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1.
Model identification and discrimination are two major statistical challenges. In this paper we consider a set of models Mk for factorial experiments with the parameters representing the general mean, main effects, and only k out of all two-factor interactions. We consider the class D of all fractional factorial plans with the same number of runs having the ability to identify all the models in Mk, i.e., the full estimation capacity.The fractional factorial plans in D with the full estimation capacity for k?2 are able to discriminate between models in Mu for u?k*, where k*=(k/2) when k is even, k*=((k-1)/2) when k is odd. We obtain fractional factorial plans in D satisfying the six optimality criterion functions AD, AT, AMCR, GD, GT, and GMCR for 2m factorial experiments when m=4 and 5. Both single stage and multi-stage (hierarchical) designs are given. Some results on estimation capacity of a fractional factorial plan for identifying models in Mk are also given. Our designs D4.1 and D10 stand out in their performances relative to the designs given in Li and Nachtsheim [Model-robust factorial designs, Technometrics 42(4) (2000) 345-352.] for m=4 and 5 with respect to the criterion functions AD, AT, AMCR, GD, GT, and GMCR. Our design D4.2 stands out in its performance relative the Li-Nachtsheim design for m=4 with respect to the four criterion functions AT, AMCR, GT, and GMCR. However, the Li-Nachtsheim design for m=4 stands out in its performance relative to our design D4.2 with respect to the criterion functions AD and GD. Our design D14 does have the full estimation capacity for k=5 but the twelve run Li-Nachtsheim design does not have the full estimation capacity for k=5.  相似文献   

2.
Estimation of regression functions from independent and identically distributed data is considered. The L2 error with integration with respect to the design measure is used as an error criterion. Usually in the analysis of the rate of convergence of estimates besides smoothness assumptions on the regression function and moment conditions on Y also boundedness assumptions on X are made. In this article we consider partitioning and nearest neighbor estimates and show that by replacing the boundedness assumption on X by a proper moment condition the same rate of convergence can be shown as for bounded data.  相似文献   

3.
In the present paper we consider a class of unequally replicated designs having concurrence range 2 and spectrum of the form μ1(μ2)v−3μ3. Now, Jacroux’s [Some sufficient conditions for the type I optimality of block designs, J. Statist. Plann. Inference 11 (1985) 385-396] Proposition 2.4 says that a design with spectrum of the above form, if satisfies some further conditions, is type 1 optimal. Unfortunately, this proposition does not apply to our designs since they have a poor status regarding E-optimality. Yet we are able to prove the A-optimality (in the general class) of these designs using majorisation technique. A method of construction of an infinite series of our A-optimal designs has also been given.The first and only known infinite series of examples of designs satisfying Jacroux’s conditions appears to be the first one in Section 4.1 of Morgan and Srivastav [On the Type-1 optimality of nearly balanced incomplete block designs with small concurrence range, Statist. Sinica 10 (2000) 1091-1116] - hitherto referred to as [MS]. In this paper, we use majorisation technique to prove stronger optimality properties of the above mentioned designs of [MS] as well as to present simpler proof of another optimality result in [MS].  相似文献   

4.
The criterion robustness of the standard likelihood ratio test (LRT) under the multivariate normal regression model and also the inference robustness of the same test under the univariate set up are established for certain nonnormal distributions of errors. Restricting attention to the normal distribution of errors in the context of univariate regression models, conditions on the design matrix are established under which the usual LRT of a linear hypothesis (under homoscedasticity of errors) remains valid if the errors have an intraclass covariance structure. The conditions hold in the case of some standard designs. The relevance of C. R. Rao's (1967 In Proceedings Fifth Berkeley Symposium on Math. Stat. and Prob., Vol. 1, pp. 355–372) and G. Zyskind's (1967, Ann. Math. Statist.38 1092–1110) conditions in this context is discussed.  相似文献   

5.
Suppose that Y=(Yi) is a normal random vector with mean Xb and covariance σ2In, where b is a p-dimensional vector (bj),X=(Xij) is an n×p matrix. A-optimal designs X are chosen from the traditional set D of A-optimal designs for ρ=0 such that X is still A-optimal in D when the components Yi are dependent, i.e., for ii′, the covariance of Yi,Yi is ρ with ρ≠0. Such designs depend on the sign of ρ. The general results are applied to X=(Xij), where Xij∈{-1,1}; this corresponds to a factorial design with -1,1 representing low level or high level respectively, or corresponds to a weighing design with -1,1 representing an object j with weight bj being weighed on the left and right of a chemical balance respectively.  相似文献   

6.
A Helmholtz equation in two dimensions discretized by a second order finite difference scheme is considered. Krylov methods such as Bi-CGSTAB and IDR(s) have been chosen as solvers. Since the convergence of the Krylov solvers deteriorates with increasing wave number, a shifted Laplace multigrid preconditioner is used to improve the convergence. The implementation of the preconditioned solver on CPU (Central Processing Unit) is compared to an implementation on GPU (Graphics Processing Units or graphics card) using CUDA (Compute Unified Device Architecture). The results show that preconditioned Bi-CGSTAB on GPU as well as preconditioned IDR(s) on GPU is about 30 times faster than on CPU for the same stopping criterion.  相似文献   

7.
We propose different nonparametric tests for multivariate data and derive their asymptotic distribution for unbalanced designs in which the number of factor levels tends to infinity (large a, small ni case). Quasi gratis, some new parametric multivariate tests suitable for the large a asymptotic case are also obtained. Finite sample performances are investigated and compared in a simulation study. The nonparametric tests are based on separate rankings for the different variables. In the presence of outliers, the proposed nonparametric methods have better power than their parametric counterparts. Application of the new tests is demonstrated using data from plant pathology.  相似文献   

8.
This paper deals with the bias correction of the cross-validation (CV) criterion to estimate the predictive Kullback-Leibler information. A bias-corrected CV criterion is proposed by replacing the ordinary maximum likelihood estimator with the maximizer of the adjusted log-likelihood function. The adjustment is just slight and simple, but the improvement of the bias is remarkable. The bias of the ordinary CV criterion is O(n-1), but that of the bias-corrected CV criterion is O(n-2). We verify that our criterion has smaller bias than the AIC, TIC, EIC and the ordinary CV criterion by numerical experiments.  相似文献   

9.
This paper is concerned with the testing problem of generalized multivariate linear hypothesis for the mean in the growth curve model(GMANOVA). Our interest is the case in which the number of the observed points p is relatively large compared to the sample size N. Asymptotic expansions of the non-null distributions of the likelihood ratio criterion, Lawley-Hotelling’s trace criterion and Bartlett-Nanda-Pillai’s trace criterion are derived under the asymptotic framework that N and p go to infinity together, while p/Nc∈(0,1). It also can be confirmed that Rothenberg’s condition on the magnitude of the asymptotic powers of the three tests is valid when p is relatively large, theoretically and numerically.  相似文献   

10.
Clear effects criterion is an important criterion for selecting fractional factorial designs[1].Tang et al.[2]derived upper and lower bounds on the maximum number of clear two-factor interactions(2fi's)in 2^n-(n-k)designs of resolution Ⅲ and Ⅳ by constructing 2^n-(n-k)designs.But the method in[2]does not perform well sometimes when the resolution is Ⅲ.This article modifies the construction method for 2^n-(n-k) designs of resolution Ⅲ in[2].The modified method is a great improvement on that used in[2].  相似文献   

11.
Outcome-dependent sampling designs are commonly used in economics, market research and epidemiological studies. Case-control sampling design is a classic example of outcome-dependent sampling, where exposure information is collected on subjects conditional on their disease status. In many situations, the outcome under consideration may have multiple categories instead of a simple dichotomization. For example, in a case-control study, there may be disease sub-classification among the “cases” based on progression of the disease, or in terms of other histological and morphological characteristics of the disease. In this note, we investigate the issue of fitting prospective multivariate generalized linear models to such multiple-category outcome data, ignoring the retrospective nature of the sampling design. We first provide a set of necessary and sufficient conditions for the link functions that will allow for equivalence of prospective and retrospective inference for the parameters of interest. We show that for categorical outcomes, prospective-retrospective equivalence does not hold beyond the generalized multinomial logit link. We then derive an approximate expression for the bias incurred when link functions outside this class are used. Most popular models for ordinal response fall outside the multiplicative intercept class and one should be cautious while performing a naive prospective analysis of such data as the bias could be substantial. We illustrate the extent of bias through a real data example, based on the ongoing Prostate, Lung, Colorectal and Ovarian (PLCO) cancer screening trial by the National Cancer Institute. The simulations based on the real study illustrate that the bias approximations work well in practice.  相似文献   

12.
The generalized information criterion (GIC) proposed by Rao and Wu [A strongly consistent procedure for model selection in a regression problem, Biometrika 76 (1989) 369-374] is a generalization of Akaike's information criterion (AIC) and the Bayesian information criterion (BIC). In this paper, we extend the GIC to select linear mixed-effects models that are widely applied in analyzing longitudinal data. The procedure for selecting fixed effects and random effects based on the extended GIC is provided. The asymptotic behavior of the extended GIC method for selecting fixed effects is studied. We prove that, under mild conditions, the selection procedure is asymptotically loss efficient regardless of the existence of a true model and consistent if a true model exists. A simulation study is carried out to empirically evaluate the performance of the extended GIC procedure. The results from the simulation show that if the signal-to-noise ratio is moderate or high, the percentages of choosing the correct fixed effects by the GIC procedure are close to one for finite samples, while the procedure performs relatively poorly when it is used to select random effects.  相似文献   

13.
This paper deals with the bias reduction of Akaike information criterion (AIC) for selecting variables in multivariate normal linear regression models when the true distribution of observation is an unknown nonnormal distribution. We propose a corrected version of AIC which is partially constructed by the jackknife method and is adjusted to the exact unbiased estimator of the risk when the candidate model includes the true model. It is pointed out that the influence of nonnormality in the bias of our criterion is smaller than the ones in AIC and TIC. We verify that our criterion is better than the AIC, TIC and EIC by conducting numerical experiments.  相似文献   

14.
We consider the fixed design regression model Yi = g(ti) + ξi, i = 1, …, n, where ξi are (not necessarily i.i.d.) no variables, ti constitute the design points where nonrepeatable measurements are to be taken and Yi are the observations from which g and its derivatives are to be estimated. The dependency of the Integrated Mean Squared Error of two different types of kernel estimates on the design {t1, …, tn} is established. This allows the derivation of asymptotically optimal designs.  相似文献   

15.
In this paper, an information-based criterion is proposed for carrying out change point analysis and variable selection simultaneously in linear models with a possible change point. Under some weak conditions, this criterion is shown to be strongly consistent in the sense that with probability one, it chooses the smallest true model for large n. Its byproducts include strongly consistent estimates of the regression coefficients regardless if there is a change point. In case that there is a change point, its byproducts also include a strongly consistent estimate of the change point parameter. In addition, an algorithm is given which has significantly reduced the computation time needed by the proposed criterion for the same precision. Results from a simulation study are also presented.  相似文献   

16.
Supersaturated designs (SSDs) have been widely used in factor screening experiments. The present paper aims to prove that the maximal balanced designs are a kind of special optimal SSDs under the E(f NOD) criterion. We also propose a new method, called the complementary design method, for constructing E(f NOD) optimal SSDs. The basic principle of this method is that for any existing E(f NOD) optimal SSD whose E(fNOD) value reaches its lower bound, its complementary design in the corresponding maximal balanced design is also E(f NOD) optimal. This method applies to both symmetrical and asymmetrical (mixed-level) cases. It provides a convenient and efficient way to construct many new designs with relatively large numbers of factors. Some newly constructed designs are given as examples.  相似文献   

17.
Doubling is a simple but powerful method of constructing two-level fractional factorial designs with high resolution. This article studies uniformity in terms of Lee discrepancy of double designs. We give some linkages between the uniformity of double design and the aberration case of the original one under different criteria. Furthermore, some analytic linkages between the generalized wordlength pattern of double design and that of the original one are firstly provided here, which extend the existing findings. The lower bound of Lee discrepancy for double designs is also given.  相似文献   

18.
Sets pooling designs   总被引:4,自引:0,他引:4  
Pooling desings have previously been used for the efficient identification of distinguished elements of a finite setU. Group testing underlies these designs: For any , a binary result is obtainable, indicating whether or not the number of distinguished elements included inS is zero. The current generalization of pooling designs will enable the efficient identification of distinguished subsets of a finite setU. In this case, for any , a binary result is obtainable, indicating whether or not the number of distinguished subsets included inS is zero. Such designs are called sets pooling designs, comprising standard pooling designs in the special case where all the distinguished subsets are elements. The new designs are similar to the standard designs but are subject to new constraints because the set of subsets included inS is its power set. To illustrate the feasibility of constructing sets pooling designs, random, non-adaptive designs are investigated for the special case where all distinguished subsets have the same size. An optimum probability for including an object in a pool is approximated as a function of the size and number of distinguished subsets, adopting the criterion of minimizing the average number of non-distinguished subsets whose status would not be resolved by the pooling design. Deterministic and adaptive designs are also described.This work was supported by the US Department of Energy under contract W-7405-ENG-36, through a Laboratory Directed Research and Development Grant at Los Alamos National Laboratory.  相似文献   

19.
In the common nonparametric regression model we consider the problem of constructing optimal designs, if the unknown curve is estimated by a smoothing spline. A special basis for the space of natural splines is introduced and the local minimax property for these splines is used to derive two optimality criteria for the construction of optimal designs. The first criterion determines the design for a most precise estimation of the coefficients in the spline representation and corresponds to D-optimality, while the second criterion is the G-optimality criterion and corresponds to an accurate prediction of the curve. Several properties of the optimal designs are derived. In general, D- and G-optimal designs are not equivalent. Optimal designs are determined numerically and compared with the uniform design.  相似文献   

20.
Summary Some new third-order rotatable designs in three dimensions are derived from some of the available third-order rotatable designs in two dimensions. When these designs are used the results of the experiments performed according to the two-dimentional designs need not be discarded. Some of these designs may be performed sequentially in all three factors, starting with a one-dimensional design. Further, these third-order rotatable designs require a smaller number of points than most of the available three-dimensional third-order rotatable designs.  相似文献   

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