Suchergebnisse

We derive uniformly most powerful (UMP) tests for simple and one-sided hypotheses for a population proportion within the framework of Differential Privacy (DP), optimizing finite sample performance. We show that in general, DP hypothesis tests can be written in terms of linear constraints, and for exchangeable data can always be expressed as a function of the empirical distribution. Using this structure, we prove a `Neyman-Pearson lemma' for binomial data under DP, where the DP-UMP only depends on the sample sum. Our tests can also be stated as a post-processing of a random variable, whose distribution we coin ``Truncated-Uniform-Laplace'' (Tulap), a generalization of the Staircase and discrete Laplace distributions. Furthermore, we obtain exact p-values, which are easily computed in terms of the Tulap random variable.
Using the above techniques, we show that our tests can be applied to give uniformly most accurate one-sided confidence intervals and optimal confidence distributions. We also derive uniformly most powerful unbiased (UMPU) two-sided tests, which lead to uniformly most accurate unbiased (UMAU) two-sided confidence intervals. We show that our results can be applied to distribution-free hypothesis tests for continuous data. Our simulation results demonstrate that all our tests have exact type I error, and are more powerful than current techniques.

Monte Carlo techniques, which require the generation of samples from some target density, are often the only alternative for performing Bayesian inference. Two classic sampling techniques to draw independent samples are the ratio of uniforms (RoU) and rejection sampling (RS). An efficient sampling algorithm is proposed combining the RoU and polar RS (i.e. RS inside a sector of a circle using polar coordinates). Its efficiency is shown in drawing samples from truncated Cauchy and Gaussian random variables, which have many important applications in signal processing and communications. ; This work has been partly financed by the Spanish government, through the DEIPRO project (TEC2009-14504-C02-01) and the CONSOLIDER-INGENIO 2010 program (CSD2008-00010). ; Publicado

Many basic questions in the social network literature center on the distribution of aggregate structural properties within and across populations of networks. Such questions are of increasing relevance given the growing availability of network data suitable for meta-analytic studies, as well as the rise of study designs that involve the collection of data on multiple networks drawn from a larger population. Despite this, little work has been done on model-based inference for the properties of graph populations, or on methods for comparing such populations. Here, we attempt to rectify this gap by introducing a family of techniques that combines an existing approach to the identification of structural biases in network data (the use of conditional uniform graph quantiles) with strategies drawn from nonparametric Bayesian analysis. Conditional uniform graph quantiles are the quantiles of an observed structural property in the reference distribution produced by evaluating that property over all graphs with certain fixed characteristics (e.g., size or density). These quantiles have long been used to measure the extent to which a property of interest on a single network deviates from what would be expected given that network's other characteristics. The methods introduced here employ such quantile information to allow for principled inference regarding the distribution of structural biases within (and comparison across) populations of networks, given data sampled at the network level. The data requirements of these methods are minimal, thus making them well-suited to meta-analytic applications for which complete network data (as opposed to summary statistics) are often unavailable. The structural biases inferred using these methods can be expressed in terms of posterior predictives for familiar and easily communicated quantities, such as p-values. In addition to the methods themselves, we present algorithms for posterior simulation from this model class, illustrating their use with applications to the analysis of social structure within urban communes and radio communications among emergency personnel. We also discuss how this approach may applied to quantiles arising from other reference distributions, such as those obtained using general exponential-family random graph models.

We study the rank of the instantaneous or spot covariance matrix ΣX(t) of a multidimensional continuous semi-martingale X(t). Given highfrequency observations X(i/n), i = 0,.,n, we test the null hypothesis rank (ΣX(t)) <= r for all t against local alternatives where the average (r + 1)st eigenvalue is larger than some signal detection rate vn. A major problem is that the inherent averaging in local covariance statistics produces a bias that distorts the rank statistics. We show that the bias depends on the regularity and a spectral gap of ΣX(t).We establish explicit matrix perturbation and concentration results that provide non-asymptotic uniform critical values and optimal signal detection rates vn. This leads to a rank estimation method via sequential testing. For a class of stochastic volatility models, we determine data-driven critical values via normed p-variations of estimated local covariance matrices. The methods are illustrated by simulations and an application to high-frequency data of U.S. government bonds.

AbstractFor each n., X1(n), X2(n), …, Xn(n) are IID, with common pdf fn(x). y1(n) < … < Yn (n) are the ordered values of X1 (n), …, Xn(n). Kn is a positive integer, with lim Kn = ∞. Under certain conditions on Kn and fn (x), it was shown in an earlier paper that the joint distribution of a special set of Kn + 1 of the variables Y1 (n), …, Yn (n) can be assumed to be normal for all asymptotic probability calculations. In another paper, it was shown that if fn (x) approaches the pdf which is uniform over (0, 1) at a certain rate as n increases, then the conditional distribution of the order statistics not in the special set can be assumed to be uniform for all asymptotic probability calculations. The present paper shows that even if fn (x) does not approach the uniform distribution as n increases, the distribution of the order statistics contained between order statistics in the special set can be assumed to be the distribution of a quadratic function of uniform random variables, for all asymptotic probability calculations. Applications to statistical inference are given.