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非参数回归中方差函数参数形式的新检验AI 解读

Summary In the common non-parametric regression model the problem of testing for the parametric form of the conditional variance is considered. A stochastic process based on the difference between the empirical processes that are obtained from the standardized non-parametric residuals under the null hypothesis (of a specific parametric form of the variance function) and the alternative is introduced and its weak convergence established. This result is used for the…

最大最小与贝叶斯最优设计的支撑点数目AI 解读

We consider maximin and Bayesian D-optimal designs for nonlinear regression models. The maximin criterion requires the specification of a region for the nonlinear parameters in the model, while the Bayesian optimality criterion assumes that a prior for these parameters is available. On interval parameter spaces, it was observed empirically by many authors that an increase of uncertainty in the prior information (i.e., a larger range for the parameter space in the maximin…

受限设计空间下剂量反应模型的最优设计AI 解读

In dose–response studies, the dose range is often restricted because of concerns over drug toxicity and/or efficacy. We derive optimal designs for estimating the underlying dose–response curve for a restricted or unrestricted dose range with respect to a broad class of optimality criteria. The underlying curve belongs to a diversified set of link functions suitable for the dose–response studies and having a common canonical form. These include the fundamental binary response…

Michaelis–Menten 酶动力学函数拟合优度的最优设计AI 解读

We construct efficient designs for the Michaelis–Menten enzyme kinetic model capable of checking model assumptions. An extended model called EMAX is also considered for this purpose. This model is widely used in pharmacokinetics and reduces to the Michaelis–Menten model for a specific choice of parameter settings. Our strategy is to find efficient designs for estimating the parameters in the EMAX model and at the same time test the validity of the Michaelis–Menten model…

二元反应生物测定中有效剂量的非参数估计札记AI 解读

For the common binary response model, we propose a direct method for the nonparametric estimation of the effective dose level. The estimator is obtained by the composition of a nonparametric estimate of the quantile response curve and a classical density estimate. The new method yields a simple and reliable monotone estimate of the effective dose-level curve α → EDα and is appealing to users of conventional smoothing methods as kernel estimators, local polynomials, series…

一类非线性回归模型的最优设计AI 解读

For a broad class of nonlinear regression models we investigate the local E- and c-optimal design problem. It is demonstrated that in many cases the optimal designs with respect to these optimality criteria are supported at the Chebyshev points, which are the local extrema of the equi-oscillating best approximation of the function f0≡0 by a normalized linear combination of the regression functions in the corresponding linearized model. The class of models includes rational,…

傅里叶回归模型中单个系数估计的最优设计AI 解读

In the common trigonometric regression model, we investigate the optimal design problem for the estimation of the individual coefficients, where the explanatory variable varies in the interval $[-a,a]$, $0 <a \le \pi$. It is demonstrated that the structure of the optimal design depends sensitively on the size of the design space. For many important cases, optimal designs can be found explicitly, where the complexity of the solution depends on the value of the parameter a and…

Monod模型中的高效实验设计AI 解读

Summary Estimation and experimental design in a non-linear regression model that is used in microbiology are studied. The Monod model is defined implicitly by a differential equation and has numerous applications in microbial growth kinetics, water research, pharmacokinetics and plant physiology. It is proved that least squares estimates are asymptotically unbiased and normally distributed. The asymptotic covariance matrix of the estimator is the basis for the construction…

回归曲线的非参数比较：经验过程方法AI 解读

We propose a new test for the comparison of two regression curves that is based on a difference of two marked empirical processes based on residuals. The large sample behavior of the corresponding statistic is studied to provide a full nonparametric comparison of regression curves. In contrast to most procedures suggested in the literature, the new procedure is applicable in the case of different design points and heteroscedasticity. Moreover, it is demonstrated that the…

非参数协方差分析AI 解读

In the problem of testing the equalityof k regression curves from independent samples, we discuss three methods using nonparametric estimators of the regression function. The first test is based on a linear combination of estimators for the integrated variance function in the individual samples and in the combined sample. The second approach transfers the classical one-way analysis of variance to the situation of comparing non-parametric curves, while the third test compares…

通过最大化最低 D 与 D₁ 效率构造多项式回归稳健设计AI 解读

In the common polynomial regression of degree m we determine the design which maximizes the minimum of the $D$-efficiency in the model of degree $m$ and the $D_1$-efficiencies in the models of degree $m-j,\dots, m +k$ ($j, k\ge 0$ given). The resulting designs allow an efficient estimation of the parameters in the chosen regression and have reasonable efficiencies for checking the goodness-of-fit of the assumed model of degree $m$ by testing the highest coefficients in the…

多项式回归的受约束 D 与 D₁ 最优设计AI 解读

In the common polynomial regression model of degree m we consider the problem of determining the $D$- and $D_1$-optimal designs subject to certain constraints for the $D_1$-efficiencies in the models of degree $m - j, m - j + 1,\dots, m + k(m > j \geq 0, k \geq 0 \text{given})$.We present a complete solution of these problems, which on the one hand allow a fast computation of the constrained optimal designs and, on the other hand, give an answer to the question of the…

有理模型与加权多项式回归的最优设计AI 解读

In this paper $D$-optimal designs for the weighted polynomial regression model of degree $p$ with efficiency function $(1 + x^2)^{-n}$ are presented. Interest in these designs stems from the fact that they are equivalent to locally $D$-optimal designs for inverse quadratic polynomial models. For the unrestricted design space $\mathbb{R}$ and $p < n$, the $D$-optimal designs put equal masses on $p + 1$ points which coincide with the zeros of an ultraspherical polynomial,…

基于方差估计量之差的回归函数形式一致检验AI 解读

In this paper we study the problem of testing the functional form of a given regression model. A consistent test is proposed which is based on the difference of the least squares variance estimator in the assumed regression model and a nonparametric variance estimator. The corresponding test statistic can be shown to be asymptotically normal under the null hypothesis and under fixed alternatives with different rates of convergence corresponding to both cases. This provides a…

多个回归函数的非参数比较：精确与渐近理论AI 解读

A new test is proposed for the comparison of two regression curves $f$ and $g$. We prove an asymptotic normal law under fixed alternatives which can be applied for power calculations, for constructing confidence regions and for testing precise hypotheses of a weighted $L_2$ distance between $f$ and $g$ . In particular, the problem of nonequal sample sizes is treated, which is related to a peculiar formula of the area between two step functions. These results are extended in…

非参数回归中的方差估计：怎样选择才合理？AI 解读

Summary The exact mean-squared error (MSE) of estimators of the variance in nonparametric regression based on quadratic forms is investigated. In particular, two classes of estimators are compared: Hall, Kay and Titterington's optimal difference-based estimators and a class of ordinary difference-based estimators which generalize methods proposed by Rice and Gasser, Sroka and Jennen-Steinmetz. For small sample sizes the MSE of the first estimator is essentially increased by…

非参数回归中的异方差检验AI 解读

Summary The importance of being able to detect heteroscedasticity in regression is widely recognized because efficient inference for the regression function requires that heteroscedasticity is taken into account. In this paper a simple consistent test for heteroscedasticity is proposed in a nonparametric regression set-up. The test is based on an estimator for the best L 2-approximation of the variance function by a constant. Under mild assumptions asymptotic normality of…

傅里叶回归阶数识别的最优设计AI 解读

For the Fourier regression model, we determine optimal designs for identifying the order of periodicity. It is shown that the optimal design problem for trigonometric regression models is equivalent to the problem of optimal design for discriminating between certain homo-and heteroscedastic polynomial regression models. These optimization problems are then solved using the theory of canonical moments, and the optimal discriminating designs for the Fourier regression model…

线性回归模型的验证AI 解读

A new test is proposed in order to verify that a regression function, say $g$, has a prescribed (linear) parametric form. This procedure is based on the large sample behavior of an empirical $L^2$-distance between $g$ and the subspace $U$ spanned by the regression functions to be verified. The asymptotic distribution of the test statistic is shown to be normal with parameters depending only on the variance of the observations and the $L^2$-distance between the regression…

非参数析因设计中的Box型近似AI 解读

Linear rank statistics in nonparametric factorial designs are asymptotically normal and, in general, heteroscedastic. In a comprehensive simulation study, the asymptotic chi-squared law of the corresponding quadratic forms is shown to be a rather poor approximation of the finite-sample distribution. Motivated by this problem, we propose simple finite-sample size approximations for the distribution of quadratic forms in factorial designs under a normal heteroscedastic error…

基于“标准化”最优性准则的实验设计AI 解读

Summary We introduce a new class of ‘standardized' optimality criteria which depend on ‘standardized' covariances of the least squares estimators and provide an alternative to the commonly used criteria in design theory. Besides a nice statistical interpretation the new criteria satisfy an extremely useful invariance property which allows an easy calculation of optimal designs on many linearly transformed design spaces.

部分指定异方差结构模型的最优贝叶斯设计AI 解读

We consider the problem of finding a nonsequential optimal design for estimating parameters in a generalized exponential growth model. This problem is solved by first considering polynomial regression models with error variances that depend on the covariate value and unknown parameters. A Bayesian approach is adopted, and optimal Bayesian designs supported on a minimal number of support points for estimating the coefficients in the polynomial model are found analytically.…

非线性回归中贝叶斯 c 与 D 最优设计札记AI 解读

We present a version of Elfving's theorem for the Bayesian D-optimality criterion in nonlinear regression models. The Bayesian optimal design can be characterized as a design which allows a representation of a (uniquely determined) boundary point of a convex subset of $L^2$-integrable functions. A similar characterization is given for the Bayesian c-optimality criterion where a (possible) nonlinear function of the unknown parameters has to be estimated. The results are…