110,321 research outputs found

### Multivariate Bernoulli distribution

In this paper, we consider the multivariate Bernoulli distribution as a model
to estimate the structure of graphs with binary nodes. This distribution is
discussed in the framework of the exponential family, and its statistical
properties regarding independence of the nodes are demonstrated. Importantly
the model can estimate not only the main effects and pairwise interactions
among the nodes but also is capable of modeling higher order interactions,
allowing for the existence of complex clique effects. We compare the
multivariate Bernoulli model with existing graphical inference models - the
Ising model and the multivariate Gaussian model, where only the pairwise
interactions are considered. On the other hand, the multivariate Bernoulli
distribution has an interesting property in that independence and
uncorrelatedness of the component random variables are equivalent. Both the
marginal and conditional distributions of a subset of variables in the
multivariate Bernoulli distribution still follow the multivariate Bernoulli
distribution. Furthermore, the multivariate Bernoulli logistic model is
developed under generalized linear model theory by utilizing the canonical link
function in order to include covariate information on the nodes, edges and
cliques. We also consider variable selection techniques such as LASSO in the
logistic model to impose sparsity structure on the graph. Finally, we discuss
extending the smoothing spline ANOVA approach to the multivariate Bernoulli
logistic model to enable estimation of non-linear effects of the predictor
variables.Comment: Published in at http://dx.doi.org/10.3150/12-BEJSP10 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

### Concentration of empirical distribution functions with applications to non-i.i.d. models

The concentration of empirical measures is studied for dependent data, whose
joint distribution satisfies Poincar\'{e}-type or logarithmic Sobolev
inequalities. The general concentration results are then applied to spectral
empirical distribution functions associated with high-dimensional random
matrices.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ254 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

### Discretized normal approximation by Stein's method

We prove a general theorem to bound the total variation distance between the
distribution of an integer valued random variable of interest and an
appropriate discretized normal distribution. We apply the theorem to 2-runs in
a sequence of i.i.d. Bernoulli random variables, the number of vertices with a
given degree in the Erd\"{o}s-R\'{e}nyi random graph, and the uniform
multinomial occupancy model.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ527 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

### The limit distribution of the maximum increment of a random walk with regularly varying jump size distribution

In this paper, we deal with the asymptotic distribution of the maximum
increment of a random walk with a regularly varying jump size distribution.
This problem is motivated by a long-standing problem on change point detection
for epidemic alternatives. It turns out that the limit distribution of the
maximum increment of the random walk is one of the classical extreme value
distributions, the Fr\'{e}chet distribution. We prove the results in the
general framework of point processes and for jump sizes taking values in a
separable Banach space.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ255 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

### The AEP algorithm for the fast computation of the distribution of the sum of dependent random variables

We propose a new algorithm to compute numerically the distribution function
of the sum of $d$ dependent, non-negative random variables with given joint
distribution.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ284 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

### A new method for obtaining sharp compound Poisson approximation error estimates for sums of locally dependent random variables

Let $X_1,X_2,...,X_n$ be a sequence of independent or locally dependent
random variables taking values in $\mathbb{Z}_+$. In this paper, we derive
sharp bounds, via a new probabilistic method, for the total variation distance
between the distribution of the sum $\sum_{i=1}^nX_i$ and an appropriate
Poisson or compound Poisson distribution. These bounds include a factor which
depends on the smoothness of the approximating Poisson or compound Poisson
distribution. This "smoothness factor" is of order $\mathrm{O}(\sigma ^{-2})$,
according to a heuristic argument, where $\sigma ^2$ denotes the variance of
the approximating distribution. In this way, we offer sharp error estimates for
a large range of values of the parameters. Finally, specific examples
concerning appearances of rare runs in sequences of Bernoulli trials are
presented by way of illustration.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ201 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

### Conditional limit laws for goodness-of-fit tests

We study the conditional distribution of goodness of fit statistics of the
Cram\'{e}r--von Mises type given the complete sufficient statistics in testing
for exponential family models. We show that this distribution is close, in
large samples, to that given by parametric bootstrapping, namely, the
unconditional distribution of the statistic under the value of the parameter
given by the maximum likelihood estimate. As part of the proof, we give uniform
Edgeworth expansions of Rao--Blackwell estimates in these models.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ366 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

### Properties and numerical evaluation of the Rosenblatt distribution

This paper studies various distributional properties of the Rosenblatt
distribution. We begin by describing a technique for computing the cumulants.
We then study the expansion of the Rosenblatt distribution in terms of shifted
chi-squared distributions. We derive the coefficients of this expansion and use
these to obtain the L\'{e}vy-Khintchine formula and derive asymptotic
properties of the L\'{e}vy measure. This allows us to compute the cumulants,
moments, coefficients in the chi-square expansion and the density and
cumulative distribution functions of the Rosenblatt distribution with a high
degree of precision. Tables are provided and software written to implement the
methods described here is freely available by request from the authors.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ421 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

### On lower limits and equivalences for distribution tails of randomly stopped sums

For a distribution $F^{*\tau}$ of a random sum
$S_{\tau}=\xi_1+...+\xi_{\tau}$ of i.i.d. random variables with a common
distribution $F$ on the half-line $[0,\infty)$, we study the limits of the
ratios of tails $\bar{F^{*\tau}}(x)/\bar{F}(x)$ as $x\to\infty$ (here, $\tau$
is a counting random variable which does not depend on $\{\xi_n\}_{n\ge1}$). We
also consider applications of the results obtained to random walks, compound
Poisson distributions, infinitely divisible laws, and subcritical branching
processes.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ111 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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