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1.
Let be a sequence of stationary positively associated random variables and a sequence of positive constants be monotonically approaching infinity and be not asymptotically equivalent to loglog n. Under some suitable conditions, a nonclassical law of the iterated logarithm is investigated, i.e.
where (f) is a real function and .  相似文献   

2.
We considern independent and identically distributed random variables with common continuous distribution functionF concentrated on (0, ∞). LetX 1∶n≤X2∶n...≤Xn∶n be the corresponding order statistics. Put $$d_s \left( x \right) = P\left( {X_{k + s:n} - X_{k:n} \geqslant x} \right) - P\left( {X_{s:n - k} \geqslant x} \right), x \geqslant 0,$$ and $$\delta _s \left( {x, \rho } \right) = P\left( {X_{k + s:n} - X_{k:n} \geqslant x} \right) - e^{ - \rho \left( {n - k} \right)x} ,\rho > 0,x \geqslant 0.$$ Fors=1 it is well known that each of the conditions d1(x)=O ?x≥0 and δ1 (x, p) = O ?x≥0 implies thatF is exponential; but the analytic tools in the proofs of these two statements are radically different. In contrast to this in the present paper we present a rather elementary method which permits us to derive the above conclusions for somes, 1≤n —k, using only asymptotic assumptions (either forx→0 orx→∞) ond s(x) and δ1 (x, p), respectively.  相似文献   

3.
4.
LetX 1,X 2, ...,X n (n≥3) be a random sample on a random variableX with distribution functionF having a unique continuous inverseF −1 over (a,b), −∞≤a<b≤∞ the support ofF. LetX 1:n <X 2:n <...<X n:n be the corresponding order statistics. Letg be a nonconstant continuous function over (a,b). Then for some functionG over (a, b) and for some positive integersr ands, 1<r+1<sn
  相似文献   

5.
LetX 1,X 2, …,X n(n ? 2) be a random sample on a random variablex with a continuous distribution functionF which is strictly increasing over (a, b), ?∞ ?a <b ? ∞, the support ofF andX 1:n ?X 2:n ? … ?X n:n the corresponding order statistics. Letg be a nonconstant continuous function over (a, b) with finiteg(a +) andE {g(X)}. Then for some positive integers, 1 <s ?n $$E\left\{ {\frac{1}{{s - 1}}\sum\limits_{i - 1}^{s - 1} {g(X_{i:n} )|X_{s:n} } = x} \right\} = 1/2(g(x) + g(a^ + )), \forall x \in (a,b)$$ iffg is bounded, monotonic and \(F(x) = \frac{{g(x) - g(a^ + )}}{{g(b^ - ) - g(a^ + )}},\forall x \in (a,b)\) . This leads to characterization of several distribution functions. A general form of this result is also stated.  相似文献   

6.
Let (X n ) be a sequence of i.i.d random variables and U n a U-statistic corresponding to a symmetric kernel function h, where h 1(x 1) = Eh(x 1, X 2, X 3, . . . , X m ), μ = E(h(X 1, X 2, . . . , X m )) and ? 1 = Var(h 1(X 1)). Denote \({\gamma=\sqrt{\varsigma_{1}}/\mu}\), the coefficient of variation. Assume that P(h(X 1, X 2, . . . , X m ) > 0) = 1, ? 1 > 0 and E|h(X 1, X 2, . . . , X m )|3 < ∞. We give herein the conditions under which
$\lim_{N\rightarrow\infty}\frac{1}{\log N}\sum_{n=1}^{N}\frac{1}{n}g\left(\left(\prod_{k=m}^{n}\frac{U_{k}}{\mu}\right)^{\frac{1}{m\gamma\sqrt{n}}}\right) =\int\limits_{-\infty}^{\infty}g(x)dF(x)\quad {\rm a.s.}$
for a certain family of unbounded measurable functions g, where F(·) is the distribution function of the random variable \({\exp(\sqrt{2} \xi)}\) and ξ is a standard normal random variable.
  相似文献   

7.
Tang Qingguo 《Metrika》2009,69(1):55-67
Suppose that the longitudinal observations (Y ij , X ij , t ij ) for i = 1, . . . ,n; j = 1, . . . ,m i are modeled by the semiparamtric model where β 0 is a k × 1 vector of unknown parameters, g(·) is an unknown estimated function and e ij are unobserved disturbances. This article consider M-type regressions which include mean, median and quantile regressions. The M-estimator of the slope parameter β 0 is obtained through piecewise local polynomial approximation of the nonparametric component. The local M-estimator of g(·) is also obtained by replacing β 0 in model with its M-estimator and using local linear approximation. The asymptotic distribution of the estimator of β 0 is derived. The asymptotic distributions of the local M-estimators of g(·) at both interior and boundary points are also established. Various applications of our main results are given. The research is supported in part by National Natural Science Foundation of China (Grant No. 10671089).  相似文献   

8.
Peng Zhao  Yiying Zhang 《Metrika》2014,77(6):811-836
In this article, we study the stochastic properties of the maxima from two independent heterogeneous gamma random variables with different both shape parameters and scale parameters. Our main purpose is to address how the heterogeneity of a random sample of size 2 affects the magnitude, skewness and dispersion of the maxima in the sense of various stochastic orderings. Let \(X_{1}\) and \(X_{2}\) be two independent gamma random variables with \(X_{i}\) having shape parameter \(r_{i}>0\) and scale parameter \(\lambda _{i}\) , \(i=1,2\) , and let \(X^{*}_{1}\) and \(X^{*}_{2}\) be another set of independent gamma random variables with \(X^{*}_{i}\) having shape parameter \(r_{i}^{*}>0\) and scale parameter \(\lambda _{i}^{*}\) , \(i=1,2\) . Denote by \(X_{2:2}\) and \(X^{*}_{2:2}\) the corresponding maxima, respectively. It is proved that, among others, if \((r_{1},r_{2})\) majorize \((r_{1}^{*},r_{2}^{*})\) and \((\lambda _{1},\lambda _{2})\) weakly majorize \((\lambda _{1}^{*},\lambda _{2}^{*})\) , then \(X_{2:2}\) is stochastically larger that \(X^{*}_{2:2}\) in the sense of the likelihood ratio order. We also study the skewness according to the star order for which a very general sufficient condition is provided, using which some useful consequences can be obtained. The new results established here strengthen and generalize some of the results known in the literature.  相似文献   

9.
P. Janssen 《Metrika》1981,28(1):35-46
This paper provides the rate of convergence in the central limit theorem and in the strong law of large numbers forvon Mises statistics , based on i.i.d. random variablesX 1 ,..., X N .The proofs rely on a decomposition ofvon Mises statistics into a linear combination ofU-statistics and then use (generalized) results on the convergence rates forU-statistics obtained byGrams/Serfling [1973] andCallaert/Janssen [1978].  相似文献   

10.
Let be independent and identically distributed random variables with continuous distribution function. Denote by the corresponding order statistics. In the present paper, the concept of -neighbourhood runs, which is an extension of the usual run concept to the continuous case, is developed for the sequence of ordered random variables   相似文献   

11.
Let {X j } be a strictly stationary sequence of negatively associated random variables with the marginal probability density function f(x). The recursive kernel estimators of f(x) are defined by
and the Rosenblatt–Parzen’s kernel estimator of f(x) is defined by , where 0  <  b n → 0 are bandwidths and K is some kernel function. In this paper, we study the uniformly Berry–Esseen bounds for these estimators of f(x). In particular, by choice of the bandwidths, the Berry–Esseen bounds of the estimators attain .  相似文献   

12.
Let U 1, U 2, . . . , U n–1 be an ordered sample from a Uniform [0,1] distribution. The non-overlapping uniform spacings of order s are defined as \({G_{i}^{(s)} =U_{is} -U_{(i-1)s}, i=1,2,\ldots,N^\prime, G_{N^\prime+1}^{(s)} =1-U_{N^\prime s}}\) with notation U 0 = 0, U n = 1, where \({N^\prime=\left\lfloor n/s\right\rfloor}\) is the integer part of n/s. Let \({ N=\left\lceil n/s\right\rceil}\) be the smallest integer greater than or equal to n/s, f m (u), m = 1, 2, . . . , N, be a sequence of real-valued Borel-measurable functions. In this article a Cramér type large deviation theorem for the statistic \({f_{1,n} (nG_{1}^{(s)})+\cdots+f_{N,n} (nG_{N}^{(s)} )}\) is proved.  相似文献   

13.
In this note we discuss the following problem. LetX andY to be two real valued independent r.v.'s with d.f.'sF and ?. Consider the d.f.F*? of the r.v.X oY, being o a binary operation among real numbers. We deal with the following equation: $$\mathcal{G}^1 (F * \phi ,s) = \mathcal{G}^2 (F,s)\square \mathcal{G}^3 (\phi ,s)\forall s \in S$$ where \(\mathcal{G}^1 ,\mathcal{G}^2 ,\mathcal{G}^3 \) are real or complex functionals, т another binary operation ands a parameter. We give a solution, that under stronger assumptions (Aczél 1966), is the only one, of the problem. Such a solution is obtained in two steps. First of all we give a solution in the very special case in whichX andY are degenerate r.v.'s. Secondly we extend the result to the general case under the following additional assumption: $$\begin{gathered} \mathcal{G}^1 (\alpha F + (1 - \alpha )\phi ,s) = H[\mathcal{G}^i (F,s),\mathcal{G}^i (\phi ,s);\alpha ] \hfill \\ \forall \alpha \in [0,1]i = 1,2,3 \hfill \\ \end{gathered} $$ .  相似文献   

14.
Let X 1, . . . , X n be independent exponential random variables with respective hazard rates λ1, . . . , λ n , and Y 1, . . . , Y n be independent and identically distributed random variables from an exponential distribution with hazard rate λ. Then, we prove that X 2:n , the second order statistic from X 1, . . . , X n , is larger than Y 2:n , the second order statistic from Y 1, . . . , Y n , in terms of the dispersive order if and only if
$\lambda\geq \sqrt{\frac{1}{{n\choose 2}}\sum_{1\leq i < j\leq n}\lambda_i\lambda_j}.$
We also show that X 2:n is smaller than Y 2:n in terms of the dispersive order if and only if
$ \lambda\le\frac{\sum^{n}_{i=1} \lambda_i-{\rm max}_{1\leq i\leq n} \lambda_i}{n-1}. $
Moreover, we extend the above two results to the proportional hazard rates model. These two results established here form nice extensions of the corresponding results on hazard rate, likelihood ratio, and MRL orderings established recently by Pǎltǎnea (J Stat Plan Inference 138:1993–1997, 2008), Zhao et al. (J Multivar Anal 100:952–962, 2009), and Zhao and Balakrishnan (J Stat Plan Inference 139:3027–3037, 2009), respectively.
  相似文献   

15.
Let \(X_{1},\ldots , X_{n}\) be lifetimes of components with independent non-negative generalized Birnbaum–Saunders random variables with shape parameters \(\alpha _{i}\) and scale parameters \(\beta _{i},~ i=1,\ldots ,n\), and \(I_{p_{1}},\ldots , I_{p_{n}}\) be independent Bernoulli random variables, independent of \(X_{i}\)’s, with \(E(I_{p_{i}})=p_{i},~i=1,\ldots ,n\). These are associated with random shocks on \(X_{i}\)’s. Then, \(Y_{i}=I_{p_{i}}X_{i}, ~i=1,\ldots ,n,\) correspond to the lifetimes when the random shock does not impact the components and zero when it does. In this paper, we discuss stochastic comparisons of the smallest order statistic arising from such random variables \(Y_{i},~i=1,\ldots ,n\). When the matrix of parameters \((h({\varvec{p}}), {\varvec{\beta }}^{\frac{1}{\nu }})\) or \((h({\varvec{p}}), {\varvec{\frac{1}{\alpha }}})\) changes to another matrix of parameters in a certain mathematical sense, we study the usual stochastic order of the smallest order statistic in such a setup. Finally, we apply the established results to two special cases: classical Birnbaum–Saunders and logistic Birnbaum–Saunders distributions.  相似文献   

16.
We give the cumulative distribution function of M n , the maximum of a sequence of n observations from an autoregressive process of order 1. Solutions are first given in terms of repeated integrals and then for the case, where the underlying random variables are absolutely continuous. When the correlation is positive, $$P \left( M_n \leq x \right)\ =a_{n,x},$$ where $$a_{n,x}= \sum_{j=1}^\infty \beta_{jx}\ \nu_{jx}^{n} = O \left( \nu_{1x}^{n}\right),$$ where {?? jx } are the eigenvalues of a non-symmetric Fredholm kernel, and ?? 1x is the eigenvalue of maximum magnitude. When the correlation is negative $$P \left( M_n \leq x \right)\ =a_{n,x} +a_{n-1,x}.$$ The weights ?? jx depend on the jth left and right eigenfunctions of the kernel. These are given formally by left and right eigenvectors of an infinite Toeplitz matrix whose eigenvalues are just {?? jx }. These results are large deviations expansions for extremes, since the maximum need not be standardized to have a limit. In fact, such a limit need not exist. The use of the derived expansion for P(M n ?? x) is illustrated using both simulated and real data sets.  相似文献   

17.
Michael Cramer 《Metrika》1997,46(1):187-211
The asymptotic distribution of a branching type recursion with non-stationary immigration is investigated. The recursion is given by , where (X l ) is a random sequence, (L n −1(1) ) are iid copies ofL n−1,K is a random number andK, (L n −1(1) ), {(X l ),Y n } are independent. This recursion has been studied intensively in the literature in the case thatX l ≥0,K is nonrandom andY n =0 ∀n. Cramer, Rüschendorf (1996b) treat the above recursion without immigration with starting conditionL 0=1, and easy to handle cases of the recursion with stationary immigration (i.e. the distribution ofY n does not depend on the timen). In this paper a general limit theorem will be deduced under natural conditions including square-integrability of the involved random variables. The treatment of the recursion is based on a contraction method. The conditions of the limit theorem are built upon the knowledge of the first two moments ofL n . In case of stationary immigration a detailed analysis of the first two moments ofL n leads one to consider 15 different cases. These cases are illustrated graphically and provide a straight forward means to check the conditions and to determine the operator whose unique fixed point is the limit distribution of the normalizedL n .  相似文献   

18.
Mariusz Bieniek 《Metrika》2007,66(2):233-242
Let , r ≥ 1, denote generalized order statistics, with arbitrary parameters , based on distribution function F. In this paper we characterize continuous distributions F by the regression of adjacent generalized order statistics, i.e. where are continuous and increasing functions and ψ is strictly increasing. Further we investigate in detail the case when ψ(x) = x and g is a linear function of the form g(x) = cx + d for some .  相似文献   

19.
Let { Xi} i 3 1{\{ X_{i}\} _{i\geq 1}} be an infinite sequence of recurrent partially exchangeable binary random variables. We study the exact distributions of two run statistics (total number of success runs and the longest success run) in { Xi} i 3 1{\{ X_{i}\} _{i\geq1}} . Since a flexible class of models for binary sequences can be obtained using the concept of partial exchangeability, as a special case of our results one can obtain the distribution of runs in ordinary Markov chains, exchangeable and independent sequences. The results also enable us to study the distribution of runs in particular urn models.  相似文献   

20.
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