On Bahadur's Representation of Sample Quantiles for Nonstationary Mixing Processes

For sequences of independent and identically distributed random variables, Bahadur (1966), obtained, under certain mild coniditions an elegant almost sure represetation of a sample quantile as an average of independent and identically distributed centered random variables plus a remainder term converging to zero almost surely at a faster rate. J. K . Ghosh (1971), obtained, under milder regularity conditions, a weaker version of the result. The present paper obtains under certain conditions Bahadur type results for non-stationary ø-mixing processes and J. K. Ghosh type results for non-stationary strongly mixed processes.

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Moderate and Large Deviations for the Smoothed Estimate of Sample Quantiles

Journal of Probability and Statistics ◽

10.1155/2015/714201 ◽

2015 ◽

Vol 2015 ◽

pp. 1-6

Author(s):

Xiaoxia He ◽

Xi Liu ◽

Chun Yao

Keyword(s):

Large Deviations ◽

Sample Quantile ◽

Large Deviations Principle ◽

Sample Quantiles ◽

Independent And Identically Distributed

We derive the moderate and large deviations principle for the smoothed sample quantile from a sequence of independent and identically distributed samples of sizen.

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MODERATE DEVIATION PRINCIPLE OF SAMPLE QUANTILES AND ORDER STATISTICS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964817000420 ◽

2017 ◽

Vol 32 (4) ◽

pp. 603-614

Author(s):

Yi Wu ◽

Xuejun Wang ◽

Shuhe Hu

Keyword(s):

Order Statistics ◽

Random Variables ◽

Moderate Deviation ◽

Dependent Random Variables ◽

Moderate Deviation Principle ◽

Deviation Principle ◽

Dependent Sequence ◽

Sample Quantiles ◽

Independent And Identically Distributed

In this paper, we mainly study the moderate deviation principle of sample quantiles and order statistics for stationary m-dependent random variables. The results obtained in this paper extend the corresponding ones for an independent and identically distributed sequence to a stationary m-dependent sequence.

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Asymptotic Distribution of Sample Quantiles for Exchangeable Random Variables

Calcutta Statistical Association Bulletin ◽

10.1177/0008068319710402 ◽

1971 ◽

Vol 20 (4) ◽

pp. 135-142

Author(s):

K. C. Chanda

Keyword(s):

Normal Distribution ◽

Asymptotic Distribution ◽

Order Statistic ◽

Multivariate Normal Distribution ◽

Random Variables ◽

Multivariate Normal ◽

Sample Quantile ◽

Exchangeable Random Variables ◽

Sample Quantiles ◽

Special Case

Summary The purpose of this article is to investigate the ‘large sample’ properties of sample quantiles when we assume that the basic random variables are exchangeable (ref. Loève (1960) p. 365). It is shown that under different conditions (to be specified below) on the nature of these exchangeable random variables the distribution of the sample quantile Xr : n where Xr : n is the rth order statistic for the first n exchangeable random variables [Formula: see text] tends, as n → ∞, to different nondegenerate forms. As an example, the special case of random variables with equicor-related multivariate normal distribution is discussed.

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The Bahadur-Kiefer Representation ofLpRegression Estimators

Econometric Theory ◽

10.1017/s0266466600006587 ◽

1996 ◽

Vol 12 (2) ◽

pp. 257-283 ◽

Cited By ~ 11

Author(s):

Miguel A. Arcones

Keyword(s):

Linear Regression ◽

Regression Model ◽

Positive Constant ◽

Linear Regression Model ◽

Random Variables ◽

Random Variable ◽

Regularity Conditions ◽

Dimensional Parameter ◽

Image Position ◽

Independent And Identically Distributed

We consider the following linear regression model:whereare independent and identically distributed random variables, Yi, is real, Zihas values in Rm, Ui, is independent of Zi, and θ0is anm-dimensional parameter to be estimated. TheLpestimator of θ0is the value 6n such thatHere, we will give the exact Bahadur-Kiefer representation of θn, for each p ≥ 1. Explicitly, we will see that, under regularity conditions,whereandcis a positive constant, which depends onpand on the random variableX.

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The number of failed components in a coherent working system when the lifetimes are discretely distributed

Metrika ◽

10.1007/s00184-021-00817-2 ◽

2021 ◽

Author(s):

Krzysztof Jasiński

Keyword(s):

Random Variables ◽

Continuous Case ◽

Coherent System ◽

Coherent Systems ◽

Discrete Random Variables ◽

Independent And Identically Distributed

AbstractIn this paper, we study the number of failed components of a coherent system. We consider the case when the component lifetimes are discrete random variables that may be dependent and non-identically distributed. Firstly, we compute the probability that there are exactly i, $$i=0,\ldots ,n-k,$$ i = 0 , … , n - k , failures in a k-out-of-n system under the condition that it is operating at time t. Next, we extend this result to other coherent systems. In addition, we show that, in the most popular model of independent and identically distributed component lifetimes, the obtained probability corresponds to the respective one derived in the continuous case and existing in the literature.

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A central limit theorem by remainder term for renewal processes

Advances in Applied Probability ◽

10.2307/1427692 ◽

1992 ◽

Vol 24 (2) ◽

pp. 267-287 ◽

Cited By ~ 5

Author(s):

Allen L. Roginsky

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Remainder Term ◽

Limit Theorems ◽

Random Variables ◽

Central Limit Theorems ◽

Renewal Processes

Three different definitions of the renewal processes are considered. For each of them, a central limit theorem with a remainder term is proved. The random variables that form the renewal processes are independent but not necessarily identically distributed and do not have to be positive. The results obtained in this paper improve and extend the central limit theorems obtained by Ahmad (1981) and Niculescu and Omey (1985).

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Prophet Inequalities for Independent and Identically Distributed Random Variables from an Unknown Distribution

Mathematics of Operations Research ◽

10.1287/moor.2021.1167 ◽

2021 ◽

Author(s):

José Correa ◽

Paul Dütting ◽

Felix Fischer ◽

Kevin Schewior

Keyword(s):

Stopping Time ◽

Random Variables ◽

Optimal Solution ◽

Random Order ◽

Secretary Problem ◽

Maximum Value ◽

Optimal Stopping Theory ◽

Long Time ◽

Object Of Study ◽

Independent And Identically Distributed

A central object of study in optimal stopping theory is the single-choice prophet inequality for independent and identically distributed random variables: given a sequence of random variables [Formula: see text] drawn independently from the same distribution, the goal is to choose a stopping time τ such that for the maximum value of α and for all distributions, [Formula: see text]. What makes this problem challenging is that the decision whether [Formula: see text] may only depend on the values of the random variables [Formula: see text] and on the distribution F. For a long time, the best known bound for the problem had been [Formula: see text], but recently a tight bound of [Formula: see text] was obtained. The case where F is unknown, such that the decision whether [Formula: see text] may depend only on the values of the random variables [Formula: see text], is equally well motivated but has received much less attention. A straightforward guarantee for this case of [Formula: see text] can be derived from the well-known optimal solution to the secretary problem, where an arbitrary set of values arrive in random order and the goal is to maximize the probability of selecting the largest value. We show that this bound is in fact tight. We then investigate the case where the stopping time may additionally depend on a limited number of samples from F, and we show that, even with o(n) samples, [Formula: see text]. On the other hand, n samples allow for a significant improvement, whereas [Formula: see text] samples are equivalent to knowledge of the distribution: specifically, with n samples, [Formula: see text] and [Formula: see text], and with [Formula: see text] samples, [Formula: see text] for any [Formula: see text].

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On the law of the iterated logarithm in the infinite variance case

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700021868 ◽

1980 ◽

Vol 30 (1) ◽

pp. 5-14 ◽

Cited By ~ 2

Author(s):

R. A. Maller

Keyword(s):

Sufficient Conditions ◽

Random Variables ◽

Necessary And Sufficient Conditions ◽

Infinite Variance ◽

Iterated Logarithm ◽

The Law ◽

Necessary And Sufficient ◽

Independent And Identically Distributed

AbstractThe main purpose of the paper is to give necessary and sufficient conditions for the almost sure boundedness of (Sn – αn)/B(n), where Sn = X1 + X2 + … + XmXi being independent and identically distributed random variables, and αnand B(n) being centering and norming constants. The conditions take the form of the convergence or divergence of a series of a geometric subsequence of the sequence P(Sn − αn > a B(n)), where a is a constant. The theorem is distinguished from previous similar results by the comparative weakness of the subsidiary conditions and the simplicity of the calculations. As an application, a law of the iterated logarithm general enough to include a result of Feller is derived.

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On the ordered partial sums of real random variables

Journal of Applied Probability ◽

10.2307/3213261 ◽

1977 ◽

Vol 14 (1) ◽

pp. 75-88 ◽

Cited By ~ 2

Author(s):

Lajos Takács

Keyword(s):

Random Variables ◽

Partial Sums ◽

Markov Sequence ◽

Stieltjes Transforms ◽

Independent And Identically Distributed

In 1952 Pollaczek discovered a remarkable formula for the Laplace-Stieltjes transforms of the distributions of the ordered partial sums for a sequence of independent and identically distributed real random variables. In this paper Pollaczek's result is proved in a simple way and is extended for a semi-Markov sequence of real random variables.

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