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.

scholarly journals Online Selection of Alternating Subsequences from a Random Sample

2011 ◽  
Vol 48 (04) ◽  
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.

scholarly journals Optimal Sequential Selection of a Unimodal Subsequence of a Random Sequence

2011 ◽  
Vol 20 (6) ◽  
pp. 799-814 ◽  
Author(s):  
ALESSANDRO ARLOTTO ◽  
J. MICHAEL STEELE
Keyword(s):  
Random Sequence ◽  
Random Variables ◽  
Square Root ◽  
Sequential Selection ◽  
Independent Identically Distributed ◽  
Selection Of

We consider the problem of selecting sequentially a unimodal subsequence from a sequence of independent identically distributed random variables, and we find that a person doing optimal sequential selection does so within a factor of the square root of two as well as a prophet who knows all of the random observations in advance of any selections. Our analysis applies in fact to selections of subsequences that have d+1 monotone blocks, and, by including the case d=0, our analysis also covers monotone subsequences.

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.

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.

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.

scholarly journals Optimal Sequential Selection of a Monotone Sequence From a Random Sample

1981 ◽  
Vol 9 (6) ◽  
pp. 937-947 ◽  
Author(s):  
Stephen M. Samuels ◽  
J. Michael Steele
Keyword(s):  
Random Sample ◽  
Monotone Sequence ◽  
Sequential Selection ◽  
Selection Of

On the maximum and its uniqueness for geometric random samples

1990 ◽  
Vol 27 (3) ◽  
pp. 598-610 ◽  
Author(s):  
F. Thomas Bruss ◽  
Colm Art O'cinneide
Keyword(s):  
Asymptotic Behavior ◽  
Random Variables ◽  
Expected Value ◽  
Asymptotic Result ◽  
Random Samples ◽  
Original Motivation ◽  
Independent Identically Distributed

Given n independent, identically distributed random variables, let ρ n denote the probability that the maximum is unique. This probability is clearly unity if the distribution of the random variables is continuous. We explore the asymptotic behavior of the ρ n's in the case of geometric random variables. We find a function Φsuch that (ρ n – Φ(n)) → 0 as n →∞. In particular, we show that ρ n does not converge as n →∞. We derive a related asymptotic result for the expected value of the maximum of the sample. These results arose out of a random depletion model due to Bajaj, which was the original motivation for this paper and which is included.

On the maximum and its uniqueness for geometric random samples

1990 ◽  
Vol 27 (03) ◽  
pp. 598-610 ◽  
Author(s):  
F. Thomas Bruss ◽  
Colm Art O'cinneide
Keyword(s):  
Asymptotic Behavior ◽  
Random Variables ◽  
Expected Value ◽  
Asymptotic Result ◽  
Random Samples ◽  
Original Motivation ◽  
Independent Identically Distributed

Given n independent, identically distributed random variables, let ρ n denote the probability that the maximum is unique. This probability is clearly unity if the distribution of the random variables is continuous. We explore the asymptotic behavior of the ρ n 's in the case of geometric random variables. We find a function Φsuch that (ρ n – Φ(n)) → 0 as n →∞. In particular, we show that ρ n does not converge as n →∞. We derive a related asymptotic result for the expected value of the maximum of the sample. These results arose out of a random depletion model due to Bajaj, which was the original motivation for this paper and which is included.

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.

Sign in / Sign up

Export Citation Format

Share Document