Goodness-of-fit tests for high-dimensional Gaussian linear models

In this study, we consider PCA for Gaussian observations X1, …, Xn with covariance Σ = ∑iλiPi in the ’effective rank’ setting with model complexity governed by r(Σ) ≔ tr(Σ)∕∥Σ∥. We prove a Berry-Essen type bound for a Wald Statistic of the spectral projector $\hat P_r$. This can be used to construct non-asymptotic goodness of fit tests and confidence ellipsoids for spectral projectors Pr. Using higher order pertubation theory we are able to show that our Theorem remains valid even when $\mathbf{r}(\Sigma) \gg \sqrt{n}$.

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Nonparametric independence testing via mutual information

Biometrika ◽

10.1093/biomet/asz024 ◽

2019 ◽

Vol 106 (3) ◽

pp. 547-566 ◽

Cited By ~ 5

Author(s):

T B Berrett ◽

R J Samworth

Keyword(s):

Mutual Information ◽

Goodness Of Fit ◽

Linear Models ◽

Real Data ◽

Nearest Neighbour ◽

Local Power ◽

Goodness Of Fit Tests ◽

Numerical Studies ◽

Independence Testing ◽

Power Analyses

Summary We propose a test of independence of two multivariate random vectors, given a sample from the underlying population. Our approach is based on the estimation of mutual information, whose decomposition into joint and marginal entropies facilitates the use of recently developed efficient entropy estimators derived from nearest neighbour distances. The proposed critical values may be obtained by simulation in the case where an approximation to one marginal is available or by permuting the data otherwise. This facilitates size guarantees, and we provide local power analyses, uniformly over classes of densities whose mutual information satisfies a lower bound. Our ideas may be extended to provide new goodness-of-fit tests for normal linear models based on assessing the independence of our vector of covariates and an appropriately defined notion of an error vector. The theory is supported by numerical studies on both simulated and real data.

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NN goodness-of-fit tests for linear models

Journal of Statistical Planning and Inference ◽

10.1016/0378-3758(95)00144-1 ◽

1996 ◽

Vol 53 (1) ◽

pp. 75-92 ◽

Cited By ~ 26

Author(s):

W. Stute ◽

W.González Manteiga

Keyword(s):

Goodness Of Fit ◽

Linear Models ◽

Goodness Of Fit Tests

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A simple yet powerful test for assessing goodness‐of‐fit of high‐dimensional linear models

Statistics in Medicine ◽

10.1002/sim.8968 ◽

2021 ◽

Author(s):

Qi Zhang ◽

Feifei Chen ◽

Shunyao Wu ◽

Hua Liang

Keyword(s):

Goodness Of Fit ◽

Linear Models ◽

High Dimensional ◽

Powerful Test

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A Global Test for the Goodness of Fit of Generalized Linear Models : An Estimating Equation Approach

Calcutta Statistical Association Bulletin ◽

10.1177/0008068320050514 ◽

2005 ◽

Vol 56 (1-4) ◽

pp. 251-282

Author(s):

R. Prabhakar Rao ◽

B.C. Sutradhar

Keyword(s):

Generalized Linear Models ◽

Goodness Of Fit ◽

Exponential Family ◽

Linear Models ◽

Estimating Equation ◽

Goodness Of Fit Test ◽

Global Test ◽

Goodness Of Fit Tests ◽

Exponential Family Of Distributions ◽

Family Of Distributions

Summary Generalized linear models are used to analyze a wide variety of discrete and continuous data with possible overdispersion under the assumption that the data follow an exponential family of distributions. The violation of this assumption may have adverse effects on the statistical inferences. The existing goodness of fit tests for checking this assumption are valid only for a standard exponential family of distributions with no overdispersion. In this paper, we develop a global goodness of fit test for the general exponential family of distributions which may or may not contain overdispersion. The proposed statistic has asymptotically standard Gaussian distribution which should be easy to implement.

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