Towards a Unified Theory of Inequality Constrained Testing in Multivariate Analysis

Shapiro, Alexander

doi:10.2307/1403361

Cited by 216 publications

(185 citation statements)

References 47 publications

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“…The results will however be different: samples that are located outside of the feasible region in the unconstrained model, are mapped on the borders of the feasible region. This is consistent with formulations of the underlying distribution of the parameters in a constrained linear regression (Shapiro 1988). …”

Section: Applying Xsample() On Overdetermined Systemssupporting

confidence: 85%

`xsample()`: AnRFunction for Sampling Linear Inverse Problems

Meersche¹,

Soetaert²,

Oevelen³

2009

J. Stat. Soft.

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An R function is implemented that uses Markov chain Monte Carlo (MCMC) algorithms to uniformly sample the feasible region of constrained linear problems. Two existing hit-and-run sampling algorithms are implemented, together with a new algorithm where an MCMC step reflects on the inequality constraints. The new algorithm is more robust compared to the hit-and-run methods, at a small cost of increased calculation time.

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Section: Applying Xsample() On Overdetermined Systemssupporting

confidence: 85%

`xsample()`: AnRFunction for Sampling Linear Inverse Problems

Meersche¹,

Soetaert²,

Oevelen³

2009

J. Stat. Soft.

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“…For more general settings, e.g., comparing models with k and k + k (k > 1) random effects, the null distribution is a mixture of χ 2 random variables (Shapiro 1988), the weights of which can only be calculated analytically in a number of special cases. Shapiro's (1988) results provide a few important special cases, not studied by Stram and Lee (1994). For example, if the null hypothesis allows for k uncorrelated random effects (with a diagonal covariance matrix D k ) versus the alternative of k + k uncorrelated random effects (with diagonal covariance matrix D k+k ), the null distribution is a mixture of the form Shapiro (1988) shows that, for a broad number of cases, determining the mixture's weights is a complex and perhaps numerical task.…”

Section: Inference For Variance Componentsmentioning

confidence: 99%

The Use of Score Tests for Inference on Variance Components

Verbeke¹,

Molenberghs²

2003

Biometrics

203

144

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Summary Whenever inference for variance components is required, the choice between one‐sided and two‐sided tests is crucial. This choice is usually driven by whether or not negative variance components are permitted. For two‐sided tests, classical inferential procedures can be followed, based on likelihood ratios, score statistics, or Wald statistics. For one‐sided tests, however, one‐sided test statistics need to be developed, and their null distribution derived. While this has received considerable attention in the context of the likelihood ratio test, there appears to be much confusion about the related problem for the score test. The aim of this paper is to illustrate that classical (two‐sided) score test statistics, frequently advocated in practice, cannot be used in this context, but that well‐chosen one‐sided counterparts could be used instead. The relation with likelihood ratio tests will be established, and all results are illustrated in an analysis of continuous longitudinal data using linear mixed models.

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“…Various types of hypothesis tests are proposed with different test statistics. However, most work in the literature has focused on observing the behavior of the test statistic as n → ∞ with a fixed value of r (Yatchew, 1992;Hall & Jeckman, 2000;Baraud et al, 2005) or imposed a condition that requires the normality of the ϵ i j 's (Bartholomew, 1959;Shapiro, 1988;Baraud et al, 2005). For example, the MSE has been studied as a test statistics in Shapiro (1988), but the behavior of the test statistic is studied only for the case where n → ∞ with r fixed.…”

Section: Test Of Positivitymentioning

confidence: 99%

“…However, most work in the literature has focused on observing the behavior of the test statistic as n → ∞ with a fixed value of r (Yatchew, 1992;Hall & Jeckman, 2000;Baraud et al, 2005) or imposed a condition that requires the normality of the ϵ i j 's (Bartholomew, 1959;Shapiro, 1988;Baraud et al, 2005). For example, the MSE has been studied as a test statistics in Shapiro (1988), but the behavior of the test statistic is studied only for the case where n → ∞ with r fixed. Empirical studies suggest that the weights used in Shapiro (1988) are difficult to compute exactly, so the test procedure proposed by Shapiro (1988) is computationally burdensome (Sen & Silvapulle, 2002).…”

Section: Test Of Positivitymentioning

confidence: 99%

“…For example, the MSE has been studied as a test statistics in Shapiro (1988), but the behavior of the test statistic is studied only for the case where n → ∞ with r fixed. Empirical studies suggest that the weights used in Shapiro (1988) are difficult to compute exactly, so the test procedure proposed by Shapiro (1988) is computationally burdensome (Sen & Silvapulle, 2002). Bartholomew (1959) used the MSE as a test statistic, but he assumed that the ϵ i j 's are normally distributed and did not consider the case where r → ∞.…”

Section: Test Of Positivitymentioning

confidence: 99%

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Nonparametric Tests for Convexity/Monotonicity/Positivity of Multivariate Functions with Noisy Observations

Lim¹,

Tavarez²

2017

IJSP

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We propose a new method of testing for a function's convexity, monotonicity, or positivity, based on some noisy observations of the function made over a finite set T of points in the domain, where the observations can be made multiple times at each point in T . One of the traditional approaches to the test of a function's shape characteristic is to fit a convex, a monotone, or a positive function, depending on the shape characteristic we wish to test for, to the data set minimizing the sum of squared errors, and to compute the sum of squared differences (SSD) between the fit and the data set. While the traditional approach proceeds by observing the SSD as the number of points in T increases to infinity, we propose observing the SSD as r, the number of observations taken at each point in T , increases to infinity. This new way of observing the asymptotic behavior of the SSD leads to a test procedure that does not require the estimation of any additional parameters, and hence, is easy to implement. The proposed test procedure is proven to achieve a prescribed power as r → ∞. Numerical examples illustrate that the proposed test successfully detects the convexity/monotonicity/positivity of a function, as well as the non-convexity/non-monotonicity/non-positivity of a function.

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Towards a Unified Theory of Inequality Constrained Testing in Multivariate Analysis

Cited by 216 publications

References 47 publications

`xsample()`: AnRFunction for Sampling Linear Inverse Problems

`xsample()`: AnRFunction for Sampling Linear Inverse Problems

The Use of Score Tests for Inference on Variance Components

Nonparametric Tests for Convexity/Monotonicity/Positivity of Multivariate Functions with Noisy Observations

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Towards a Unified Theory of Inequality Constrained Testing in Multivariate Analysis

Cited by 216 publications

References 47 publications

xsample(): AnRFunction for Sampling Linear Inverse Problems

xsample(): AnRFunction for Sampling Linear Inverse Problems

The Use of Score Tests for Inference on Variance Components

Nonparametric Tests for Convexity/Monotonicity/Positivity of Multivariate Functions with Noisy Observations

Contact Info

Product

Resources

About

`xsample()`: AnRFunction for Sampling Linear Inverse Problems

`xsample()`: AnRFunction for Sampling Linear Inverse Problems