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1.
Zheng Zukang 《数学学报(英文版)》1994,10(4):337-347
LetX
1,X
2, ...,X
n
be a sequence of nonnegative independent random variables with a common continuous distribution functionF. LetY
1,Y
2, ...,Y
n
be another sequence of nonnegative independent random variables with a common continuous distribution functionG, also independent of {X
i
}. We can only observeZ
i
=min(X
i
,Y
i
), and
. LetH=1−(1−F)(1−G) be the distribution function ofZ. In this paper, the limit theorems for the ratio of the Kaplan-Meier estimator
or the Altshuler estimator
to the true survival functionS(t) are given. It is shown that (1)P(δ(n)=1 i.o.)=0 ifF(τ
H
) < 1 andP(δ
n
=0 i.o. )=0 ifG(τH) > 1 where δ(n) is the corresponding indicator function of
and
have the same order
a.s., where {T
n
} is a sequence of constants such that 1−H(T
n
)=n
−α(logn)β(log logn)γ. 相似文献
2.
David Pollard 《Probability Theory and Related Fields》1981,57(2):181-195
Summary The empirical measure P
n
for iid sampling on a distribution P is formed by placing mass n
–1 at each of the first n observations. Generalizations of the classical Glivenko-Cantelli theorem for empirical measures have been proved by Vapnik and ervonenkis using combinatorial methods. They found simple conditions on a class C to ensure that sup {|P
n
(C) – P(C)|: C C} converges in probability to zero. They used a randomization device that reduced the problem to finding exponential bounds on the tails of a hypergeometric distribution. In this paper an alternative randomization is proposed. The role of the hypergeometric distribution is thereby taken over by the binomial distribution, for which the elementary Bernstein inequalities provide exponential boundson the tails. This leads to easier proofs of both the basic results of Vapnik-ervonenkis and the extensions due to Steele. A similar simplification is made in the proof of Dudley's central limit theorem forn
1/2(P P
n
–P)— a result that generalizes Donsker's functional central limit theorem for empirical distribution functions.This research was supported in part by the Air Force Office of Scientific Research, Contract No. F49620-79-C-0164 相似文献
3.
《Journal of multivariate analysis》1986,18(1):32-45
Nonsingular limit distributions are determined for sequences of affine transformations of random vectors whose distributions are multivariate binomial. Each of these limit distributions is that of an affine transformation of a random vector having a multivariate normal distribution or a multivariate Possion distribution or a joint distribution of two independent random vectors, one normal and the other Poisson. 相似文献
4.
Theorems of approximation of Gaussian processes for the sequential empirical process of the permutations of independent random variables are established. The results are applied to simulate critical values for the functionals of sequential empirical processes used in change point analysis. The proofs are based on the properties of rank statistics and negatively associated random variables. 相似文献
5.
6.
P. W. Millar 《Probability Theory and Related Fields》1979,48(3):233-252
Summary Fix a family C of continuous distributions on the line. Sufficient and (different) necessary conditions on C are given in order that the sample distribution function be an optimal estimator in the asymptotic minimax sense. The abstract results are illustrated by a variety of concrete families C that have arisen in the literature; some of these illustrations settle known, but previously unsolved, problems. Methods involve systematic consideration of statistical experiments whose parameter lies in a Hilbert space, and the theory of abstract Wiener spaces.Supported by NSF grant MCS75-10376 相似文献
7.
Hsien-Kuei Hwang 《Random Structures and Algorithms》1996,8(4):319-336
Central and local limit theorems (including large deviations) are established for the number of comparisons used by the standard top-down recursive mergesort under the uniform permutation model. The method of proof utilizes Dirichlet series, Mellin transforms, and standard analytic methods in probability theory. © 1996 John Wiley & Sons, Inc. 相似文献
8.
9.
Summary In this paper, we obtain a strong law and central limit theorem for the median deviation under only very mild smoothness conditions
on the underlying distribution. Under an additional condition implied by symmetry, we derive a weak Bahadur representation
for the median deviation and establish the asymptotic equivalence of the median deviation and the semi-interquartile range. 相似文献
10.
Ya. M. L'vovskii 《Journal of Mathematical Sciences》1996,81(4):2843-2850
Distributions are found of independent nonnegative integer valued random variables under linear constraints. Limit theorems
for these distributions are proved.
Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipolez, pp. 136–148, Perm, 1993. 相似文献
11.
V. I. Rotar' 《Journal of multivariate analysis》1979,9(4):511-530
The limit theorems for polylinear forms are obtained. Conditions are found under which the distribution of the polylinear form of many random variables is essentially the same as if all the distributions of arguments were normal. 相似文献
12.
13.
This paper studies the heavily trimmed sums (*)
[ns] + 1
[nt]
X
j
(n)
, where {X
j
(n)
}
j = 1
n
are the order statistics from independent random variables {X
1,...,X
n
} having a common distributionF. The main theorem gives the limiting process of (*) as a process oft. More smoothly trimmed sums like
j = 1
[nt]
J(j/n)X
j
(n)
are also discussed. 相似文献
14.
Alexander Bendikov Wojciech Cygan Bartosz Trojan 《Stochastic Processes and their Applications》2017,127(10):3268-3290
We consider a random walk which is obtained from the simple random walk by a discrete time version of Bochner’s subordination. We prove that under certain conditions on the subordinator appropriately scaled random walk converges in the Skorohod space to the symmetric -stable process . We also prove asymptotic formula for the transition function of similar to the Pólya’s asymptotic formula for . 相似文献
15.
Piet Groeneboom 《Probability Theory and Related Fields》1988,79(3):327-368
It is shown that the process of vertices of the convex hull of a uniform sample from the interior of a convex polygon converges locally, after rescaling, to a strongly mixing Markov process, as the sample size tends to infinity. The structure of the limiting Markov process is determined explicitly, and from this a central limit theorem for the number of vertices of the convex hull is derived. Similar results are given for uniform samples from the unit disk. 相似文献
16.
Let T(K1,r,Gn) be the number of monochromatic copies of the r‐star K1,r in a uniformly random coloring of the vertices of the graph Gn. In this paper we provide a complete characterization of the limiting distribution of T(K1,r,Gn), in the regime where is bounded, for any growing sequence of graphs Gn. The asymptotic distribution is a sum of mutually independent components, each term of which is a polynomial of a single Poisson random variable of degree at most r. Conversely, any limiting distribution of T(K1,r,Gn) has a representation of this form. Examples and connections to the birthday problem are discussed. 相似文献
17.
Douglas P. Kennedy 《Stochastic Processes and their Applications》1973,1(3):269-278
Limit distributions are given for both the dam content at time n and the limiting dam content in the nth dam, as n tends to infinity, for a sequence of finite dams in discrete time, under assumptions which correspond to the various cases of heavy traffic in queueing theory. The proof employed is an application of the theory of weak convergence of probability measures. 相似文献
18.
19.
Limit theorems for numbers of near-records 总被引:1,自引:0,他引:1
Anthony G. Pakes 《Extremes》2007,10(4):207-224
Observations occurring between successive record times and within a distance a > 0 of the current record value are called near-records. Limit theorems for the number ξ
n
(a) of near records are found for cases in which the parent distribution lies in a maximal domain of attraction and a is a function of n. Corollaries are indicated for numbers of near-k-records and sums of near-records. If the parent law is thin-tailed and a is constant, then a centered and normed version of logξ
n
(a) has a limit law under appropriate conditions.
相似文献
20.
Representation theorem and local asymptotic minimax theorem are derived for nonparametric estimators of the distribution function on the basis of randomly truncated data. The convolution-type representation theorem asserts that the limiting process of any regular estimator of the distribution function is at least as dispersed as the limiting process of the product-limit estimator. The theorems are similar to those results for the complete data case due to Beran (1977, Ann. Statist., 5, 400–404) and for the censored data case due to Wellner (1982, Ann. Statist., 10, 595–602). Both likelihood and functional approaches are considered and the proofs rely on the method of Begun et al. (1983, Ann. Statist., 11, 432–452) with slight modifications.Division of Biostatistics, School of Public Health, Columbia Univ. 相似文献