We propose a new one-sample test for normality in a Reproducing Kernel Hilbert Space (RKHS). Namely, we test the null-hypothesis of belonging to a given family of Gaussian distributions. Hence our procedure may be applied either to test data for norm
ality or to test parameters (mean and covariance) if data are assumed Gaussian. Our test is based on the same principle as the MMD (Maximum Mean Discrepancy) which is usually used for two-sample tests such as homogeneity or independence testing. Our method makes use of a special kind of parametric bootstrap (typical of goodness-of-fit tests) which is computationally more efficient than standard parametric bootstrap. Moreover, an upper bound for the Type-II error highlights the dependence on influential quantities. Experiments illustrate the practical improvement allowed by our test in high-dimensional settings where common normality tests are known to fail. We also consider an application to covariance rank selection through a sequential procedure.
We analyze the performance of cross-validation (CV) in the density estimation framework with two purposes: (i) risk estimation and (ii) model selection. The main focus is given to the so-called leave-$p$-out CV procedure (Lpo), where $p$ denotes the
cardinality of the test set. Closed-form expressions are settled for the Lpo estimator of the risk of projection estimators. These expressions provide a great improvement upon $V$-fold cross-validation in terms of variability and computational complexity. From a theoretical point of view, closed-form expressions also enable to study the Lpo performance in terms of risk estimation. The optimality of leave-one-out (Loo), that is Lpo with $p=1$, is proved among CV procedures used for risk estimation. Two model selection frameworks are also considered: estimation, as opposed to identification. For estimation with finite sample size $n$, optimality is achieved for $p$ large enough [with $p/n=o(1)$] to balance the overfitting resulting from the structure of the model collection. For identification, model selection consistency is settled for Lpo as long as $p/n$ is conveniently related to the rate of convergence of the best estimator in the collection: (i) $p/nto1$ as $nto+infty$ with a parametric rate, and (ii) $p/n=o(1)$ with some nonparametric estimators. These theoretical results are validated by simulation experiments.
A new goodness-of-fit test for normality in high-dimension (and Reproducing Kernel Hilbert Space) is proposed. It shares common ideas with the Maximum Mean Discrepancy (MMD) it outperforms both in terms of computation time and applicability to a wide
r range of data. Theoretical results are derived for the Type-I and Type-II errors. They guarantee the control of Type-I error at prescribed level and an exponentially fast decrease of the Type-II error. Synthetic and real data also illustrate the practical improvement allowed by our test compared with other leading approaches in high-dimensional settings.
MPAgenomics, standing for multi-patients analysis (MPA) of genomic markers, is an R-package devoted to: (i) efficient segmentation, and (ii) genomic marker selection from multi-patient copy number and SNP data profiles. It provides wrappers from comm
only used packages to facilitate their repeated (sometimes difficult) use, offering an easy-to-use pipeline for beginners in R. The segmentation of successive multiple profiles (finding losses and gains) is based on a new automatic choice of influential parameters since default ones were misleading in the original packages. Considering multiple profiles in the same time, MPAgenomics wraps efficient penalized regression methods to select relevant markers associated with a given response.
This paper tackles the problem of detecting abrupt changes in the mean of a heteroscedastic signal by model selection, without knowledge on the variations of the noise. A new family of change-point detection procedures is proposed, showing that cross
-validation methods can be successful in the heteroscedastic framework, whereas most existing procedures are not robust to heteroscedasticity. The robustness to heteroscedasticity of the proposed procedures is supported by an extensive simulation study, together with recent theoretical results. An application to Comparative Genomic Hybridization (CGH) data is provided, showing that robustness to heteroscedasticity can indeed be required for their analysis.
In the multiple testing context, a challenging problem is the estimation of the proportion $pi_0$ of true-null hypotheses. A large number of estimators of this quantity rely on identifiability assumptions that either appear to be violated on real dat
a, or may be at least relaxed. Under independence, we propose an estimator $hat{pi}_0$ based on density estimation using both histograms and cross-validation. Due to the strong connection between the false discovery rate (FDR) and $pi_0$, many multiple testing procedures (MTP) designed to control the FDR may be improved by introducing an estimator of $pi_0$. We provide an example of such an improvement (plug-in MTP) based on the procedure of Benjamini and Hochberg. Asymptotic optimality results may be derived for both $hat{pi}_0$ and the resulting plug-in procedure. The latter ensures the desired asymptotic control of the FDR, while it is more powerful than the BH-procedure. Finally, we compare our estimator of $pi_0$ with other widespread estimators in a wide range of simulations. We obtain better results than other tested methods in terms of mean square error (MSE) of the proposed estimator. Finally, both asymptotic optimality results and the interest in tightly estimating $pi_0$ are confirmed (empirically) by results obtained with the plug-in MTP.
