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.

Optimal Sequential selection of n random variables under a constraint

1984 ◽  
Vol 21 (3) ◽  
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.

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.

Sequential selection of random vectors under a sum constraint

2004 ◽  
Vol 41 (1) ◽  
pp. 131-146
Author(s):  
Mario Stanke
Keyword(s):  
Continuous Distribution ◽  
Center Of Gravity ◽  
Continuous Measure ◽  
Random Vectors ◽  
Expected Number ◽  
Absolutely Continuous ◽  
Selection Policy ◽  
Sequential Selection ◽  
Absolutely Continuous Measure ◽  
Selection Of

We observe a sequence X1, X2,…, Xn of independent and identically distributed coordinatewise nonnegative d-dimensional random vectors. When a vector is observed it can either be selected or rejected but once made this decision is final. In each coordinate the sum of the selected vectors must not exceed a given constant. The problem is to find a selection policy that maximizes the expected number of selected vectors. For a general absolutely continuous distribution of the Xi we determine the maximal expected number of selected vectors asymptotically and give a selection policy which asymptotically achieves optimality. This problem raises a question closely related to the following problem. Given an absolutely continuous measure μ on Q = [0,1]d and a τ ∈ Q, find a set A of maximal measure μ(A) among all A ⊂ Q whose center of gravity lies below τ in all coordinates. We will show that a simplicial section {x ∈ Q | 〈x, θ〉 ≤ 1}, where θ ∈ ℝd, θ ≥ 0, satisfies a certain additional property, is a solution to this problem.

Sequential selection of random vectors under a sum constraint

2004 ◽  
Vol 41 (01) ◽  
pp. 131-146
Author(s):  
Mario Stanke
Keyword(s):  
Continuous Distribution ◽  
Center Of Gravity ◽  
Continuous Measure ◽  
Random Vectors ◽  
Expected Number ◽  
Absolutely Continuous ◽  
Selection Policy ◽  
Sequential Selection ◽  
Absolutely Continuous Measure ◽  
Selection Of

We observe a sequence X 1, X 2,…, X n of independent and identically distributed coordinatewise nonnegative d-dimensional random vectors. When a vector is observed it can either be selected or rejected but once made this decision is final. In each coordinate the sum of the selected vectors must not exceed a given constant. The problem is to find a selection policy that maximizes the expected number of selected vectors. For a general absolutely continuous distribution of the X i we determine the maximal expected number of selected vectors asymptotically and give a selection policy which asymptotically achieves optimality. This problem raises a question closely related to the following problem. Given an absolutely continuous measure μ on Q = [0,1] d and a τ ∈ Q, find a set A of maximal measure μ(A) among all A ⊂ Q whose center of gravity lies below τ in all coordinates. We will show that a simplicial section { x ∈ Q | 〈 x , θ 〉 ≤ 1}, where θ ∈ ℝ d , θ ≥ 0, satisfies a certain additional property, is a solution to this problem.

Sequential selection of an increasing subsequence from a sample of random size

1999 ◽  
Vol 36 (04) ◽  
pp. 1074-1085 ◽  
Author(s):  
Alexander V. Gnedin
Keyword(s):  
Random Number ◽  
Random Variables ◽  
Random Size ◽  
Sequential Selection ◽  
Expected Length ◽  
Decision Policies ◽  
Independent Identically Distributed ◽  
Selection Of

A random number of independent identically distributed random variables is inspected in strict succession. As a variable is inspected, it can either be selected or rejected and this decision becomes final at once. The selected sequence must increase. The problem is to maximize the expected length of the selected sequence. We demonstrate decision policies which approach optimality when the number of observations becomes in a sense large and show that the maximum expected length is close to an easily computable value.

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.

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

1991 ◽  
Vol 28 (04) ◽  
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 (X 1, · ··, 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.

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.

Sequential selection of an increasing subsequence from a sample of random size

1999 ◽  
Vol 36 (4) ◽  
pp. 1074-1085 ◽  
Author(s):  
Alexander V. Gnedin
Keyword(s):  
Random Number ◽  
Random Variables ◽  
Random Size ◽  
Sequential Selection ◽  
Expected Length ◽  
Decision Policies ◽  
Independent Identically Distributed ◽  
Selection Of

A random number of independent identically distributed random variables is inspected in strict succession. As a variable is inspected, it can either be selected or rejected and this decision becomes final at once. The selected sequence must increase. The problem is to maximize the expected length of the selected sequence.We demonstrate decision policies which approach optimality when the number of observations becomes in a sense large and show that the maximum expected length is close to an easily computable value.

scholarly journals Online Selection of Alternating Subsequences from a Random Sample

2011 ◽  
Vol 48 (4) ◽  
pp. 1114-1132 ◽  
Author(s):  
Alessandro Arlotto ◽  
Robert W. Chen ◽  
Lawrence A. Shepp ◽  
J. Michael Steele
Keyword(s):  
Asymptotic Behavior ◽  
Random Sample ◽  
Random Sequence ◽  
Random Variables ◽  
Sequential Selection ◽  
Exact Asymptotic Behavior ◽  
Full Knowledge ◽  
Independent Identically Distributed ◽  
Selection Of

We consider sequential selection of an alternating subsequence from a sequence of independent, identically distributed, continuous random variables, and we determine the exact asymptotic behavior of an optimal sequentially selected subsequence. Moreover, we find (in a sense we make precise) that a person who is constrained to make sequential selections does only about 12 percent worse than a person who can make selections with full knowledge of the random sequence.

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