scholarly journals Two-choice optimal stopping

2004 ◽  
Vol 36 (4) ◽  
pp. 1116-1147 ◽  
Author(s):  
David Assaf ◽  
Larry Goldstein ◽  
Ester Samuel-Cahn
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Large Class ◽  
Optimal Stopping ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Indicator Function ◽  
Stopping Rules ◽  
A Value ◽  
Independent Identically Distributed

Let Xn,…,X1 be independent, identically distributed (i.i.d.) random variables with distribution function F. A statistician, knowing F, observes the X values sequentially and is given two chances to choose Xs using stopping rules. The statistician's goal is to stop at a value of X as small as possible. Let equal the expectation of the smaller of the two values chosen by the statistician when proceeding optimally. We obtain the asymptotic behaviour of the sequence for a large class of Fs belonging to the domain of attraction (for the minimum) 𝒟(Gα), where Gα(x) = [1 - exp(-xα)]1(x ≥ 0) (with 1(·) the indicator function). The results are compared with those for the asymptotic behaviour of the classical one-choice value sequence , as well as with the ‘prophet value’ sequence

scholarly journals Two-choice optimal stopping

2004 ◽  
Vol 36 (04) ◽  
pp. 1116-1147 ◽  
Author(s):  
David Assaf ◽  
Larry Goldstein ◽  
Ester Samuel-Cahn
Keyword(s):  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Large Class ◽  
Optimal Stopping ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Indicator Function ◽  
Stopping Rules ◽  
A Value ◽  
Independent Identically Distributed

Let X n ,…,X 1 be independent, identically distributed (i.i.d.) random variables with distribution function F. A statistician, knowing F, observes the X values sequentially and is given two chances to choose Xs using stopping rules. The statistician's goal is to stop at a value of X as small as possible. Let equal the expectation of the smaller of the two values chosen by the statistician when proceeding optimally. We obtain the asymptotic behaviour of the sequence for a large class of Fs belonging to the domain of attraction (for the minimum) 𝒟(G α), where G α(x) = [1 - exp(-x α)]1(x ≥ 0) (with 1(·) the indicator function). The results are compared with those for the asymptotic behaviour of the classical one-choice value sequence , as well as with the ‘prophet value’ sequence

scholarly journals A Double-indexed Functional Hill Process and Applications

2016 ◽  
Vol 8 (4) ◽  
pp. 144
Author(s):  
Modou Ngom ◽  
Gane Samb Lo
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Stochastic Processes ◽  
Order Statistics ◽  
Finite Dimension ◽  
Domain Of Attraction ◽  
Extreme Value ◽  
Extreme Value Index

<div>Let $X_{1,n} \leq .... \leq X_{n,n}$ be the order statistics associated with a sample $X_{1}, ...., X_{n}$ whose pertaining distribution function (\textit{df}) is $F$. We are concerned with the functional asymptotic behaviour of the sequence of stochastic processes</div><div> </div><div>\begin{equation}<br />T_{n}(f,s)=\sum_{j=1}^{j=k}f(j)\left( \log X_{n-j+1,n}-\log<br />X_{n-j,n}\right)^{s} ,  \label{fme}<br />\end{equation}</div><div> </div><div>indexed by some classes $\mathcal{F}$ of functions $f:\mathbb{N}%^{\ast}\longmapsto \mathbb{R}_{+}$ and $s \in ]0,+\infty[$ and where $k=k(n)$ satisfies</div><div> </div><div>\begin{equation*}<br />1\leq k\leq n,k/n\rightarrow 0\text{ as }n\rightarrow \infty .<br />\end{equation*}</div><div> </div><div>We show that this is a stochastic process whose margins generate estimators of the extreme value index when $F$ is in the extreme domain of attraction. We focus in this paper on its finite-dimension asymptotic law and provide a class of new estimators of the extreme value index whose performances are compared to analogous ones. The results are next particularized for one explicit class $\mathcal{F}$.</div>

A clustering law for some discrete order statistics

2003 ◽  
Vol 40 (01) ◽  
pp. 226-241 ◽  
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.

The convex hull of a normal sample

1994 ◽  
Vol 26 (04) ◽  
pp. 855-875 ◽  
Author(s):  
Irene Hueter
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Convex Hull ◽  
Asymptotic Behaviour ◽  
Central Limit ◽  
Random Variables ◽  
Poisson Approximation ◽  
Normal Sample ◽  
Central Importance ◽  
Independent Identically Distributed

Consider the convex hull of n independent, identically distributed points in the plane. Functionals of interest are the number of vertices Nn , the perimeter Ln and the area An of the convex hull. We study the asymptotic behaviour of these three quantities when the points are standard normally distributed. In particular, we derive the variances of Nn, Ln and An for large n and prove a central limit theorem for each of these random variables. We enlarge on a method developed by Groeneboom (1988) for uniformly distributed points supported on a bounded planar region. The process of vertices of the convex hull is of central importance. Poisson approximation and martingale techniques are used.

The fair charge on a car ferry

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.

The fair charge on a car ferry

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.

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

1970 ◽  
Vol 7 (02) ◽  
pp. 432-439 ◽  
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].

A martingale approach to strong convergence of the number of records

2001 ◽  
Vol 33 (4) ◽  
pp. 864-873 ◽  
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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