Optimal selection of stochastic intervals under a sum constraint

1987 ◽  
Vol 19 (2) ◽  
pp. 454-473 ◽  
Author(s):  
E. G. Coffman ◽  
L. Flatto ◽  
R. R. Weber
Keyword(s):  
Decision Rule ◽  
Selection Process ◽  
Random Variables ◽  
Optimal Threshold ◽  
Optimal Selection ◽  
Expected Number ◽  
Common Distribution ◽  
The Common ◽  
Image Position ◽  
Selection Of

We model a selection process arising in certain storage problems. A sequence (X1, · ··, Xn) of non-negative, independent and identically distributed random variables is given. F(x) denotes the common distribution of the Xi′s. With F(x) given we seek a decision rule for selecting a maximum number of the Xi′s subject to the following constraints: (1) the sum of the elements selected must not exceed a given constant c > 0, and (2) the Xi′s must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected.We prove first that there exists such a rule of threshold type, i.e. the ith element inspected is accepted if and only if it is no larger than a threshold which depends only on i and the sum of the elements already accepted. Next, we prove that if F(x) ~ Axα as x → 0 for some A, α> 0, then for fixed c the expected number, En(c), selected by an optimal threshold is characterized by Asymptotics as c → ∞and n → ∞with c/n held fixed are derived, and connections with several closely related, well-known problems are brought out and discussed.

Optimal selection of stochastic intervals under a sum constraint

1987 ◽  
Vol 19 (02) ◽  
pp. 454-473 ◽  
Author(s):  
E. G. Coffman ◽  
L. Flatto ◽  
R. R. Weber
Keyword(s):  
Decision Rule ◽  
Selection Process ◽  
Random Variables ◽  
Optimal Threshold ◽  
Optimal Selection ◽  
Expected Number ◽  
Common Distribution ◽  
The Common ◽  
Image Position ◽  
Selection Of

We model a selection process arising in certain storage problems. A sequence (X 1, · ··, Xn ) of non-negative, independent and identically distributed random variables is given. F(x) denotes the common distribution of the Xi′s. With F(x) given we seek a decision rule for selecting a maximum number of the Xi′s subject to the following constraints: (1) the sum of the elements selected must not exceed a given constant c > 0, and (2) the Xi′s must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected. We prove first that there exists such a rule of threshold type, i.e. the ith element inspected is accepted if and only if it is no larger than a threshold which depends only on i and the sum of the elements already accepted. Next, we prove that if F(x) ~ Axα as x → 0 for some A, α> 0, then for fixed c the expected number, En (c), selected by an optimal threshold is characterized by Asymptotics as c → ∞and n → ∞with c/n held fixed are derived, and connections with several closely related, well-known problems are brought out and discussed.

An extension of a lemma of Wald

1966 ◽  
Vol 3 (01) ◽  
pp. 272-273 ◽  
Author(s):  
H. Robbins ◽  
E. Samuel
Keyword(s):  
Natural Extension ◽  
Random Variables ◽  
Random Variable ◽  
Common Distribution ◽  
The Common ◽  
Image Position ◽  
Independent Identically Distributed

We define a natural extension of the concept of expectation of a random variable y as follows: M(y) = a if there exists a constant − ∞ ≦ a ≦ ∞ such that if y 1, y 2, … is a sequence of independent identically distributed (i.i.d.) random variables with the common distribution of y then

An extension of a lemma of Wald

1966 ◽  
Vol 3 (1) ◽  
pp. 272-273 ◽  
Author(s):  
H. Robbins ◽  
E. Samuel
Keyword(s):  
Natural Extension ◽  
Random Variables ◽  
Random Variable ◽  
Common Distribution ◽  
The Common ◽  
Image Position ◽  
Independent Identically Distributed

We define a natural extension of the concept of expectation of a random variable y as follows: M(y) = a if there exists a constant − ∞ ≦ a ≦ ∞ such that if y1, y2, … is a sequence of independent identically distributed (i.i.d.) random variables with the common distribution of y then

A Note on Normal Attraction to a Stable Law

1971 ◽  
Vol 14 (3) ◽  
pp. 451-452
Author(s):  
M. V. Menon ◽  
V. Seshadri
Keyword(s):  
Distribution Function ◽  
Sufficient Conditions ◽  
Characteristic Exponent ◽  
Random Variables ◽  
Stable Law ◽  
Common Distribution ◽  
Normal Attraction ◽  
Common Distribution Function ◽  
The Common ◽  
Necessary And Sufficient

Let X1, X2, …, be a sequence of independent and identically distributed random variables, with the common distribution function F(x). The sequence is said to be normally attracted to a stable law V with characteristic exponent α, if for some an (converges in distribution to V). Necessary and sufficient conditions for normal attraction are known (cf [1, p. 181]).

Optimal Sequential selection of n random variables under a constraint

1984 ◽  
Vol 21 (03) ◽  
pp. 537-547 ◽  
Author(s):  
R. W. Chen ◽  
V. N. Nair ◽  
A. M. Odlyzko ◽  
L. A. Shepp ◽  
Y. Vardi
Keyword(s):  
Random Variables ◽  
Selection Procedure ◽  
Expected Number ◽  
Sequential Selection ◽  
Selection Of

We observe a sequence {Xk } k≧1 of i.i.d. non-negative random variables one at a time. After each observation, we select or reject the observed variable. A variable that is rejected may not be recalled. We want to select N variables as soon as possible subject to the constraint that the sum of the N selected variables does not exceed some prescribed value C > 0. In this paper, we develop a sequential selection procedure that minimizes the expected number of observed variables, and we study some of its properties. We also consider the situation where N → ∞and C/N → α > 0. Some applications are briefly discussed.

Asymptotic distributions of extremes of extremal Markov sequences

1994 ◽  
Vol 31 (01) ◽  
pp. 256-261
Author(s):  
S. R. Adke ◽  
C. Chandran
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Asymptotic Distributions ◽  
Arbitrary Distribution ◽  
Extreme Statistics ◽  
Common Distribution ◽  
Common Distribution Function ◽  
The Common

Let {ξ n , n ≧1} be a sequence of independent real random variables, F denote the common distribution function of identically distributed random variables ξ n , n ≧1 and let ξ 1 have an arbitrary distribution. Define Xn+ 1 = k max(Xn, ξ n +1), Yn + 1 = max(Yn, ξ n +1) – c, Un +1 = l min(Un, ξ n +1), Vn+ 1 = min(Vn, ξ n +1) + c, n ≧ 1, 0 < k < 1, l > 1, 0 < c < ∞, and X 1 = Υ 1 = U 1 = V 1 = ξ 1. We establish conditions under which the limit law of max(X 1, · ··, Xn ) coincides with that of max(ξ 2, · ··, ξ n+ 1) when both are appropriately normed. A similar exercise is carried out for the extreme statistics max(Y 1, · ··, Yn ), min(U 1,· ··, Un ) and min(V 1, · ··, Vn ).

A note on the selection of random variables under a sum constraint

1991 ◽  
Vol 28 (4) ◽  
pp. 919-923 ◽  
Author(s):  
Wansoo Rhee ◽  
Michel Talagrand
Keyword(s):  
Decision Rules ◽  
Stopping Time ◽  
Random Variables ◽  
Expected Number ◽  
Time Argument ◽  
Selection Of

Consider an i.i.d. sequence of non-negative random variables (X1, · ··, Xn) with known distribution F. Consider decision rules for selecting a maximum number of the subject to the following constraints: (1) the sum of the elements selected must not exceed a given constant c > 0, and (2) the must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected. Coffman et al. (1987) proved that there exists such a rule that maximizes the expected number En(c) of variables selected, and determined the asymptotics of En(c) for special distributions. Here we determine the asymptotics of En(cn) for very general choices of sequences (cn) and of F, by showing that En(c) is very close to an easily computable number. Our proofs are (somewhat deceptively) very simple, and rely on an appropriate stopping-time argument.

scholarly journals A coincidence problem in telephone traffic with non-recurrent arrival process

1963 ◽  
Vol 3 (2) ◽  
pp. 237-240 ◽  
Author(s):  
P. D. Finch
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Holding Time ◽  
Arrival Process ◽  
Coincidence Problem ◽  
Transient Behaviour ◽  
Telephone Exchange ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Image Position

We consider the following problem. Calls arrive at a telephone exchange at the instants t0, t1, … tm …. The telephone exchange contains a denumerable infinity of channels. The holding times of calls are non-negative random variables distributed independently of the times at which calls arrive, independently of which channel a call engages and independently of each other with a common distribution function B(x). Takacs [3Τm = tm+1 − tm, m ≧ are identically and independently distributed non-negative random variables with common distribution function, A (x). Finch [1] has studied the transient behaviour in the case of a recurrent arrival process and exponential holding time, that is when the common distribution of holding time is given by In this paper we make no assumption about the arrival process {tm}. The underlying principle of this paper is the same as that of Finch [2]. We consider the instants of arrival t0, t1,…, tm,… as given and determine various probabilities of interest conditionally as functions of the inter-arrival intervals Τ1,…, Τm,… When the arrival process is a stochastic process we can then determine the relevant unconditional probabilities by integration.

A Method for Selecting Measurement Points on Planar Faces of Objects With Holes

1992 ◽  
Author(s):  
Hayri O. Vardar ◽  
Tulga M. Ozsoy
Keyword(s):  
Finite Element ◽  
Mesh Generation ◽  
Selection Process ◽  
Optimal Selection ◽  
Element Mesh ◽  
Computation Algorithm ◽  
Convex Hull Computation ◽  
Set Of Points ◽  
Special Case ◽  
Selection Of

Abstract In this paper, a method facilitating the automatic selection of measurement points on planar faces of objects with holes will be presented. The objective is to further automate the process of generation of inspection programs for coordinate measuring machines. The method developed aims for the creation of candidate points on a planar face of an object and the optimal selection of a required number of measurement points among the candidate points. A triangulation technique based on a finite element mesh generation method has been implemented in creating the candidate points. Except for the special case of four points, a random selection process is utilized for choosing among the candidate points. The effects of randomness are compensated by applying a convex hull computation algorithm to find the set of points among other candidate sets that covers the largest area. The method has been successfully incorporated into a software demonstrating the feasibility of the approach.

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