1,429 research outputs found
Gaussian limits for generalized spacings
Nearest neighbor cells in , are used to define
coefficients of divergence (-divergences) between continuous multivariate
samples. For large sample sizes, such distances are shown to be asymptotically
normal with a variance depending on the underlying point density. In ,
this extends classical central limit theory for sum functions of spacings. The
general results yield central limit theorems for logarithmic -spacings,
information gain, log-likelihood ratios and the number of pairs of sample
points within a fixed distance of each other.Comment: Published in at http://dx.doi.org/10.1214/08-AAP537 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Spectral fluctuations of tridiagonal random matrices from the beta-Hermite ensemble
A time series delta(n), the fluctuation of the nth unfolded eigenvalue was
recently characterized for the classical Gaussian ensembles of NxN random
matrices (GOE, GUE, GSE). It is investigated here for the beta-Hermite ensemble
as a function of beta (zero or positive) by Monte Carlo simulations. The
fluctuation of delta(n) and the autocorrelation function vary logarithmically
with n for any beta>0 (1<<n<<N). The simple logarithmic behavior reported for
the higher-order moments of delta(n) for the GOE (beta=1) and the GUE (beta=2)
is valid for any positive beta and is accounted for by Gaussian distributions
whose variances depend linearly on ln(n). The 1/f noise previously demonstrated
for delta(n) series of the three Gaussian ensembles, is characterized by
wavelet analysis both as a function of beta and of N. When beta decreases from
1 to 0, for a given and large enough N, the evolution from a 1/f noise at
beta=1 to a 1/f^2 noise at beta=0 is heterogeneous with a ~1/f^2 noise at the
finest scales and a ~1/f noise at the coarsest ones. The range of scales in
which a ~1/f^2 noise predominates grows progressively when beta decreases.
Asymptotically, a 1/f^2 noise is found for beta=0 while a 1/f noise is the rule
for beta positive.Comment: 35 pages, 10 figures, corresponding author: G. Le Cae
Linear Estimation of Location and Scale Parameters Using Partial Maxima
Consider an i.i.d. sample X^*_1,X^*_2,...,X^*_n from a location-scale family,
and assume that the only available observations consist of the partial maxima
(or minima)sequence, X^*_{1:1},X^*_{2:2},...,X^*_{n:n}, where
X^*_{j:j}=max{X^*_1,...,X^*_j}. This kind of truncation appears in several
circumstances, including best performances in athletics events. In the case of
partial maxima, the form of the BLUEs (best linear unbiased estimators) is
quite similar to the form of the well-known Lloyd's (1952, Least-squares
estimation of location and scale parameters using order statistics, Biometrika,
vol. 39, pp. 88-95) BLUEs, based on (the sufficient sample of) order
statistics, but, in contrast to the classical case, their consistency is no
longer obvious. The present paper is mainly concerned with the scale parameter,
showing that the variance of the partial maxima BLUE is at most of order
O(1/log n), for a wide class of distributions.Comment: This article is devoted to the memory of my six-years-old, little
daughter, Dionyssia, who leaved us on August 25, 2010, at Cephalonia isl. (26
pages, to appear in Metrika
The Hellinger Correlation
In this paper, the defining properties of a valid measure of the dependence
between two random variables are reviewed and complemented with two original
ones, shown to be more fundamental than other usual postulates. While other
popular choices are proved to violate some of these requirements, a class of
dependence measures satisfying all of them is identified. One particular
measure, that we call the Hellinger correlation, appears as a natural choice
within that class due to both its theoretical and intuitive appeal. A simple
and efficient nonparametric estimator for that quantity is proposed. Synthetic
and real-data examples finally illustrate the descriptive ability of the
measure, which can also be used as test statistic for exact independence
testing
From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond
The discovery of connections between the distribution of energy levels of
heavy nuclei and spacings between prime numbers has been one of the most
surprising and fruitful observations in the twentieth century. The connection
between the two areas was first observed through Montgomery's work on the pair
correlation of zeros of the Riemann zeta function. As its generalizations and
consequences have motivated much of the following work, and to this day remains
one of the most important outstanding conjectures in the field, it occupies a
central role in our discussion below. We describe some of the many techniques
and results from the past sixty years, especially the important roles played by
numerical and experimental investigations, that led to the discovery of the
connections and progress towards understanding the behaviors. In our survey of
these two areas, we describe the common mathematics that explains the
remarkable universality. We conclude with some thoughts on what might lie ahead
in the pair correlation of zeros of the zeta function, and other similar
quantities.Comment: Version 1.1, 50 pages, 6 figures. To appear in "Open Problems in
Mathematics", Editors John Nash and Michael Th. Rassias. arXiv admin note:
text overlap with arXiv:0909.491
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