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1.
This article studies the asymptotic properties of the random weighted empirical distribution function of independent random variables. Suppose X1, X2, ???, Xn is a sequence of independent random variables, and this sequence is not required to be identically distributed. Denote the empirical distribution function of the sequence by Fn(x). Based on the random weighting method and Fn(x), the random weighted empirical distribution function Hn(x) is constructed and the asymptotic properties of Hn are discussed. Under weak conditions, the Glivenko–Cantelli theorem and the central limit theorem for the random weighted empirical distribution function are obtained. The obtained results have also been applied to study the distribution functions of random errors of multiple sensors.  相似文献   

2.
In this article, a semi-Markovian random walk with delay and a discrete interference of chance (X(t)) is considered. It is assumed that the random variables ζ n , n = 1, 2,…, which describe the discrete interference of chance form an ergodic Markov chain with ergodic distribution which is a gamma distribution with parameters (α, λ). Under this assumption, the asymptotic expansions for the first four moments of the ergodic distribution of the process X(t) are derived, as λ → 0. Moreover, by using the Riemann zeta-function, the coefficients of these asymptotic expansions are expressed by means of numerical characteristics of the summands, when the process considered is a semi-Markovian Gaussian random walk with small drift β.  相似文献   

3.
Assume that X 1, X 2,…, X n is a sequence of i.i.d. random variables with α-stable distribution (α ∈ (0,2], the stable exponent, is the unknown parameter). We construct minimum distance estimators for α by minimizing the Kolmogorov distance or the Cramér–von-Mises distance between the empirical distribution function G n , and a class of distributions defined based on the sum-preserving property of stable random variables. The minimum distance estimators can also be obtained by minimizing a U-statistic estimate of an empirical distribution function involving the stable exponent. They share the same invariance property with the maximum likelihood estimates. In this article, we prove the strong consistency of the minimum distance estimators. We prove the asymptotic normality of our estimators. Simulation study shows that the new estimators are competitive to the existing ones and perform very closely even to the maximum likelihood estimator.  相似文献   

4.
Let X 1, X 2,… be a sequence of independent and identically distributed random variables, and let Y n , n = K, K + 1, K + 2,… be the corresponding backward moving average of order K. At epoch n ≥ K, the process Y n will be off target by the input X n if it exceeds a threshold. By introducing a two-state Markov chain, we define a level of significance (1 ? a)% to be the percentage of times that the moving average process stays on target. We establish a technique to evaluate, or estimate, a threshold, to guarantee that {Y n } will stay (1 ? a)% of times on target, for a given (1 ? a)%. It is proved that if the distribution of the inputs is exponential or normal, then the threshold will be a linear function in the mean of the distribution of inputs μ X . The slope and intercept of the line, in each case, are specified. It is also observed that for the gamma inputs, the threshold is merely linear in the reciprocal of the scale parameter. These linear relationships can be easily applied to estimate the desired thresholds by samples from the inputs.  相似文献   

5.
Summary Let {X n } be a sequence of random variables conditionally independent and identically distributed given the random variable Θ. The aim of this paper is to show that in many interesting situations the conditional distribution of Θ, given (X 1,…,X n ), can be approximated by means of the bootstrap procedure proposed by Efron and applied to a statisticT n (X 1,…,X n ) sufficient for predictive purposes. It will also be shown that, from the predictive point of view, this is consistent with the results obtained following a common Bayesian approach.  相似文献   

6.
In this paper we consider a sequence of independent continuous symmetric random variables X1, X2, …, with heavy-tailed distributions. Then we focus on limiting behavior of randomly weighted averages Sn = R(n)1X1 + ??? + R(n)nXn, where the random weights R(n)1, …, Rn(n) which are independent of X1, X2, …, Xn, are the cuts of (0, 1) by the n ? 1 order statistics from a uniform distribution. Indeed we prove that cnSn converges in distribution to a symmetric α-stable random variable with cn = n1 ? 1/α1/α(α + 1).  相似文献   

7.
Let X1, X2, … be a strictly stationary sequence of observations, and g be the joint density of (X1, …, Xd) for some fixed d ? 1. We consider kernel estimators of the density g. The asymptotic behaviour of the mean integrated squared error of the kernel estimators is obtained under an assumption of weak dependence between the observations.  相似文献   

8.
Let {Xn} be a generalized autoregressive process of order ρ defined by Xnn(Xn-ρ,…,Xn-1)-ηm, where {φn} is a sequence of i.i.d. random maps taking values on H, and {ηn} is a sequence of i.i.d. random variables. Let H be a collection of Borel measurable functions on RP to R. By considering the associated Markov process, we obtain sufficient conditions for stationarity, (geometric) ergodicity of {Xn}.  相似文献   

9.
Let {Xn, n ? 1} be a sequence of asymptotically almost negatively associated (AANA, for short) random variables which is stochastically dominated by a random variable X, and {dni, 1 ? i ? n, n ? 1} be a sequence of real function, which is defined on a compact set E. Under some suitable conditions, we investigate some convergence properties for weighted sums of AANA random variables, especially the Lp convergence and the complete convergence. As an application, the Marcinkiewicz–Zygmund-type strong law of large numbers for AANA random variables is obtained.  相似文献   

10.
In this article, we consider a partially linear single-index model Y = g(Z τθ0) + X τβ0 + ? when the covariate X may be missing at random. We propose weighted estimators for the unknown parametric and nonparametric part by applying weighted estimating equations. We establish normality of the estimators of the parameters and asymptotic expansion for the estimator of the nonparametric part when the selection probabilities are unknown. Simulation studies are also conducted to illustrate the finite sample properties of these estimators.  相似文献   

11.
We discuss some problems connected with the role of record values and maximal values generated by sequences of random variables X1, X2,…, X n in the process of the growth of sums X1 +···+ Xn, n = 1, 2,….  相似文献   

12.
In this paper a generalization of the semi-Pareto autoregressive minification process of the first order is given. The necessary and sufficient condition for stationarity of the process is determined. It is shown that the process is ergodic and uniformly mixing. The joint survival function and the joint density function of the random variables X n+h and X n are determined. The extremes of the random variables X 1, X 2, ..., X n and the geometric extremes of random variables X 1, X 2, ..., X N are derived and their asymptotic distributions are discussed. The estimation of the parameters is discussed and some numerical results are given.  相似文献   

13.
Let {X j , j ≥ 1} be a strictly stationary negatively or positively associated sequence of real valued random variables with unknown distribution function F(x). On the basis of the random variables {X j , j ≥ 1}, we propose a smooth recursive kernel-type estimate of F(x), and study asymptotic bias, quadratic-mean consistency and asymptotic normality of the recursive kernel-type estimator under suitable conditions.  相似文献   

14.
In this paper, by considering a (3n+1) -dimensional random vector (X0, XT, YT, ZT)T having a multivariate elliptical distribution, we derive the exact joint distribution of (X0, aTX(n), bTY[n], cTZ[n])T, where a, b, c∈?n, X(n)=(X(1), …, X(n))T, X(1)<···<X(n), is the vector of order statistics arising from X, and Y[n]=(Y[1], …, Y[n])T and Z[n]=(Z[1], …, Z[n])T denote the vectors of concomitants corresponding to X(n) ((Y[r], Z[r])T, for r=1, …, n, is the vector of bivariate concomitants corresponding to X(r)). We then present an alternate approach for the derivation of the exact joint distribution of (X0, X(r), Y[r], Z[r])T, for r=1, …, n. We show that these joint distributions can be expressed as mixtures of four-variate unified skew-elliptical distributions and these mixture forms facilitate the prediction of X(r), say, based on the concomitants Y[r] and Z[r]. Finally, we illustrate the usefulness of our results by a real data.  相似文献   

15.
The supremum of random variables representing a sequence of rewards is of interest in establishing the existence of optimal stopping rules. Necessary and sufficient conditions are given for existence of moments of supn(Xn ? cn) and supn(Sn ? cn) where X1, X2, … are i.i.d. random variables, Sn = X1 + … + Xn, and cn = (nL(n))1/r, 0 < r < 2, L = 1, L = log, and L = log log. Following Cohn (1974), “rates of convergence” results are used in the proof.  相似文献   

16.
Let X1,…,X2n be independent and identically distributed copies of the non-negative integer valued random variable X distributed according to the unknown frequency function f(x). A total of 2n disjoint sequences of urns, each consisting of k urns, are given. Xj balls are placed in urn sequence j (1 ≤ j ≤ 2n). Each ball is placed in an urn of a given sequence with a certain known probability independently of the other balls. The variables X1,…,X2n are not observed; rather we observe whether certain pairs of urns are both empty or not. Our object is to estimate the mean μ of the number of balls X. Two different kinds of estimators of μ are investigated. One of the estimators studied is a method of moments type estimator while the other is motivated by the maximum likelihood principle. These estimators are compared on the basis of their asymptotic mean squared error as k tends to infinity. An application of these results to a problem in genetics involved with estimating codon substitution rates is discussed.  相似文献   

17.
Let X1,…,Xn be exchangeable normal variables with a common correlation p, and let X(1) > … > X(n) denote their order statistics. The random variable σni=nk+1xi, called the selection differential by geneticists, is of particular interest in genetic selection and related areas. In this paper we give results concerning a conjecture of Tong (1982) on the distribution of this random variable as a function of ρ. The same technique used can be applied to yield more general results for linear combinations of order statistics from elliptical distributions.  相似文献   

18.
Let X1:n ≤ X2:n ≤···≤ Xn:n denote the order statistics of a sample of n independent random variables X1, X2,…, Xn, all identically distributed as some X. It is shown that if X has a log-convex [log-concave] density function, then the general spacing vector (Xk1:n, Xk2:n ? Xk1:n,…, Xkr:n ? Xkr?1:n) is MTP2 [S-MRR2] whenever 1 ≤ k1 < k2 <···< kr ≤ n and 1 ≤ r ≤ n. Multivariate likelihood ratio ordering of such general spacing vectors corresponding to two random samples is also considered. These extend some of the results in the literature for usual spacing vectors.  相似文献   

19.
Let X 1,X 2,…,X n be independent exponential random variables such that X i has hazard rate λ for i = 1,…,p and X j has hazard rate λ* for j = p + 1,…,n, where 1 ≤ p < n. Denote by D i:n (λ, λ*) = X i:n  ? X i?1:n the ith spacing of the order statistics X 1:n  ≤ X 2:n  ≤ ··· ≤ X n:n , i = 1,…,n, where X 0:n ≡ 0. It is shown that the spacings (D 1,n ,D 2,n ,…,D n:n ) are MTP2, strengthening one result of Khaledi and Kochar (2000), and that (D 1:n 2, λ*),…,D n:n 2, λ*)) ≤ lr (D 1:n 1, λ*),…,D n:n 1, λ*)) for λ1 ≤ λ* ≤ λ2, where ≤ lr denotes the multivariate likelihood ratio order. A counterexample is also given to show that this comparison result is in general not true for λ* < λ1 < λ2.  相似文献   

20.
Let X be a non-negative random variable with cumulative probability distribution function F. Suppse X1, X2, ..., Xn be a random sample of size n from F and Xi,n is the i-th smallest order statistics. We define the standardized spacings Dr,n=(n-r) (Xr+1,n-Xr,n), 1≤r≤n, with DO,n=nX1,n and Dn,n=0. Characterizations of the exponential distribution are given by considering the expectation and hazard rates of Dr,n.  相似文献   

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