Small Confidence Sets for the Mean of a Spherically Symmetric Distribution

Samworth, Richard J.

doi:10.1111/j.1467-9868.2005.00505.x

Cited by 19 publications

(15 citation statements)

References 35 publications

(88 reference statements)

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“…In the bivariate case when a =0 and

v_{E}^{2} = p F_{α, p, ν}

, the confidence region C C H ( X , s ) reduces to the standard region and is centred at the ML estimator

\overset{θ}{true}

. Related approaches have been proposed that do not use empirical Bayes arguments to obtain the radius function

v_{E}^{2}

but Taylor series or parametric bootstrapping or a piecewise cubic Hermite interpolating polynomial function …”

Section: Methodsmentioning

confidence: 99%

Simultaneous confidence regions for multivariate bioequivalence

Pallmann

Jaki

2017

Statistics in Medicine

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Demonstrating bioequivalence of several pharmacokinetic (PK) parameters, such as AUC and C , that are calculated from the same biological sample measurements is in fact a multivariate problem, even though this is neglected by most practitioners and regulatory bodies, who typically settle for separate univariate analyses. We believe, however, that a truly multivariate evaluation of all PK measures simultaneously is clearly more adequate. In this paper, we review methods to construct joint confidence regions around multivariate normal means and investigate their usefulness in simultaneous bioequivalence problems via simulation. Some of them work well for idealised scenarios but break down when faced with real-data challenges such as unknown variance and correlation among the PK parameters. We study the shapes of the confidence regions resulting from different methods, discuss how marginal simultaneous confidence intervals for the individual PK measures can be derived, and illustrate the application to data from a trial on ticlopidine hydrochloride. An R package is available.

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“…In the bivariate case when a =0 and

v_{E}^{2} = p F_{α, p, ν}

, the confidence region C C H ( X , s ) reduces to the standard region and is centred at the ML estimator

\overset{θ}{true}

. Related approaches have been proposed that do not use empirical Bayes arguments to obtain the radius function

v_{E}^{2}

but Taylor series or parametric bootstrapping or a piecewise cubic Hermite interpolating polynomial function …”

Section: Methodsmentioning

confidence: 99%

Simultaneous confidence regions for multivariate bioequivalence

Pallmann

Jaki

2017

Statistics in Medicine

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“…The attracting vector ν 2 is dependent not just on y but also on δ, through the two attractors a 1 and a 2 . In Lemma 2, for the deviation probability in (20) to fall exponentially in n, δ needs to be held constant and independent of n. From a practical design point of view, what is needed is nδ 2 1. Indeed , for σ 2 2nδ n i=0 1 {|yi−ȳ|≤δ} to be a reliable approximation of the term σ…”

Section: B Notationmentioning

confidence: 99%

Cluster-Seeking James–Stein Estimators

Srinath

Venkataramanan

2018

IEEE Trans. Inform. Theory

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This paper considers the problem of estimating a high-dimensional vector of parameters θ ∈ R n from a noisy observation. The noise vector is i.i.d. Gaussian with known variance. For a squared-error loss function, the James-Stein (JS) estimator is known to dominate the simple maximum-likelihood (ML) estimator when the dimension n exceeds two. The JSestimator shrinks the observed vector towards the origin, and the risk reduction over the ML-estimator is greatest for θ that lie close to the origin. JS-estimators can be generalized to shrink the data towards any target subspace. Such estimators also dominate the ML-estimator, but the risk reduction is significant only when θ lies close to the subspace. This leads to the question: in the absence of prior information about θ, how do we design estimators that give significant risk reduction over the MLestimator for a wide range of θ?In this paper, we propose shrinkage estimators that attempt to infer the structure of θ from the observed data in order to construct a good attracting subspace. In particular, the components of the observed vector are separated into clusters, and the elements in each cluster shrunk towards a common attractor. The number of clusters and the attractor for each cluster are determined from the observed vector. We provide concentration results for the squared-error loss and convergence results for the risk of the proposed estimators. The results show that the estimators give significant risk reduction over the ML-estimator for a wide range of θ, particularly for large n.Simulation results are provided to support the theoretical claims.An estimatorθ 1 is said to dominate another estimatorθ 2 if R(θ,θ 1 ) ≤ R(θ,θ 2 ), ∀θ ∈ R n , with the inequality being strict for at least one θ. Thus (4) implies that the James-Stein estimator (JS-estimator) dominates the ML-estimator. Unlike the ML-estimator, the JS-estimator is non-linear and biased. However, the risk reduction over the ML-estimator can be significant, making it an attractive option in many situations -see, for example, [3].By evaluating the expression in (3), it can be shown that the risk of the JS-estimator depends on θ only via θ [1]. Further, the risk decreases as θ decreases. (For intuition about this, note in (3) that for large n, y 2 ≈ nσ 2 + θ 2 .) The dependence of the risk on θ is illustrated in Fig. 1, where the average loss of the JS-estimator is plotted versus θ , for two different choices of θ.The JS-estimator in (2) shrinks each element of y towards the origin. Extending this idea, JS-like estimators can be defined by shrinking y towards any vector, or more generally, towards a target subspace V ⊂ R n . Let P V (y) denote the projection of y onto V, so that y−P V (y) 2 = min v∈V y−v 2 . Then the JS-estimator that shrinks y towards the subspace V arXiv:1602.00542v4 [cs.IT]

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“…x ∈ R + ; inverse trigonometric functions tan −1 , sin −1 (see Samworth (2005) In practical applications, such as the resampling example mentioned earlier, there will be a tradeoff between reducing the magnitude of the remainder terms (requires larger m) and the imprecision it introduces in the evaluation of the required moments empirically;…”

Section: Choice Of Deterministic M and Implications For Gmentioning

confidence: 99%

Link of Moments Before and After Transformations, With an Application to Resampling From Fat-Tailed Distributions

Abadir

Cornea‐Madeira

2018

Econom. Theory

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Let x be a transformation of y, whose distribution is unknown. We derive an expansion formulating the expectations of x in terms of the expectations of y. Apart from the intrinsic interest in such a fundamental relation, our results can be applied to calculating E(x) by the low-order moments of a transformation which can be chosen to give a good approximation for E(x). To do so, we generalize the approach of bounding the terms in expansions of characteristic functions, and use our result to derive an explicit and accurate bound for the remainder when a finite number of terms is taken. We illustrate one of the implications of our method by providing * We thank Essie Maasoumi, Natalia Markovich, Peter Phillips, Robert Taylor, Michael Wolf and three anonymous referees for their comments. This paper was invited at the 2nd conference of the International

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Small Confidence Sets for the Mean of a Spherically Symmetric Distribution

Cited by 19 publications

References 35 publications

Simultaneous confidence regions for multivariate bioequivalence

Simultaneous confidence regions for multivariate bioequivalence

Cluster-Seeking James–Stein Estimators

Link of Moments Before and After Transformations, With an Application to Resampling From Fat-Tailed Distributions

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