Asymptotics of the Joint Distribution of Multivariate Extrema

Let ZN be a maximum of independent identically distributed random variables. In this paper, a nonuniform estimate of convergence rate in the transfer theorem max-scheme is obtained. Presented results make the estimates, given in [1] and [2], more precise.

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The nonuniform estimate of the convergence rate of the k-th maxima

Lietuvos matematikos rinkinys ◽

10.15388/lmr.2009.74 ◽

2009 ◽

Vol 50 ◽

Author(s):

Arvydas Jokimaitis

Keyword(s):

Convergence Rate ◽

Random Variables ◽

Nonuniform Estimate ◽

Kth Maxima ◽

Independent Identically Distributed

In this paper the nonuniform estimate of the convergence rate for the kth maxima of the independent identically distributed random variables is obtained.

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Nonuniform estimate of maximum density convergence rate for independent random variables

Nonlinear Analysis Modelling and Control ◽

10.15388/na.1999.4.0.15246 ◽

1999 ◽

Vol 4 ◽

pp. 3-9

Author(s):

A. Aksomaitis ◽

A. Jokimaitis

Keyword(s):

Limit Theorem ◽

Convergence Rate ◽

Random Variables ◽

Maximum Density ◽

Independent Random Variables ◽

Nonuniform Estimate ◽

Density Limit ◽

Estimate Of Convergence Rate

The nonuniform estimate of convergence rate in the maximum density limit theorem of independent nonidentically distributed random variables is obtained. This result is generalization of the work presented in [1].

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A clustering law for some discrete order statistics

Journal of Applied Probability ◽

10.1017/s002190020002235x ◽

2003 ◽

Vol 40 (01) ◽

pp. 226-241 ◽

Cited By ~ 1

Author(s):

Sunder Sethuraman

Keyword(s):

Distribution Function ◽

Positive Integer ◽

Order Statistics ◽

Law Of Large Numbers ◽

Asymptotic Properties ◽

Random Variables ◽

Large Numbers ◽

The Law ◽

Sample Maximum ◽

Independent Identically Distributed

Let X 1, X 2, …, X n be a sequence of independent, identically distributed positive integer random variables with distribution function F. Anderson (1970) proved a variant of the law of large numbers by showing that the sample maximum moves asymptotically on two values if and only if F satisfies a ‘clustering’ condition, In this article, we generalize Anderson's result and show that it is robust by proving that, for any r ≥ 0, the sample maximum and other extremes asymptotically cluster on r + 2 values if and only if Together with previous work which considered other asymptotic properties of these sample extremes, a more detailed asymptotic clustering structure for discrete order statistics is presented.

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Representations and limit theorems for extreme value distributions

Journal of Applied Probability ◽

10.1017/s0021900200046027 ◽

1978 ◽

Vol 15 (03) ◽

pp. 639-644 ◽

Cited By ~ 2

Author(s):

Peter Hall

Keyword(s):

Stochastic Process ◽

Order Statistics ◽

Joint Distribution ◽

Limit Theorems ◽

Random Variables ◽

Canonical Representation ◽

Extreme Value ◽

Limiting Distribution ◽

Extreme Value Distributions ◽

Independent Identically Distributed

LetXn1≦Xn2≦ ··· ≦Xnndenote the order statistics from a sample ofnindependent, identically distributed random variables, and suppose that the variablesXnn, Xn,n–1, ···, when suitably normalized, have a non-trivial limiting joint distributionξ1,ξ2, ···, asn → ∞. It is well known that the limiting distribution must be one of just three types. We provide a canonical representation of the stochastic process {ξn,n≧ 1} in terms of exponential variables, and use this representation to obtain limit theorems forξnasn →∞.

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The fair charge on a car ferry

Journal of Applied Probability ◽

10.2307/3212959 ◽

1980 ◽

Vol 17 (3) ◽

pp. 654-661

Author(s):

D. R. Grey

Keyword(s):

Distribution Function ◽

Random Variables ◽

Expected Number ◽

Expected Revenue ◽

Independent Identically Distributed

Vehicles whose lengths are independent identically distributed random variables with known distribution function are loaded onto ferries of fixed known length, each ferry departing as soon as it can no longer accommodate the next vehicle in the queue. We work out how much a vehicle of any particular length ought to pay for use of the ferry, as well as the expected number of vehicles per ferry and expected revenue per ferry in equilibrium.

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The fair charge on a car ferry

Journal of Applied Probability ◽

10.1017/s0021900200033763 ◽

1980 ◽

Vol 17 (03) ◽

pp. 654-661

Author(s):

D. R. Grey

Keyword(s):

Distribution Function ◽

Random Variables ◽

Expected Number ◽

Expected Revenue ◽

Independent Identically Distributed

Vehicles whose lengths are independent identically distributed random variables with known distribution function are loaded onto ferries of fixed known length, each ferry departing as soon as it can no longer accommodate the next vehicle in the queue. We work out how much a vehicle of any particular length ought to pay for use of the ferry, as well as the expected number of vehicles per ferry and expected revenue per ferry in equilibrium.

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On the law of the iterated logarithm for inter-record times

Journal of Applied Probability ◽

10.1017/s0021900200034999 ◽

1970 ◽

Vol 7 (02) ◽

pp. 432-439 ◽

Cited By ~ 10

Author(s):

William E. Strawderman ◽

Paul T. Holmes

Keyword(s):

Distribution Function ◽

Probability Space ◽

Continuous Distribution ◽

Random Variables ◽

Continuous Distribution Function ◽

Record Times ◽

Iterated Logarithm ◽

The Law ◽

Image Position ◽

Independent Identically Distributed

Let X 1, X2, X 3 , ··· be independent, identically distributed random variables on a probability space (Ω, F, P); and with a continuous distribution function. Let the sequence of indices {Vr } be defined as Also define The following theorem is due to Renyi [5].

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A martingale approach to strong convergence of the number of records

Advances in Applied Probability ◽

10.1239/aap/1011994033 ◽

2001 ◽

Vol 33 (4) ◽

pp. 864-873 ◽

Cited By ~ 7

Author(s):

Raúl Gouet ◽

F. Javier López ◽

Miguel San Miguel

Keyword(s):

Distribution Function ◽

Strong Convergence ◽

Wide Class ◽

Random Variables ◽

Strong Law ◽

Martingale Approach ◽

Common Distribution ◽

Common Distribution Function ◽

Independent Identically Distributed

Let (Xn) be a sequence of independent, identically distributed random variables, with common distribution function F, possibly discontinuous. We use martingale arguments to connect the number of upper records from (Xn) with sums of minima of related random variables. From this relationship we derive a general strong law for the number of records for a wide class of distributions F, including geometric and Poisson.

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The Convergence Rate with Bootstrap for Multidimensional Density Functional Kernel Estimation

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.591-593.2559 ◽

2012 ◽

Vol 591-593 ◽

pp. 2559-2563

Author(s):

De Wang Li

Keyword(s):

Distribution Function ◽

Convergence Rate ◽

Density Function ◽

Density Functional ◽

Bootstrap Method ◽

Kernel Estimation ◽

Random Variable ◽

University Professor ◽

The Common ◽

Independent Identically Distributed

Bootstrap method is a statistical method proposed by the American Stanford University professor of Statistics Efron, which belongs to the parameters of statistical methods. According to a given sub-sample, we do not need its distributional assumptions or increase the sample information which can be described the overall distribution characteristics of statistical inference. The basic idea of the Bootstrap statistics is unknown and can not repeat the sampling distribution function instead of using a repeat sampling of the distribution function estimates. The independent identically distributed random variable series ,have the common probability density function, with .In the paper, combining with multidimensional density function, we discuss the convergence rate with Bootstrap method for the kernel estimation of the density functional .

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