arXiv subjects

Derek Bean

Publications and source records attributed to Derek Bean.

3 recordsLinked to original sources

arXiv2024

Nonparametric Evaluation of Noisy ICA Solutions

Independent Component Analysis (ICA) was introduced in the 1980's as a model for Blind Source Separation (BSS), which refers to the process of recovering the sources underlying a mixture of signals, with little knowledge about the source signals or the mixing process. While there are many sophisticated algorithms for estimation, different methods have different shortcomings. In this paper, we develop a nonparametric score to adaptively pick the right algorithm for ICA with arbitrary Gaussian noise. The novelty of this score stems from the fact that it just assumes a finite second moment of the data and uses the characteristic function to evaluate the quality of the estimated mixing matrix without any knowledge of the parameters of the noise distribution. In addition, we propose some new contrast functions and algorithms that enjoy the same fast computability as existing algorithms like FASTICA and JADE but work in domains where the former may fail. While these also may have weaknesses, our proposed diagnostic, as shown by our simulations, can remedy them. Finally, we propose a theoretical framework to analyze the local and global convergence properties of our algorithms.

math.ST↗

arXiv2012

Two-Sample Inference in Highly Dispersed Negative Binomial Models

Two-sample inference for the difference of population means typically relies upon a Central Limit Theorem approximation. When data are drawn from a Negative Binomial distribution, previous work of Shilane et al. (2010) showed that a Normal approximation is often unreliable in one-sample inference and proposed alternative techniques. We seek to extend these methods to the problem of two-sample inference on the difference of sample means in highly dispersed Negative Binomial models. We demonstrate that the Normal approximation is considerably more robust in the two-sample setting and may often be applied even when it is not appropriate for either sample individually. We also provide an intuitive extension of Bernstein's Inequality to the two-sample case. A simple mixture of these two methods may improve coverage in borderline and small sample settings. We subsequently investigate the coverage quality of confidence intervals and sample size considerations in a wide variety of simulation studies. Overall, we demonstrate that the Normal approximation is considerably more robust in approximating the distribution of the mean difference than the corresponding one-sample theory would suggest.

stat.ME↗

arXiv2012

Growth Estimators and Confidence Intervals for the Mean of Negative Binomial Random Variables with Unknown Dispersion

The Negative Binomial distribution becomes highly skewed under extreme dispersion. Even at moderately large sample sizes, the sample mean exhibits a heavy right tail. The standard Normal approximation often does not provide adequate inferences about the data's mean in this setting. In previous work, we have examined alternative methods of generating confidence intervals for the expected value. These methods were based upon Gamma and Chi Square approximations or tail probability bounds such as Bernstein's Inequality. We now propose growth estimators of the Negative Binomial mean. Under high dispersion, zero values are likely to be overrepresented in the data. A growth estimator constructs a Normal-style confidence interval by effectively removing a small, pre--determined number of zeros from the data. We propose growth estimators based upon multiplicative adjustments of the sample mean and direct removal of zeros from the sample. These methods do not require estimating the nuisance dispersion parameter. We will demonstrate that the growth estimators' confidence intervals provide improved coverage over a wide range of parameter values and asymptotically converge to the sample mean. Interestingly, the proposed methods succeed despite adding both bias and variance to the Normal approximation.

stat.ME↗