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We determine a credible set A that is the “best” with respect to the variation of the prior distribution in a neighborhood Γ of the starting prior π0(θ). Among the class of sets with credibility γ under π0, the “optimally robust” set will be the one which maximizes the minimum probability of including θ as the prior varies over Γ. This procedure is also Γ-minimax with respect to the risk function, probability of non-inclusion. We find the optimally robust credible set for three neighborhood classes Γ, the ε-contamination class, the density ratio class and the density bounded class. A consequence of this investigation is that the maximum likelihood set is seen to be an optimal credible set from a robustness perspective. 相似文献
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We consider the problem of estimating the mean θ of an Np(θ,Ip) distribution with squared error loss ∥δ−θ∥2 and under the constraint ∥θ∥≤m, for some constant m>0. Using Stein's identity to obtain unbiased estimates of risk, Karlin's sign change arguments, and conditional risk analysis, we compare the risk performance of truncated linear estimators with that of the maximum likelihood estimator δmle. We obtain for fixed (m,p) sufficient conditions for dominance. An asymptotic framework is developed, where we demonstrate that the truncated linear minimax estimator dominates δmle, and where we obtain simple and accurate measures of relative improvement in risk. Numerical evaluations illustrate the effectiveness of the asymptotic framework for approximating the risks for moderate or large values of p. 相似文献
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We consider a linear regression model with regression parameter β=(β1,…,βp) and independent and identically N(0,σ2) distributed errors. Suppose that the parameter of interest is θ=aTβ where a is a specified vector. Define the parameter τ=cTβ-t where the vector c and the number t are specified and a and c are linearly independent. Also suppose that we have uncertain prior information that τ=0. We present a new frequentist 1-α confidence interval for θ that utilizes this prior information. We require this confidence interval to (a) have endpoints that are continuous functions of the data and (b) coincide with the standard 1-α confidence interval when the data strongly contradict this prior information. This interval is optimal in the sense that it has minimum weighted average expected length where the largest weight is given to this expected length when τ=0. This minimization leads to an interval that has the following desirable properties. This interval has expected length that (a) is relatively small when the prior information about τ is correct and (b) has a maximum value that is not too large. The following problem will be used to illustrate the application of this new confidence interval. Consider a 2×2 factorial experiment with 20 replicates. Suppose that the parameter of interest θ is a specified simple effect and that we have uncertain prior information that the two-factor interaction is zero. Our aim is to find a frequentist 0.95 confidence interval for θ that utilizes this prior information. 相似文献
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The generalized order-restricted information criterion (goric) is a model selection criterion which can, up to now, solely be applied to the analysis of variance models and, so far, only evaluate restrictions of the form Rθ≤0, where θ is a vector of k group means and R a cm×k matrix. In this paper, we generalize the goric in two ways: (i) such that it can be applied to t -variate normal linear models and (ii) such that it can evaluate a more general form of order restrictions: Rθ≤r, where θ is a vector of length tk, r a vector of length cm, and R a cm×tk matrix of full rank (when r≠0). At the end, we illustrate that the goric is easy to implement in a multivariate regression model. 相似文献
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Consider the model where there are I independent multivariate normal treatment populations with p×1 mean vectors μi, i=1,…,I, and covariance matrix Σ. Independently the (I+1)st population corresponds to a control and it too is multivariate normal with mean vector μI+1 and covariance matrix Σ. Now consider the following two multiple testing problems. 相似文献
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José Antonio Moler Fernando Plo Miguel San Miguel 《Journal of statistical planning and inference》2007
We study a randomized adaptive design to assign one of the L treatments to patients who arrive sequentially by means of an urn model. At each stage n, a reward is distributed between treatments. The treatment applied is rewarded according to its response, 0?Yn?1, and 1-Yn is distributed among the other treatments according to their performance until stage n-1. Patients can be classified in K+1 levels and we assume that the effect of this level in the response to the treatments is linear. We study the asymptotic behavior of the design when the ordinary least square estimators are used as a measure of performance until stage n-1. 相似文献
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We consider m×m covariance matrices, Σ1 and Σ2, which satisfy Σ2-Σ1=Δ, where Δ has a specified rank. Maximum likelihood estimators of Σ1 and Σ2 are obtained when sample covariance matrices having Wishart distributions are available and rank(Δ) is known. The likelihood ratio statistic for a test about the value of rank(Δ) is also given and some properties of its null distribution are obtained. The methods developed in this paper are illustrated through an example. 相似文献
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Joseph P.S. Kung Anna de Mier Xinyu Sun Catherine Yan 《Journal of statistical planning and inference》2009
We consider paths in the plane with (1,0), (0,1), and (a,b)-steps that start at the origin, end at height n, and stay strictly to the left of a given non-decreasing right boundary. We show that if the boundary is periodic and has slope at most b/a, then the ordinary generating function for the number of such paths ending at height n is algebraic. Our argument is in two parts. We use a simple combinatorial decomposition to obtain an Appell relation or “umbral” generating function, in which the power zn is replaced by a power series of the form znφn(z), where φn(0)=1. Then we convert (in an explicit way) the umbral generating function to an ordinary generating function by solving a system of linear equations and a polynomial equation. This conversion implies that the ordinary generating function is algebraic. We give several concrete examples, including an alternative way to solve the tennis ball problem. 相似文献
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In this paper, we consider the following simple linear Errors-in-Variables (EV) regression model ηi=θ+βxi+?i, ξi=xi+δi, 1?i?n. The moderate deviation principle for the least squares (LS) estimators of the unknown parameters θ, β in the model are obtained. 相似文献
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In this paper, we study a random field U?(t,x) governed by some type of stochastic partial differential equations with an unknown parameter θ and a small noise ?. We construct an estimator of θ based on the continuous observation of N Fourier coefficients of U?(t,x), and prove the strong convergence and asymptotic normality of the estimator when the noise ? tends to zero. 相似文献
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A ridge function with shape function g in the horizontal direction is a function of the form g(x)h(y,0). Along each horizontal line it has the shape g(x), multiplied by a function h(y,0) which depends on the y-value of the horizontal line. Similarly a ridge function with shape function g in the vertical direction has the form g(y)h(x,π/2). For a given shape function g it may or may not be possible to represent an arbitrary function f(x,y) as a superposition over all angles of a ridge function with shape g in each direction, where h=hf=hf,g depends on the functions f and g and also on the direction, θ:h=hf,g(·,θ). We show that if g is Gaussian centered at zero then this is always possible and we give the function hf,g for a given f(x,y). For highpass or for odd shapes g , we show it is impossible to represent an arbitrary f(x,y), i.e. in general there is no hf,g. Note that our problem is similar to tomography, where the problem is to invert the Radon transform, except that the use of the word inversion is here somewhat “inverted”: in tomography f(x,y) is unknown and we find it by inverting the projections of f ; here, f(x,y) is known, g(z) is known, and hf(·,θ)=hf,g(·,θ) is the unknown. 相似文献
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Consider the partially balanced one-way layout for comparing k treatments μi,1?i?k, with a control μ0. We propose a new test which is similar to the test statistics of Marcus [1976. The powers of some tests of the equality of normal means against an ordered alternative. Biometrika 63, 177–183]. By simulation we find that the proposed test has a good power performance when compared with other tests. Moreover, it can produce confidence intervals for μi-μ0,1?i?k. 相似文献
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In common with other non-linear models, the optimal design for a limiting dilution assay (LDA) depends on the value of the unknown parameter, θ, in the model. Consequently optimal designs cannot be specified unless some assumptions are made about the possible values of θ. If a prior distribution can be specified then a Bayesian approach can be adopted. A proper specification of the Bayesian approach requires the aim of the experiment to be described and quantified through an appropriate utility function. This paper addresses the problem of finding optimal designs for LDAs when the aim is to determine whether θ is above or below a specified threshold, θ0. 相似文献
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