Sample Path Large Deviations for Order Statistics

We consider the sample paths of the order statistics of independent and identically distributed random variables with common distribution function F. If F is strictly increasing but possibly having discontinuities, we prove that the sample paths of the order statistics satisfy the large deviation principle in the Skorokhod M 1 topology. Sanov's theorem is deduced in the Skorokhod M'1 topology as a corollary to this result. A number of illustrative examples are presented, including applications to the sample paths of trimmed means and Hill plots.

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Sample Path Large Deviations of Poisson Shot Noise with Heavy-Tailed Semiexponential Distributions

Journal of Applied Probability ◽

10.1017/s002190020000824x ◽

2011 ◽

Vol 48 (03) ◽

pp. 688-698 ◽

Cited By ~ 2

Author(s):

Ken R. Duffy ◽

Giovanni Luca Torrisi

Keyword(s):

Large Deviations ◽

Shot Noise ◽

Large Deviation Principle ◽

Large Deviation ◽

Sample Path ◽

Rate Function ◽

Sample Paths ◽

Heavy Tailed ◽

Poisson Shot Noise ◽

Sample Path Large Deviations

It is shown that the sample paths of Poisson shot noise with heavy-tailed semiexponential distributions satisfy a large deviation principle with a rate function that is insensitive to the shot shape. This demonstrates that, on the scale of large deviations, paths to rare events do not depend on the shot shape.

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Sample Path Large Deviations of Poisson Shot Noise with Heavy-Tailed Semiexponential Distributions

Journal of Applied Probability ◽

10.1239/jap/1316796907 ◽

2011 ◽

Vol 48 (3) ◽

pp. 688-698

Author(s):

Ken R. Duffy ◽

Giovanni Luca Torrisi

Keyword(s):

Large Deviations ◽

Shot Noise ◽

Large Deviation Principle ◽

Large Deviation ◽

Sample Path ◽

Rate Function ◽

Sample Paths ◽

Heavy Tailed ◽

Poisson Shot Noise ◽

Sample Path Large Deviations

It is shown that the sample paths of Poisson shot noise with heavy-tailed semiexponential distributions satisfy a large deviation principle with a rate function that is insensitive to the shot shape. This demonstrates that, on the scale of large deviations, paths to rare events do not depend on the shot shape.

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Sample-Path Large Deviations in Credit Risk

Journal of Applied Mathematics ◽

10.1155/2011/354171 ◽

2011 ◽

Vol 2011 ◽

pp. 1-28 ◽

Cited By ~ 3

Author(s):

V. J. G. Leijdekker ◽

M. R. H. Mandjes ◽

P. J. C. Spreij

Keyword(s):

Credit Risk ◽

Large Deviation Principle ◽

Large Deviation ◽

Sample Path ◽

Rare Event ◽

Asymptotic Results ◽

Loss Process ◽

Logarithmic Decay ◽

Sample Path Large Deviations ◽

Deviation Theory

The event of large losses plays an important role in credit risk. As these large losses are typically rare, and portfolios usually consist of a large number of positions, large deviation theory is the natural tool to analyze the tail asymptotics of the probabilities involved. We first derive a sample-path large deviation principle (LDP) for the portfolio's loss process, which enables the computation of the logarithmic decay rate of the probabilities of interest. In addition, we derive exact asymptotic results for a number of specific rare-event probabilities, such as the probability of the loss process exceeding some given function.

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Large deviations of heavy-tailed random sums with applications in insurance and finance

Journal of Applied Probability ◽

10.2307/3215371 ◽

1997 ◽

Vol 34 (2) ◽

pp. 293-308 ◽

Cited By ~ 94

Author(s):

C. Klüppelberg ◽

T. Mikosch

Keyword(s):

Distribution Function ◽

Large Deviation ◽

Random Variables ◽

Compound Poisson Process ◽

Random Sum ◽

Random Sums ◽

Common Distribution ◽

Common Distribution Function ◽

Heavy Tailed ◽

The Right

We prove large deviation results for the random sum , , where are non-negative integer-valued random variables and are i.i.d. non-negative random variables with common distribution function F, independent of . Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.

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STOCHASTIC MONOTONICITY OF CONDITIONAL ORDER STATISTICS IN MULTIPLE-OUTLIER SCALE POPULATION

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964816000383 ◽

2016 ◽

Vol 31 (3) ◽

pp. 366-380

Author(s):

Ebrahim Amini-Seresht ◽

Yiying Zhang

Keyword(s):

Distribution Function ◽

Order Statistics ◽

Likelihood Ratio ◽

Monotonicity Property ◽

Independent Random Variables ◽

Likelihood Ratio Order ◽

Stochastic Monotonicity ◽

Common Distribution ◽

Common Distribution Function ◽

Parametric Families

This paper discusses the stochastic monotonicity property of the conditional order statistics from independent multiple-outlier scale variables in terms of the likelihood ratio order. Let X1, …, Xn be a set of non-negative independent random variables with Xi, i=1, …, p, having common distribution function F(λ1x), and Xj, j=p+1, …, n, having common distribution function F(λ2x), where F(·) denotes the baseline distribution. Let Xi:n(p, q) be the ith smallest order statistics from this sample. Denote by $X_{i,n}^{s}(p,q)\doteq [X_{i:n}(p,q)|X_{i-1:n}(p,q)=s]$. Under the assumptions that xf′(x)/f(x) is decreasing in x∈ℛ+, λ1≤λ2 and s1≤s2, it is shown that $X_{i:n}^{s_{1}}(p+k,q-k)$ is larger than $X_{i:n}^{s_{2}}(p,q)$ according to the likelihood ratio order for any 2≤i≤n and k=1, 2, …, q. Some parametric families of distributions are also provided to illustrate the theoretical results.

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Large deviations of heavy-tailed random sums with applications in insurance and finance

Journal of Applied Probability ◽

10.1017/s0021900200100956 ◽

1997 ◽

Vol 34 (02) ◽

pp. 293-308 ◽

Cited By ~ 21

Author(s):

C. Klüppelberg ◽

T. Mikosch

Keyword(s):

Distribution Function ◽

Large Deviation ◽

Random Variables ◽

Compound Poisson Process ◽

Random Sum ◽

Random Sums ◽

Common Distribution ◽

Common Distribution Function ◽

Heavy Tailed ◽

The Right

We prove large deviation results for the random sum , , where are non-negative integer-valued random variables and are i.i.d. non-negative random variables with common distribution function F, independent of . Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.

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On the asymptotic normality of Hill's estimator

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100073710 ◽

1995 ◽

Vol 118 (2) ◽

pp. 375-382 ◽

Cited By ~ 4

Author(s):

Sándor Csörgő ◽

László Viharos

Keyword(s):

Distribution Function ◽

Asymptotic Normality ◽

Order Statistics ◽

Random Variables ◽

Fixed Number ◽

Distribution Functions ◽

Independent Random Variables ◽

Hill’S Estimator ◽

Common Distribution ◽

Common Distribution Function

Let X, X1, X2, …, be independent random variables with a common distribution function F(x) = P {X ≤ x}, x∈ℝ, and for each n∈ℕ, let X1, n ≤ … ≤ Xn, n denote the order statistics pertaining to the sample X1, …, Xn. We assume that 1–F(x) = x−1/cl(x), 0 < x < ∞, where l is some function slowly varying at infinity and c > 0 is any fixed number. The class of all such distribution functions will be denoted by .

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An extremal markovian sequence

Journal of Applied Probability ◽

10.1017/s0021900200027236 ◽

1989 ◽

Vol 26 (02) ◽

pp. 219-232 ◽

Cited By ~ 6

Author(s):

M. Teresa Alpuim

Keyword(s):

Distribution Function ◽

Order Statistics ◽

Stationary Distribution ◽

Random Variable ◽

Stationary Sequence ◽

Left Endpoint ◽

Limit Laws ◽

Marginal Distributions ◽

Common Distribution ◽

Common Distribution Function

In this paper we consider an independent and identically distributed sequence {Yn } with common distribution function F(x) and a random variable X 0, independent of the Yi 's, and define a Markovian sequence {Xn } as Xi = X 0, if i = 0, Xi = k max{Xi − 1, Yi }, if i ≧ 1, k ∈ R, 0 < k < 1. For this sequence we evaluate basic distributional formulas and give conditions on F(x) for the sequence to possess a stationary distribution. We prove that for any distribution function H(x) with left endpoint greater than or equal to zero for which log H(ex ) is concave it is possible to construct such a stationary sequence with marginal distributions equal to it. We study the limit laws for extremes and kth order statistics.

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THE SECOND-ORDER REGULAR VARIATION OF ORDER STATISTICS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964813000430 ◽

2013 ◽

Vol 28 (2) ◽

pp. 209-222 ◽

Cited By ~ 1

Author(s):

Qing Liu ◽

Tiantian Mao ◽

Taizhong Hu

Keyword(s):

Distribution Function ◽

Order Statistics ◽

Regular Variation ◽

Random Variables ◽

Second Order ◽

Tail Probabilities ◽

Common Distribution ◽

Common Distribution Function ◽

Second Order Regular Variation ◽

Independent And Identically Distributed

Let X1, …, Xn be non-negative, independent and identically distributed random variables with a common distribution function F, and denote by X1:n ≤ ··· ≤ Xn:n the corresponding order statistics. In this paper, we investigate the second-order regular variation (2RV) of the tail probabilities of Xk:n and Xj:n − Xi:n under the assumption that $\bar {F}$ is of the 2RV, where 1 ≤ k ≤ n and 1 ≤ i < j ≤ n.

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