance estimation for volume when D2H is the covariate in regression

For two realistic (but not real) populations of loblolly pine, variance estimators for weighed regression estimates of total volume are compared. The traditional variance estimators and two robust variance estimators proposed by Royall and Cumberland (R. M. Royall and W. G. Cumberland. 1981. JASA, J. Am. Stat. Assoc. 76: 924–930) are found to be quite unreliable. In contrast, the jackknife variance estimate and confidence intervals based on it are found to be much more reliable.

Assessing the Accuracy of the Maximum Likelihood Estimator: Observed Versus Expected Fisher Information

This paper concerns normal approximations to the distribution of the maximum likelihood estimator in one-parameter families. The traditional variance approximation is 1/§, where θ is the maximum likelihood estimator and § is the expected total Fisher information. Many writers, including R. A. Fisher, have argued in favour of the variance estimate 1/I(x), where I(x) is the observed information, i.e. minus the second derivative of the log likelihood function at θ given data x. We give a frequentist justification for preferring 1/I(x) to 1/§. The former is shown to approximate the conditional variance of 8 given an appropriate ancillary statistic which to a first approximation is I(x). The theory may be seen to flow naturally from Fisher's pioneering papers on likelihood estimation. A large number of examples are used to supplement a small amount of theory. Our evidence indicates preference for the likelihood ratio method of obtaining confidence limits.

Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information

SUMMARY This paper concerns normal approximations to the distribution of the maximum likelihood estimator in one-parameter families. The traditional variance approximation is 1/1.I, where 0 is the maximum likelihood estimator and fo is the expected total Fisher information. Many writers, including R. A. Fisher, have argued in favour of the variance estimate I/I(x), where I(x) is the observed information, i.e. minus the second derivative of the log likelihood function at # given data x. We give a frequentist justification for preferring I/I(x) to 1/.Io. The former is shown to approximate the conditional variance of # given an appropriate ancillary statistic which to a first approximation is I(x). The theory may be seen to flow naturally from Fisher's pioneering papers on likelihood estimation. A large number of examples are used to supplement a small amount of theory. Our evidence indicates preference for the likelihood ratio method of obtaining confidence limits.

A Generalization of the Response Surface Mean Square Error Criterion with a Specific Application to the Scope

A response surface criterion for design selection is developed in this paper for the case of estimation of a set of parametric functions rather than a single response. A specific application is emphasized, that in which interest, centers on the partial derivat. ives of the response function. Both bias and variance of the parametric functions are taken into consideration. For the case of the slope of the response function two situations are given special emphasis; the fitting of a first order model in the presence of a true second order structure, and the fitting of a second order function in the presence of a third order alternative. The traditional response surface bias and variance criteria arise as a special case for the situation in which a single parametric function is merely the ordinary response surface.

Application of a Measure of Information to the Design and Comparison of Regression Experiments

A normal regression experiment can be represented by \begin{equation*}\tag{1.1} Y_i = \sum_{j=1}^k X_{ij} \theta_j + \eta_i\qquad (i = 1, \cdots, n)\end{equation*} where $\{\eta_i/i = 1, \cdots, n\}$ is a set of normally distributed random variables with zero means and non-singular dispersion matrix $C, \theta = (\theta_1, \cdots, \theta_k)$ is the parameter-vector of interest and $X = (X_{ij})$ is a known $n \times k$ matrix which will be called the allocation matrix. The rows of $X$ will be called the allocation vectors. We denote the experiment by $\varepsilon(X, C)$. We assume that $C$ is known; generally it will be a function of $X, C(X)$. The particular realisation of $Y$ will be denoted $y$. The matrix $F = X'C^{-1}X$ is the Fisher-information-matrix of $\varepsilon(X, C)$. When $F$ is non-singular, one answer to the question What information does $y$ give about $\theta$? is to quote $F^{-1}$, the dispersion matrix of the maximum-likelihood-estimates of $\theta$. A strong argument in favour of this is that $F^{-1}$ is independent of both $\theta$ and $y$. The fact that it is independent of $\theta$ means that the answer is not local; the fact that it is independent of $y$ leads to simplicity. This approach is taken by Box and Hunter [1] in their work on rotatable designs. However, we must accept the fact that many experimenters wish to have a one-dimensional answer to the question i.e. we must associate with $\varepsilon(X, C)$ a single number which we call the information. For instance Elfving [5] has developed the use of trace $F^{-1}$. In this paper we adopt the measure of information introduced by Lindley [7]. In Section 2 we generalise Lindley's treatment of the regression situation to include the singular case, explain the uses of the measure and compare it with that of Elfving. Section 3 deals with the analogue of Elfving's main theorem. Theorems 4.1 and 4.2 of Section 4 provide links with the traditional variance approach. In Section 5 we derive the asymptotic form of the measure as the $n$ of (1.1) increases and show that this form can be derived also from Neyman-Pearsonian theory. In Section 6 the influence of nuisance parameters is discussed and an analogue of a theorem of Chernoff [2] is established.