46,212 research outputs found
Concentration inequalities for order statistics
This note describes non-asymptotic variance and tail bounds for order
statistics of samples of independent identically distributed random variables.
Those bounds are checked to be asymptotically tight when the sampling
distribution belongs to a maximum domain of attraction. If the sampling
distribution has non-decreasing hazard rate (this includes the Gaussian
distribution), we derive an exponential Efron-Stein inequality for order
statistics: an inequality connecting the logarithmic moment generating function
of centered order statistics with exponential moments of Efron-Stein
(jackknife) estimates of variance. We use this general connection to derive
variance and tail bounds for order statistics of Gaussian sample. Those bounds
are not within the scope of the Tsirelson-Ibragimov-Sudakov
Gaussian concentration inequality. Proofs are elementary and combine
R\'enyi's representation of order statistics and the so-called entropy approach
to concentration inequalities popularized by M. Ledoux.Comment: 13 page
Order Statistics and Benford's Law
Fix a base B and let zeta have the standard exponential distribution; the
distribution of digits of zeta base B is known to be very close to Benford's
Law. If there exists a C such that the distribution of digits of C times the
elements of some set is the same as that of zeta, we say that set exhibits
shifted exponential behavior base B (with a shift of log_B C \bmod 1). Let X_1,
>..., X_N be independent identically distributed random variables. If the X_i's
are drawn from the uniform distribution on [0,L], then as N\to\infty the
distribution of the digits of the differences between adjacent order statistics
converges to shifted exponential behavior (with a shift of \log_B L/N \bmod 1).
By differentiating the cumulative distribution function of the logarithms
modulo 1, applying Poisson Summation and then integrating the resulting
expression, we derive rapidly converging explicit formulas measuring the
deviations from Benford's Law. Fix a delta in (0,1) and choose N independent
random variables from any compactly supported distribution with uniformly
bounded first and second derivatives and a second order Taylor series expansion
at each point. The distribution of digits of any N^\delta consecutive
differences \emph{and} all N-1 normalized differences of the order statistics
exhibit shifted exponential behavior. We derive conditions on the probability
density which determine whether or not the distribution of the digits of all
the un-normalized differences converges to Benford's Law, shifted exponential
behavior, or oscillates between the two, and show that the Pareto distribution
leads to oscillating behavior.Comment: 14 pages, 2 figures, version 4: Version 3: most of the numerical
simulations on shifted exponential behavior have been suppressed (though are
available from the authors upon request). Version 4: a referee pointed out
that we need epsilon > 1/3 - delta/2 in the proof of Theorem 1.5; this has
now been adde
On Order Statistics for GS-Distributions
In this article, a class of distributions is used to establish several recurrence relations satisfied by single and product moments of order statistics and progressive Type-II right censoring. The recurrence relations for moments of some specific distributions including uniform (a;b); exponential (λ); generalized exponential (α;λ;ν); beta (1;b); beta (b;1); logistic (α;β) and other distributions from order statistics and progressive Type-II right censoring can be obtained as special cases. A short explanation of GS-distribution can be found in reference [27]. As an example, means, variances and covariances for standard exponential distribution of progressive Type-II right censored order statistics are computed. Various characterizations of the recently introduced GS-distributions are presented. These characterizations are based on a simple relationship between two truncated moments ; on hazard function ; and on functions of order statistics. A characterization of the GS-distributions based on conditional moment of order statistics is extended to truncated moment of order statistics
Order statistics of the trapping problem
When a large number N of independent diffusing particles are placed upon a
site of a d-dimensional Euclidean lattice randomly occupied by a concentration
c of traps, what is the m-th moment of the time t_{j,N} elapsed
until the first j are trapped? An exact answer is given in terms of the
probability Phi_M(t) that no particle of an initial set of M=N, N-1,..., N-j
particles is trapped by time t. The Rosenstock approximation is used to
evaluate Phi_M(t), and it is found that for a large range of trap
concentracions the m-th moment of t_{j,N} goes as x^{-m} and its variance as
x^{-2}, x being ln^{2/d} (1-c) ln N. A rigorous asymptotic expression (dominant
and two corrective terms) is given for for the one-dimensional
lattice.Comment: 11 pages, 7 figures, to be published in Phys. Rev.
Order statistics of 1/f^{\alpha} signals
Order statistics of periodic, Gaussian noise with 1/f^{\alpha} power spectrum
is investigated. Using simulations and phenomenological arguments, we find
three scaling regimes for the average gap d_k= between the k-th
and (k+1)-st largest values of the signal. The result d_k ~ 1/k known for
independent, identically distributed variables remains valid for 0<\alpha<1.
Nontrivial, \alpha-dependent scaling exponents d_k ~ k^{(\alpha -3)/2} emerge
for 1<\alpha<5 and, finally, \alpha-independent scaling, d_k ~ k is obtained
for \alpha>5. The spectra of average ordered values \epsilon_k= ~
k^{\beta} is also examined. The exponent {\beta} is derived from the gap
scaling as well as by relating \epsilon_k to the density of near extreme
states. Known results for the density of near extreme states combined with
scaling suggest that \beta(\alpha=2)=1/2, \beta(4)=3/2, and beta(infinity)=2
are exact values. We also show that parallels can be drawn between \epsilon_k
and the quantum mechanical spectra of a particle in power-law potentials.Comment: 8 pages, 5 figure
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