A new acceptance-rejection method is proposed and investigated for the Bingham distribution on the sphere using the angular central Gaussian distribution as an envelope. It is shown to have high efficiency and to be straightfoward to use. The method can also be extended to Fisher and Fisher-Bingham distributions on spheres and related manifolds.
In this work we study the estimation of the density of a totally positive random vector. Total positivity of the distribution of a random vector implies a strong form of positive dependence between its coordinates and, in particular, it implies posit
ive association. Since estimating a totally positive density is a non-parametric problem, we take on a (modified) kernel density estimation approach. Our main result is that the sum of scaled standard Gaussian bumps centered at a min-max closed set provably yields a totally positive distribution. Hence, our strategy for producing a totally positive estimator is to form the min-max closure of the set of samples, and output a sum of Gaussian bumps centered at the points in this set. We can frame this sum as a convolution between the uniform distribution on a min-max closed set and a scaled standard Gaussian. We further conjecture that convolving any totally positive density with a standard Gaussian remains totally positive.
In many statistical problems the hypotheses are naturally divided into groups, and the investigators are interested to perform group-level inference, possibly along with inference on individual hypotheses. We consider the goal of discovering groups c
ontaining $u$ or more signals with group-level false discovery rate (FDR) control. This goal can be addressed by multiple testing of partial conjunction hypotheses with a parameter $u,$ which reduce to global null hypotheses for $u=1.$ We consider the case where the partial conjunction $p$-values are combinations of within-group $p$-values, and obtain sufficient conditions on (1) the dependencies among the $p$-values within and across the groups, (2) the combining method for obtaining partial conjunction $p$-values, and (3) the multiple testing procedure, for obtaining FDR control on partial conjunction discoveries. We consider separately the dependencies encountered in the meta-analysis setting, where multiple features are tested in several independent studies, and the $p$-values within each study may be dependent. Based on the results for this setting, we generalize the procedure of Benjamini, Heller, and Yekutieli (2009) for assessing replicability of signals across studies, and extend their theoretical results regarding FDR control with respect to replicability claims.
Many popular robust estimators are $U$-quantiles, most notably the Hodges-Lehmann location estimator and the $Q_n$ scale estimator. We prove a functional central limit theorem for the sequential $U$-quantile process without any moment assumptions and
under weak short-range dependence conditions. We further devise an estimator for the long-run variance and show its consistency, from which the convergence of the studentized version of the sequential $U$-quantile process to a standard Brownian motion follows. This result can be used to construct CUSUM-type change-point tests based on $U$-quantiles, which do not rely on bootstrapping procedures. We demonstrate this approach in detail at the example of the Hodges-Lehmann estimator for robustly detecting changes in the central location. A simulation study confirms the very good robustness and efficiency properties of the test. Two real-life data sets are analyzed.
We consider the fitting of heavy tailed data and distribution with a special attention to distributions with a non--standard shape in the body of the distribution. To this end we consider a dense class of heavy tailed distributions introduced recentl
y, employing an EM algorithm for the the maximum likelihood estimates of its parameters. We present methods for fitting to observed data, histograms, censored data, as well as to theoretical distributions. Numerical examples are provided with simulated data and a benchmark reinsurance dataset. We empirically demonstrate that our model can provide excellent fits to heavy--tailed data/distributions with minimal assumptions
In this paper, we propose to obtain the skewed version of a unimodal symmetric density using a skewing mechanism that is not based on a cumulative distribution function. Then we disturb the unimodality of the resulting skewed density. In order to int
roduce skewness we use the general method which transforms any continuous unimodal and symmetric distribution into a skewed one by changing the scale at each side of the mode.
John T. Kent
,Asaad M. Ganeiber
,Kanti V. Mardia
.
(2013)
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"A new method to simulate the Bingham and related distributions in directional data analysis with applications"
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John Kent
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