A Comment on Locally Most Powerful Tests in the Presence of Nuisance Parameters

Marohn, Frank

doi:10.1081/sta-120002852

Cited by 9 publications

(13 citation statements)

References 9 publications

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“…The distribution of

S_{\overset{false^}{bold-italicγ}}

can be substantially different from that of

S_{{bold-italicγ}_{0}}

: the score based on the true nuisance parameters. Indeed, under the null hypothesis, the asymptotic variance of

S_{\overset{false^}{bold-italicγ}}

is not the Fisher information, but the effective Fisher information (Rippon and Rayner (), Rayner (), Hall and Mathiason (), Marohn () and Cox and Hinkley (), section 9.3), which is also the asymptotic variance of the effective score , which is defined below. The effective information is smaller than the information, given that the score for the parameter of interest and the nuisance score are correlated.…”

Section: Taking Into Account Nuisance Estimationmentioning

confidence: 99%

“…In case nuisance parameters are estimated, the individual score contributions become dependent and our basic sign flipping test is no longer asymptotically exact. To deal with this problem, we consider the effective score , which is less dependent on the nuisance estimate than is the basic score (Hall and Mathiason, ; Marohn, ). In this case we need slightly more assumptions: the variance misspecification is not always allowed to depend on the covariates.…”

Section: Introductionmentioning

confidence: 99%

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Robust Testing in Generalized Linear Models by Sign Flipping Score Contributions

Hemerik

Goeman

Finos

2020

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary Generalized linear models are often misspecified because of overdispersion, heteroscedasticity and ignored nuisance variables. Existing quasi‐likelihood methods for testing in misspecified models often do not provide satisfactory type I error rate control. We provide a novel semiparametric test, based on sign flipping individual score contributions. The parameter tested is allowed to be multi‐dimensional and even high dimensional. Our test is often robust against the mentioned forms of misspecification and provides better type I error control than its competitors. When nuisance parameters are estimated, our basic test becomes conservative. We show how to take nuisance estimation into account to obtain an asymptotically exact test. Our proposed test is asymptotically equivalent to its parametric counterpart.

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“…The distribution of

S_{\overset{false^}{bold-italicγ}}

can be substantially different from that of

S_{{bold-italicγ}_{0}}

: the score based on the true nuisance parameters. Indeed, under the null hypothesis, the asymptotic variance of

S_{\overset{false^}{bold-italicγ}}

Section: Taking Into Account Nuisance Estimationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Robust Testing in Generalized Linear Models by Sign Flipping Score Contributions

Hemerik

Goeman

Finos

2020

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…Consider the approximate likelihood function for the negative binomial model defined in . We then construct an asymptotically and locally most powerful test for testing H 0 : θ = 0. The null hypothesis should be rejected when the value of the score function based on the approximate likelihood function as θ tends to zero, T a m l , is large, where

T_{aml} = \frac{1}{2} \sum_{i = 1}^{n} [{(y_{i} - λ_{m_{i}})}^{2} - y_{i}] - \frac{1}{2} β_{1}^{2} τ^{2} \sum_{i = 1}^{n} \frac{λ_{m_{i}} [y_{i} - 2 λ_{m_{i}} + 2 {(y_{i} - λ_{m_{i}})}^{2}]}{1 + \frac{1}{2} β_{1}^{2} τ^{2} [{(y_{i} - λ_{m_{i}})}^{2} - λ_{m_{i}}]} .

Moreover, V a r ( T a m l | w ) is approximately equal to

\frac{1}{2} \sum_{i = 1}^{n} λ_{m_{i}}^{2} - β_{1}^{2} τ^{2} \sum_{i = 1}^{n} <...>

…”

Section: Tests For Overdispersionmentioning

confidence: 99%

“…Consider the approximate likelihood function for the negative binomial model defined in (8). We then construct an asymptotically and locally most powerful test [23,24] for testing H 0 ∶ = 0. The null hypothesis should be rejected when the value of the score function based on the approximate likelihood function as tends to zero, T aml , is large, where…”

Section: Score Test Statisticsmentioning

confidence: 99%

Tests for detecting overdispersion in models with measurement error in covariates

Yang

Wong

2015

Statistics in Medicine

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Measurement error in covariates can affect the accuracy in count data modeling and analysis. In overdispersion identification, the true mean-variance relationship can be obscured under the influence of measurement error in covariates. In this paper, we propose three tests for detecting overdispersion when covariates are measured with error: a modified score test and two score tests based on the proposed approximate likelihood and quasi-likelihood, respectively. The proposed approximate likelihood is derived under the classical measurement error model, and the resulting approximate maximum likelihood estimator is shown to have superior efficiency. Simulation results also show that the score test based on approximate likelihood outperforms the test based on quasi-likelihood and other alternatives in terms of empirical power. By analyzing a real dataset containing the health-related quality-of-life measurements of a particular group of patients, we demonstrate the importance of the proposed methods by showing that the analyses with and without measurement error correction yield significantly different results.

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“…In addition, the demonstrated equivalence also provides further motivation for a procedure introduced in [15] where a test statistic is derived by replacing nuisance parameters in the locally most powerful test statistic by their conditional MLEs. Results in [15] show this procedure produces an asymptotically uniformly most powerful test. Our equivalence finding shows that when we used this approach to derive the R test, in a finite sample context, we arrive at a locally most powerful invariant test.…”

Section: Proposed Test For Heteroscedasticitymentioning

confidence: 99%

Tolerance Intervals in a Heteroscedastic Linear Regression Context with Applications to Aerospace Equipment Surveillance

Myhre

Jeske

Rennie³

et al. 2009

International Journal of Quality, Statistics, and Reliability

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A heteroscedastic linear regression model is developed from plausible assumptions that describe the time evolution of performance metrics for equipment. The inherited motivation for the related weighted least squares analysis of the model is an essential and attractive selling point to engineers with interest in equipment surveillance methodologies. A simple test for the significance of the heteroscedasticity suggested by a data set is derived and a simulation study is used to evaluate the power of the test and compare it with several other applicable tests that were designed under different contexts. Tolerance intervals within the context of the model are derived, thus generalizing well-known tolerance intervals for ordinary least squares regression. Use of the model and its associated analyses is illustrated with an aerospace application where hundreds of electronic components are continuously monitored by an automated system that flags components that are suspected of unusual degradation patterns.

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A Comment on Locally Most Powerful Tests in the Presence of Nuisance Parameters

Cited by 9 publications

References 9 publications

Robust Testing in Generalized Linear Models by Sign Flipping Score Contributions

Robust Testing in Generalized Linear Models by Sign Flipping Score Contributions

Tests for detecting overdispersion in models with measurement error in covariates

Tolerance Intervals in a Heteroscedastic Linear Regression Context with Applications to Aerospace Equipment Surveillance

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