Wolfgang Bischoff

On D-optimal designs for linear models under correlated observations with an application to a linear model with multiple response

Abstract In the general linear model we set conditions under which an exact D-optimal design for uncorrelated observations with common variance is also D-optimal for correlated observations. Further we determine conditions under which approximate D-optimal designs can be considered as approximate D-optimal designs for correlated observations. Then these results are applied to a regression model with multiple response generalizing Theorem 1 of Krafft and Schaefer (J. Multivariate Anal. 42, 1992). In the above context, however, a serious problem may arise if the covariance matrix is not known; for the Gauss-Markov estimator with respect to a D-optimal design does not need to be calculable for the correlated case. This leads to D-optimal-invariant designs introduced by Bischoff (Ann. Inst. Statist. Math., 44, 1992); such a design τ∗ remains D-optimal when the covariance matrix is changed, and additionally the Gauss-Markov estimator with respect to the design τ∗ stays fixed. For regression models with multiple response we determine classes of covariance matrices for which a D-optimal design for uncorrelated observations with common variance is D-optimal-invariant. As examples we consider linear models where each response belongs to a regression model with intercept term.

Asymptotics of a boundary crossing probability of a Brownian bridge with general trend

Let us consider a signal-plus-noise model γh(z)+B0(z), z ∈ [0,1], where γ > 0, h: [0,1] → ℝ, and B0 is a Brownian bridge. We establish the asymptotics for the boundary crossing probability of the weighted signal-plus-noise model for γ→∞, that is P (supzε [0,1]w(z)(γ h(z)+B0(z))>c), for γ→∞, (1) where w: [0,1]→ [0,∞ is a weight function and c>0 is arbitrary. By the large deviation principle one gets a result with a constant which is the solution of a minimizing problem. In this paper we show an asymptotic expansion that is stronger than large deviation. As byproduct of our result we obtain the solution of the minimizing problem occurring in the large deviation expression. It is worth mentioning that the probability considered in (1) appears as power of the weighted Kolmogorov test applied to the test problem H0: h≡ 0 against the alternative K: h>0 in the signal-plus-noise model.

Exact asymptotics for Boundary crossings of the brownian bridge with trend with application to the Kolmogorov test

We consider a boundary crossing probability of a Brownian bridgeB0 and a piecewise linear boundary functionu(t)−γh(t). The main result of this paper is an asymptotic expansion for γ→∞ of the boundary crossing probability thatB0(t) is larger than the piecewise linear boundary functionu(t)−γh(t) for somet. Such probabilities occur for instance in the context of change point problems when the Kolmogorov test is used. Examples are discussed showing that the approximation is rather accurate even for small positive γ values.

Optimal designs which are efficient for lack of fit tests

Linear regression models are among the models most used in practice, although the practitioners are often not sure whether their assumed linear regression model is at least approximately true. In such situations, only designs for which the linear model can be checked are accepted in practice. For important linear regression models such as polynomial regression, optimal designs do not have this property. To get practically attractive designs, we suggest the following strategy. One part of the design points is used to allow one to carry out a lack of fit test with good power for practically interesting alternatives. The rest of the design points are determined in such a way that the whole design is optimal for inference on the unknown parameter in case the lack of fit test does not reject the linear regression model. To solve this problem, we introduce efficient lack of fit designs. Then we explicitly determine the e k -optimal design in the class of efficient lack of fit designs for polynomial regression of degree k - 1.

ON EXACT D-OPTIMAL DESIGNS FOR REGRESSION MODELS WITH CORRELATED OBSERVATIONS

Let τ* be an exact D-optimal design for a given regression model Yτ = Xτβ + Zτ. In this paper sufficient conditions are given for sesigning how the covariance matrix of Zτ may be changed so that not only τ* remains D-optimal but also that the best linear unbiased estimator (BLUE) of β stays fixed for the design τ*, although the covariance matrix of Zτ* is changed. Hence under these conditions a best, according to D-optimality, BLUE of β is known for the model with the changed covariance matrix. The results may also be considered as determination of exact D-optimal designs for regression models with special correlated observations where the covariance matrices are not fully known. Various examples are given, especially for regression with intercept term, polynomial regression, and straight-line regression. A real example in electrocardiography is treated shortly.

A note on change point estimation in dose-response trials

An often used model for the shape of a drugs dose-response relationship is the following: the curve increases to a certain dose until a plateau is reached. In this situation, the minimum dose with maximum effect can be seen as a change point in a regression model. We present four estimators for the change point and compare their performance with respect to selection rates and loss function based criteria by extensive simulations. The choice of the loss function depending on the practical situation is discussed. Application of the estimators is illustrated by an example.

Determinant formulas with applications to designing when the observations are correlated

In the general linear model consider the designing problem for the Gauß-Markov estimator or for the least squares estimator when the observations are correlated. Determinant formulas are proved being useful for theD-criterion. They allow, for example, a (nearly) elementary proof and a generalization of recent results for an important linear model with multiple response. In the second part of the paper the determinant formulas are used for deriving lower bounds for the efficiency of a design. These bounds are applied in examples for tridiagonal covariance matrices. For these examples maximin designs are determined.

Cusum techniques for timeslot sequences with applications to network surveillance

We develop two cusum change-point detection algorithms for data network monitoring applications where numerous and various performance and reliability metrics are available to aid with the early identification of realized or impending failures. We confront three significant challenges with our cusum algorithms: (1) the need for nonparametric techniques so that a wide variety of metrics can be included in the monitoring process, (2) the need to handle time varying distributions for the metrics that reflect natural cycles in work load and traffic patterns, and (3) the need to be computationally efficient with the massive amounts of data that are available for processing. The only critical assumption we make when developing the algorithms is that suitably transformed observations within a defined timeslot structure are independent and identically distributed under normal operating conditions. To facilitate practical implementations of the algorithms, we present asymptotically valid thresholds. Our research was motivated by a real-world application and we use that context to guide the design of a simulation study that examines the sensitivity of the cusum algorithms.

On maximin designs for correlated observations

In the linear model, we consider the problem of finding optimal or efficient designs with respect to the D-criterion when the covariance matrix is an unknown element of a class . In general, designs that are efficient for each do not exist. Therefore, maximin designs are of interest. These designs maximize the minimal efficiency where the minimum is taken over all possible covariance matrices and the maximum is taken over all feasible designs. Efficient maximin designs are derived for tridiagonal covariance matrices.

MINIMAX- AND Γ-MINIMAX ESTIMATION OF A BOUNDED NORMAL MEAN UNDER LINEX LOSS

Estimating a bounded normal mean is considered under LINEX loss