Principal component analysis (PCA) is a well-known tool in multivariate statistics. One significant challenge in using PCA is the choice of the number of components. In order to address this challenge, we propose an exact distribution-based method for hypothesis testing and construction of confidence intervals for signals in a noisy matrix. Assuming Gaussian noise, we use the conditional distribution of the singular values of a Wishart matrix and derive exact hypothesis tests and confidence intervals for the true signals. Our paper is based on the approach of Taylor, Loftus and Tibshirani (2013) for testing the global null: we generalize it to test for any number of principal components, and derive an integrated version with greater power. In simulation studies we find that our proposed methods compare well to existing approaches.
Our method is motivated by the Kac-Rice test , an exact method for testing and constructing confidence intervals for signals under the global null hypothesis in adaptive regression. Our work corresponds to the Kac-Rice test under the global null scenario, in a penalized regression minimizing the Frobenius norm with a nuclear norm penalty. In this paper, we extend the test and confidence intervals to the general case with improved power. The resulting statistic uses the survival function of the conditional distribution of the eigenvalues of a Wishart matrix. In the context of inference based on the distribution of eigenvalues, @cite_7 and, more recently, @cite_0 have proposed methods for testing essentially the same hypothesis as in this paper. Both @cite_7 and @cite_0 benefit from an asymptotic distribution of the test statistic: @cite_7 forms a likelihood ratio test with the asymptotic Chi-square distribution. @cite_0 use the Tracy-Widom law, which is the asymptotic distribution of the largest eigenvalue of a Wishart matrix, incorporating the result of @cite_3 . We provide a method for constructing confidence intervals in addition to hypothesis testing, with the procedures being exact.
@cite_0 @cite_3 @cite_7
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We propose a new method for pointwise estimation of monotone, uni- modal and U-shaped failure rates, under a right-censoring mechanism, using non- parametric likelihood ratios. The asymptotic distribution of the likelihood ratio is pivotal, though non-standard, and can therefore be used to construct asymptotic confidence intervals for the failure rate at a point of interest, via inversion. Major advantages of the new method lie in the facts that it completely avoids estima- tion of nuisance parameters, or the choice of a bandwidth tuning parameter, and is extremely easy to implement. The new method is shown to perform competi- tively in simulations, and is illustrated on a data set involving time to diagnosis of schizophrenia in the Jerusalem Perinatal Cohort Schizophrenia Study. Abstract A possible alternative to the hypothesis that the sequence X 1, X 2, …, Xn are iid N(ξ, σ2) random variables is that at some unknown instant the expectation ξ shifts. The likelihood ratio test for the alternative of a location shift is studied and its distribution under the null hypothesis found. Tables of standard fractiles are given, along with asymptotic results. A new non parametric approach to the problem of testing the independence of two random process is developed. The test statistic is the Hilbert Schmidt Independence Criterion (HSIC), which was used previously in testing independence for i.i.d pairs of variables. The asymptotic behaviour of HSIC is established when computed from samples drawn from random processes. It is shown that earlier bootstrap procedures which worked in the i.i.d. case will fail for random processes, and an alternative consistent estimate of the p-values is proposed. Tests on artificial data and real-world Forex data indicate that the new test procedure discovers dependence which is missed by linear approaches, while the earlier bootstrap procedure returns an elevated number of false positives. The code is available online: this https URL . We study the asymptotic-capacity-achieving score function that was recently proposed by for bias-based traitor tracing codes. For the bias function, we choose the Dirichlet distribution with a cutoff. Using Bernstein's inequality and Bennett's inequality, we upper bound the false-positive and false-negative error probabilities. From these bounds we derive sufficient conditions for the scheme parameters. We solve these conditions in the limit of large coalition size c0 and obtain asymptotic solutions for the cutoff, the sufficient code length, and the corresponding accusation threshold. We find that the code length converges to its asymptote approximately as c0?1 2 @math , which is faster than the c0?1 3 @math of Tardos' score function. MSC:94B60