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2000, Statistica Neerlandica
We investigate the validity of the bootstrap method for the elementary random variables X 1 , ... ,Xn. For both fix~d and'incre~sing brder k, as n __, oo the cases where µ = EX 1 f= 0, the nondegenerate case, and where µ = EX 1 = 0, the degenerate case, are considered.
Mathematics of Operations Research, 2014
Random sampling is a simple but powerful method in statistics and in the design of randomized algorithms. In a typical application, random sampling can be applied to estimate an extreme value, say maximum, of a function f over a set S ⊆ ℝn. To do so, one may select a simpler (even finite) subset S0 ⊆ S, randomly take some samples over S0 for a number of times, and pick the best sample. The hope is to find a good approximate solution with reasonable chance. This paper sets out to present a number of scenarios for f, S and S0 where certain probability bounds can be established, leading to a quality assurance of the procedure. In our setting, f is a multivariate polynomial function. We prove that if f is a d-th order homogeneous polynomial in n variables and F is its corresponding super-symmetric tensor, and ξi (i = 1, 2, …, n) are i.i.d. Bernoulli random variables taking 1 or −1 with equal probability, then Prob{f(ξ1, ξ2, …, ξn) ≥ τn−d/2 ‖F‖1} ≥ θ, where τ, θ > 0 are two universal ...
TEST, 2009
In this work, we give a complete picture of the behavior of the low intensity bootstrap of linear statistics. Our setup is given by triangular arrays of independent identically distributed random variables and different normalizations related to the rates of bootstrap intensities. We show that the behavior of this low intensity bootstrap coincides with that of partial sums of a number of summands equal to the bootstrap resampling size. Agreement on the limit laws for different (small) bootstrap sizes is thus shown to be closely related to domains of attraction of α-stable laws. As a byproduct, we obtain local distributional properties of Lévy processes.
Statistics & Probability Letters, 1993
Using Edgeworth expansions we compare the rates of convergence of the normal approximation and hvo bootstrap approaches for the sample mean of a symmetric population. In a simulation study we see how bad is the Monte Carlo approximation when bootstrapping the Edgeworth expansions.
Mathematics
This paper presents a comprehensive exploration of a probabilistic adaptation of the Eneström–Kakeya theorem, applied to random polynomials featuring various coefficient distributions. Unlike the deterministic rendition of the theorem, our study dispenses with the necessity of any specific coefficient order. Instead, we consider coefficients drawn from a spectrum of sets with diverse probability distributions, encompassing finite, countable, and uncountable sets. Furthermore, we provide a result concerning the probability of failure of Schur stability for a random polynomial with coefficients distributed independently and identically as standard normal variates. We also provide simulations to corroborate our results.
Journal of Computational and Applied Mathematics, 2005
This paper deals with the classes S 3 (, , b) of strong distribution functions defined on the interval [ 2 /b, b], 0 < < b ∞, where 2 ∈ Z. The classification is such that the distribution function ∈ S 3 (, , b) has a (reciprocal) symmetry, depending on , about the point. We consider properties of the L-orthogonal polynomials associated with ∈ S 3 (, , b). Through linear combination of these polynomials we relate them to the L-orthogonal polynomials associated with some ∈ S 3 (1/2, , b).
Proceedings of the London Mathematical Society, 2015
Let Pn(x) = n i=0 ξix i be a Kac random polynomial where the coefficients ξi are iid copies of a given random variable ξ. Our main result is an optimal quantitative bound concerning real roots repulsion. This leads to an optimal bound on the probability that there is a double root. As an application, we consider the problem of estimating the number of real roots of Pn, which has a long history and in particular was the main subject of a celebrated series of papers by Littlewood and Offord from the 1940s. We show, for a large and natural family of atom variables ξ, that the expected number of real roots of Pn(x) is exactly 2 π log n + C + o(1), where C is an absolute constant depending on the atom variable ξ. Prior to this paper, such a result was known only for the case when ξ is Gaussian.
Institute of Mathematical Statistics Lecture Notes - Monograph Series, 2004
This paper gives new proofs for many known results about the convergence in law of the bootstrap distribution to the true distribution of smooth statistics, whether the samples studied come from independent realizations of a random variable or dependent realizations with weak dependence. Moreover it suggests a novel bootstrap procedure and provides a proof that this new bootstrap works under uniform local dependence. The techniques employed are based on Stein's method for empirical processes as developed by . The last section provides some simulations and applications for which the relevant matlab functions are available from the first author.
The Annals of Probability, 2000
Let be a symmetric function, nondecreasing on [0, ∞) and satisfying a 2 growth condition, X 1 Y 1 X 2 Y 2 X n Y n be independent random vectors such that (for each 1 ≤ i ≤ n) either Y i = X i or Y i is independent of all the other variates, and the marginal distributions of X i and Y j are otherwise arbitrary. Let f ij x y 1≤i j≤n be any array of real valued measurable functions. We present a method of obtaining the order of magnitude of E 1≤i j≤n f ij X i Y j
Electronic Journal of Probability
We determine the asymptotics for the variance of the number of zeros of random linear combinations of orthogonal polynomials of degree ≤ n in subintervals [a, b] of the support of the underlying orthogonality measure µ. We show that, as n → ∞, this variance is asymptotic to cn, for some explicit constant c > 0.
Journal of Symbolic Computation, 1990
Real polynomials have very often very few real roots, and when algorithms depend on the number of real roots of polynomials rather than on their degrees, this fact has consequences on average complexity of algorithms. In this paper we recall some classical results on the average number of real roots (which is in O(log n) where n is the degree of the polynomial for many natural random distributions) and use them to get estimates on the average complexity of various algorithms characterizing real algebraic numbers.
Vietnam Journal of Mathematics, 2014
It is proved that the central limit theorem for general bootstrap empirical process with random resample size indexed by a class of functions F and based on a probability measure P holds a.s. if and F 2 denote the classes of squared functions and squared differences of functions from F, respectively. The bootstrap general empirical process with random resample size is also considered in the case where the resample size is independent of the original sample and of the bootstrap sample.
1996
We first analyze some results by Csrrg6 (1992). Then, by taking into account the different rates of convergence of the resampling size, we give new, simple proofs of those results. We provide examples that show that the sizes of resampling required by our results to ensure a.s. convergence are not far from being optimal.
Journal of Multivariate Analysis, 2013
This paper is mainly concerned with asymptotic studies of weighted bootstrap for u− and v−statistics. We derive the consistency of the weighted bootstrap u− and v−statistics, based on i.i.d. and non i.i.d. observations, from some more general results which we first establish for sums of randomly weighted arrays of random variables. Some of the results in this paper significantly extend some well-known results on consistency of u-statistics and also consistency of sums of arrays of random variables. We also employ a new approach to conditioning to derive a conditional CLT for weighted bootstrap u− and v−statistics, assuming the same conditions as the classical central limit theorems for regular u− and v−statistics.
Statistics & Probability Letters, 2008
This paper provides an asymptotic formula for the expected number of zeros of a polynomial of the form a 0 (ω) + a 1 (ω) n 1 1/2 x + a 2 (ω) n 2 1/2 x 2 + · · · + a n (ω) n n 1/2 x n for large n. The coefficients {a j (ω)} n j=0 are assumed to be a sequence of independent normally distributed random variables with fixed mean µ and variance one. It is shown that for µ non-zero this expected number is half of that for µ = 0. This behavior is similar to that of classical random algebraic polynomials but differs from that of random trigonometric polynomials.
The Annals of Statistics, 2012
This paper provides conditions under which subsampling and the bootstrap can be used to construct estimators of the quantiles of the distribution of a root that behave well uniformly over a large class of distributions P. These results are then applied (i) to construct confidence regions that behave well uniformly over P in the sense that the coverage probability tends to at least the nominal level uniformly over P and (ii) to construct tests that behave well uniformly over P in the sense that the size tends to no greater than the nominal level uniformly over P. Without these stronger notions of convergence, the asymptotic approximations to the coverage probability or size may be poor, even in very large samples. Specific applications include the multivariate mean, testing moment inequalities, multiple testing, the empirical process and U-statistics.
Statistical Science, 2008
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation - ISSAC '10, 2010
Our probabilistic analysis sheds light to the following questions: Why do random polynomials seem to have few, and well separated real roots, on the average? Why do exact algorithms for real root isolation may perform comparatively well or even better than numerical ones?
Journal of Applied Mathematics and Stochastic Analysis, 2001
The problem on asymptotic of the valueπ(m,n)=m!σm(p(1,n),p(2,n),…,p(n,n))is considered, whereσm(x1,x2,…,xn)is themth elementary symmetric function ofnvariables. The result is interpreted in the context of nonequiprobable random mappings theory.
We study the asymptotic properties of a class of multiple orthogonal polynomials with respect to a Nikishin system generated by two measures (σ 1 , σ 2 ) with unbounded supports (supp(σ 1 ) ⊂ R + , supp(σ 2 ) ⊂ R − ), and such that the second measure σ 2 is discrete. The weak asymptotics for these polynomials was obtained by Sorokin in . We use his result and the Riemann-Hilbert analysis to derive the strong asymptotics of these polynomials and of the reproducing kernel.
2016
Let {X n , n ≥ 1} be a sequence of stationary associated random variables. In this paper, we obtain consistent estimators of the distribution function and the variance of the sample mean based on {g(X n), n ≥ 1}, g : R → R using Circular Block Bootstrap (CBB). We extend these results to derive consistent estimators of the distribution function and the variance of U-statistics. As applications, we obtain interval estimators for L-moments. We also discuss consistent point estimators for L-moments. Finally, as an illustration, we obtain point estimators and confidence intervals for L-moments of a stationary autoregressive process with a minification structure which is fitted to a hydrological dataset.
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