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1.
Let X be a normally distributed p-dimensional column vector with mean μ and positive definite covariance matrix σ. and let X α, α = 1,…, N, be a random sample of size N from this distribution. Partition X as ( X 1, X (2)', X '(3))', where X1 is one-dimension, X(2) is p2- dimensional, and so 1 + p1 + p2 = p. Let ρ1 and ρ be the multiple correlation coefficients of X1 with X(2) and with ( X '(2), X '(3))', respectively. Write ρ2/2 = ρ2 - ρ2/1. We shall cosider the following two problems  相似文献   

2.
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   

3.
Suppose that we have a nonparametric regression model Y = m(X) + ε with XRp, where X is a random design variable and is observed completely, and Y is the response variable and some Y-values are missing at random. Based on the “complete” data sets for Y after nonaprametric regression imputation and inverse probability weighted imputation, two estimators of the regression function m(x0) for fixed x0Rp are proposed. Asymptotic normality of two estimators is established, which is used to construct normal approximation-based confidence intervals for m(x0). We also construct an empirical likelihood (EL) statistic for m(x0) with limiting distribution of χ21, which is used to construct an EL confidence interval for m(x0).  相似文献   

4.
This paper deals with the derivation of the probability distribution of the rank order statistic N1μ,n(r), the number of crossings of heightr(≥0) in a generalized random walk with steps 1 and ?μ by using the modified Dwass technique.  相似文献   

5.
For X with binomial (n, p) distribution the usual measure of the error of X/n as an estimator of p is its standard error Sn(p) = √{E(X/n – p)2} = √{p(1 – p)/n}. A somewhat more natural measure is the average absolute error Dn(p) = E‖X/n – p‖. This article considers use of Dn(p) instead of Sn(p) in a student's first introduction to statistical estimation. Exact and asymptotic values of Dn(p), and the appearance of its graph, are described in detail. The same is done for the Poisson distribution.  相似文献   

6.

We consider the regression model yi = ?(xi ) + ε in which the function ? or its pth derivative ?(p) may have a discontinuity at some unknown point τ. By fitting local polynomials from the left and right, we test the null that ?(p) is continuous against the alternative that ?(p)(τ?) ≠ ?(p)(τ+). We obtain Darling-Erdös type limit theorems for the test statistics under the null hypothesis of no change, as well as their limits in probability under the alternative. Consistency of the related change-point estimators is also established.  相似文献   

7.
Consider an ergodic Markov chain X(t) in continuous time with an infinitesimal matrix Q = (qij) defined on a finite state space {0, 1,…, N}. In this note, we prove that if X(t) is skip-free positive (negative, respectively), i.e., qij, = 0 for j > i+ 1 (i > j+ 1), then the transition probability pij(t) = Pr[X(t)=j | X(0) =i] can be represented as a linear combination of p0N(t) (p(m)(N0)(t)), 0 ≤ m ≤N, where f(m)(t) denotes the mth derivative of a function f(t) with f(0)(t) =f(t). If X(t) is a birth-death process, then pij(t) is represented as a linear combination of p0N(m)(t), 0 ≤mN - |i-j|.  相似文献   

8.
A simple random sample is observed from a population with a large number‘K’ of alleles, to test for random mating. Of n couples, nijkl have female genotype ij and male genotype kl (i, j, k, l{1,…, A‘}). The large contingency table is collapsed into three counts, n0, n1 and n2 where np is the number of couples with s alleles in common (s = 0,1, 2). The counts are estimated by np?o where n0, is the estimated probability of a couple having s alleles in common under the hypothesis of random mating. The usual chi-square goodness of fit statistic X2 compares observed (ns) with expected (np?) over the three categories, s = 0,1,2. An empirical observation has suggested that X2 is close to having a chi-square distribution with two degrees of freedom (X) despite a large number of parameters implicitly estimated in e. This paper gives two theorems which show that x is indeed the approximate distribution of X2 for large n and K1“, provided that no allele type over-dominates the others.  相似文献   

9.
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   

10.
In the standard linear regression model with independent, homoscedastic errors, the Gauss—Markov theorem asserts that = (X'X)-1(X'y) is the best linear unbiased estimator of β and, furthermore, that is the best linear unbiased estimator of c'β for all p × 1 vectors c. In the corresponding random regressor model, X is a random sample of size n from a p-variate distribution. If attention is restricted to linear estimators of c'β that are conditionally unbiased, given X, the Gauss—Markov theorem applies. If, however, the estimator is required only to be unconditionally unbiased, the Gauss—Markov theorem may or may not hold, depending on what is known about the distribution of X. The results generalize to the case in which X is a random sample without replacement from a finite population.  相似文献   

11.
Given an unknown function (e.g. a probability density, a regression function, …) f and a constant c, the problem of estimating the level set L(c) ={fc} is considered. This problem is tackled in a very general framework, which allows f to be defined on a metric space different from . Such a degree of generality is motivated by practical considerations and, in fact, an example with astronomical data is analyzed where the domain of f is the unit sphere. A plug‐in approach is followed; that is, L(c) is estimated by Ln(c) ={fnc} , where fn is an estimator of f. Two results are obtained concerning consistency and convergence rates, with respect to the Hausdorff metric, of the boundaries ?Ln(c) towards ?L(c) . Also, the consistency of Ln(c) to L(c) is shown, under mild conditions, with respect to the L1 distance. Special attention is paid to the particular case of spherical data.  相似文献   

12.
In this article, we study large deviations for non random difference ∑n1(t)j = 1X1j ? ∑n2(t)j = 1X2j and random difference ∑N1(t)j = 1X1j ? ∑N2(t)j = 1X2j, where {X1j, j ? 1} is a sequence of widely upper orthant dependent (WUOD) random variables with non identical distributions {F1j(x), j ? 1}, {X2j, j ? 1} is a sequence of independent identically distributed random variables, n1(t) and n2(t) are two positive integer-valued functions, and {Ni(t), t ? 0}2i = 1 with ENi(t) = λi(t) are two counting processes independent of {Xij, j ? 1}2i = 1. Under several assumptions, some results of precise large deviations for non random difference and random difference are derived, and some corresponding results are extended.  相似文献   

13.
The authors give easy‐to‐check sufficient conditions for the geometric ergodicity and the finiteness of the moments of a random process xt = ?(xt‐1,…, xt‐p) + ?tσ(xt‐1,…, xt‐q) in which ?: Rp → R, σ Rq → R and (?t) is a sequence of independent and identically distributed random variables. They deduce strong mixing properties for this class of nonlinear autoregressive models with changing conditional variances which includes, among others, the ARCH(p), the AR(p)‐ARCH(p), and the double‐threshold autoregressive models.  相似文献   

14.
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   

15.
For non-negative integral valued interchangeable random variables v1, v2,…,vn, Takács (1967, 70) has derived the distributions of the statistics ?n' ?1n' ?(c)n and ?(-c)n concerning the partial sums Nr = v1 + v2 + ··· + vrr = 1,…,n. This paper deals with the joint distributions of some other statistics viz., (α(c)n, δ(c)n, Zn), (β(c)n, Zn) and (β(-c)n, Zn) concerning the partial sums Nr = ε1 + ··· + εrr = 1,2,…,n, of geometric random variables ε1, ε2,…,εn.  相似文献   

16.
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   

17.
Let X ? (r), r ≥ 1, denote generalized order statistics based on an arbitrary distribution function F with finite pth absolute moment for some 1 ≤ p ≤ ∞. We present sharp upper bounds on E(X ? (s) ? X ? (r)), 1 ≤ r < s, for F being either general or life distribution. The bounds are expressed in various scale units generated by pth central absolute or raw moments of F, respectively. The distributions achieving the bounds are specified.  相似文献   

18.
For a continuous random variable X with support equal to (a, b), with c.d.f. F, and g: Ω1 → Ω2 a continuous, strictly increasing function, such that Ω1∩Ω2?(a, b), but otherwise arbitrary, we establish that the random variables F(X) ? F(g(X)) and F(g? 1(X)) ? F(X) have the same distribution. Further developments, accompanied by illustrations and observations, address as well the equidistribution identity U ? ψ(U) = dψ? 1(U) ? U for UU(0, 1), where ψ is a continuous, strictly increasing and onto function, but otherwise arbitrary. Finally, we expand on applications with connections to variance reduction techniques, the discrepancy between distributions, and a risk identity in predictive density estimation.  相似文献   

19.
One of the basic parameters in survival analysis is the mean residual life M 0. For right censored observation, the usual empirical likelihood based log-likelihood ratio leads to a scaled c12{\chi_1^2} limit distribution and estimating the scaled parameter leads to lower coverage of the corresponding confidence interval. To solve the problem, we present a log-likelihood ratio l(M 0) by methods of Murphy and van der Vaart (Ann Stat 1471–1509, 1997). The limit distribution of l(M 0) is the standard c12{\chi_1^2} distribution. Based on the limit distribution of l(M 0), the corresponding confidence interval of M 0 is constructed. Since the proof of the limit distribution does not offer a computational method for the maximization of the log-likelihood ratio, an EM algorithm is proposed. Simulation studies support the theoretical result.  相似文献   

20.
Let X (n) and X (1) be the largest and smallest order statistics, respectively, of a random sample of fixed size n. Quite generally, X (1) and X (n) are approximately independent for n sufficiently large. In this article, we study the dependence properties of random extremes in terms of their copula, when the sample size has a left-truncated binomial distribution and show that they tend to be more dependent in this case. We also give closed-form formulas for the measures of association Kendall's τ and Spearman's ρ to measure the amount of dependence between two extremes.  相似文献   

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