404 research outputs found
Characterization of Probability Law by Absolute Moments of Its Partial Sums
If Sn = X1 + . . . + Xn, where Xi are independent and identically distributed (i.i.d.) standard normal, then E|Sn| ⥠â2n/Ï, n ⧠0. We show that no other symmetric law has exactly these âmomentsâ; the general case remains (stubbornly) open. If X is standard two-sided exponential, then E|Sn| = 2n2-2n(2n/n). We show the latter moments are obtained exactly for all n also for Xi ~ B(2;0.5), the sum of two standard (± 1-valued) Bernoulliâs as well as for many other laws including unsymmetrical ones: Xi ~ G - 1, where G is geometric with mean 1, is one example.
Our interest in this delicate nonlinear inverse problem (which was initiated by Klebanov, cf. [12]) of inverting the moments to recover the law was also drawn by the fact that it gives a way to study positive definite functions through the formula E|Sn| = (2/Ï) â«0âRe(1 - Ïn(1 / u))du, n ⧠0, expressing E|Sn| in terms of the moments of Ï, where Ï is the characteristic function of X, Ï(u) = Eexp(iuX). We show that if for some b \u3e 0, Ïb (u) = Ï (btan (u / b)) is a positive definite function then the distributions corresponding to Ï and Ïb have the same E|Sn| moments for all n.
We show that if X is Bernoulli with zero mean and values ±1 then the moments characterize the distribution uniquely even among nonsymmetric laws. In general however we expect that the moments do not characterize the law, and this may well be the only nontrivial case of uniqueness.
We extend some of our results to the case of pth moments, p different from an even integer
Executive Summary: Young people, education, employment and ESOL
The report, âYoung people, education, employment and ESOLâ reviewed 47 studies1 to examine how current ESOL provision2 meets the needs of young people aged 16-25 years who use English as an Additional Language (EAL)3, and who need time and support to develop their English language skills in order to progress in education, training and employment
An historical survey of the development of the vibraphone as an alternative accompanying instrument in jazz
Includes bibliographical references (p. 150-159).Based on commercially available recordings of performances where the vibraphone alone performs the role of harmonic accompanist in the jazz ensemble, this study looks at the historical development of the use of the vibraphone as an alternative harmonic accompanying instrument. Descriptive analyses of key recordings made by leading and influential vibraphonists are given with regard to the use of the vibraphone in the role of an harmonic accompanying instrument
Beam-induced Background Simulations for the CMS Experiment at the LHC
Beam-induced background comes from interactions of the beam and beam halo particles with either the residual gas in the vacuum chamber of accelerator or the collimators that define the beam aperture. Beam-induced processes can potentially be a significant source of background for physics analyses at the LHC. This contribution describes the simulation software environment used for this part of the CMS experiment activity and recent beam-induced background simulation results for the Phase-2 CMS operation design
Analysis of airplane boarding via space-time geometry and random matrix theory
We show that airplane boarding can be asymptotically modeled by 2-dimensional
Lorentzian geometry. Boarding time is given by the maximal proper time among
curves in the model. Discrepancies between the model and simulation results are
closely related to random matrix theory. We then show how such models can be
used to explain why some commonly practiced airline boarding policies are
ineffective and even detrimental.Comment: 4 page
Limit Distributions of Self-Normalized Sums
If Xi are i.i.d. and have zero mean and arbitrary finite variance the limiting probability distribution of Sn(2) =(âni=1 Xi)/(ânj=1Xj2)1/2 as nââ has density f(t) = (2Ï)â1/2 exp(ât2/2) by the central limit theorem and the law of large numbers. If the tails of Xi are sufficiently smooth and satisfy P(Xi \u3e t) ⌠rtâα and P(Xi \u3c ât) ⌠ltâα as tââ, where 0 \u3c α \u3c 2, r \u3e 0, l \u3e 0, Sn(2) still has a limiting distribution F even though Xi has infinite variance. The density f of F depends on α as well as on r/l. We also study the limiting distribution of the more general Sn(p) = (âni=1Xi)/(ânj=1 |Xj|p)1/p where Xi are i.i.d. and in the domain of a stable law G with tails as above. In the cases p = 2 (see (4.21)) and p = 1 (see (3.7)) we obtain exact, computable formulas for f(t) = f(t,α,r/l), and give graphs of f for a number of values of α and r/l. For p = 2, we find that f is always symmetric about zero on (â1,1), even though f is symmetric on (ââ,â) only when r = l
Run Generation Revisited: What Goes Up May or May Not Come Down
In this paper, we revisit the classic problem of run generation. Run
generation is the first phase of external-memory sorting, where the objective
is to scan through the data, reorder elements using a small buffer of size M ,
and output runs (contiguously sorted chunks of elements) that are as long as
possible.
We develop algorithms for minimizing the total number of runs (or
equivalently, maximizing the average run length) when the runs are allowed to
be sorted or reverse sorted. We study the problem in the online setting, both
with and without resource augmentation, and in the offline setting.
(1) We analyze alternating-up-down replacement selection (runs alternate
between sorted and reverse sorted), which was studied by Knuth as far back as
1963. We show that this simple policy is asymptotically optimal. Specifically,
we show that alternating-up-down replacement selection is 2-competitive and no
deterministic online algorithm can perform better.
(2) We give online algorithms having smaller competitive ratios with resource
augmentation. Specifically, we exhibit a deterministic algorithm that, when
given a buffer of size 4M , is able to match or beat any optimal algorithm
having a buffer of size M . Furthermore, we present a randomized online
algorithm which is 7/4-competitive when given a buffer twice that of the
optimal.
(3) We demonstrate that performance can also be improved with a small amount
of foresight. We give an algorithm, which is 3/2-competitive, with
foreknowledge of the next 3M elements of the input stream. For the extreme case
where all future elements are known, we design a PTAS for computing the optimal
strategy a run generation algorithm must follow.
(4) Finally, we present algorithms tailored for nearly sorted inputs which
are guaranteed to have optimal solutions with sufficiently long runs
Linear regression for numeric symbolic variables: an ordinary least squares approach based on Wasserstein Distance
In this paper we present a linear regression model for modal symbolic data.
The observed variables are histogram variables according to the definition
given in the framework of Symbolic Data Analysis and the parameters of the
model are estimated using the classic Least Squares method. An appropriate
metric is introduced in order to measure the error between the observed and the
predicted distributions. In particular, the Wasserstein distance is proposed.
Some properties of such metric are exploited to predict the response variable
as direct linear combination of other independent histogram variables. Measures
of goodness of fit are discussed. An application on real data corroborates the
proposed method
Adaptive density estimation for stationary processes
We propose an algorithm to estimate the common density of a stationary
process . We suppose that the process is either or
-mixing. We provide a model selection procedure based on a generalization
of Mallows' and we prove oracle inequalities for the selected estimator
under a few prior assumptions on the collection of models and on the mixing
coefficients. We prove that our estimator is adaptive over a class of Besov
spaces, namely, we prove that it achieves the same rates of convergence as in
the i.i.d framework
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