The secretary problem: minimizing the expected rank with I.I.D. random variables

n candidates, represented by n i.i.d. continuous random variables X 1, …, Xn with known distribution arrive sequentially, and one of them must be chosen, using a non-anticipating stopping rule. The objective is to minimize the expected rank (among the ranks of X 1, …, Xn ) of the candidate chosen, where the best candidate, i.e. the one with smallest X-value, has rank one, etc. Let the value of the optimal rule be Vn , and lim Vn = V. We prove that V > 1.85. Limiting consideration to the class of threshold rules of the form tn = min {k: Xk ≦ ak for some constants ak , let Wn be the value of the expected rank for the optimal threshold rule, and lim Wn = W. We show 2.295 < W < 2.327.

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Odds Theorem with Multiple Selection Chances

Journal of Applied Probability ◽

10.1017/s0021900200007397 ◽

2010 ◽

Vol 47 (04) ◽

pp. 1093-1104 ◽

Cited By ~ 1

Author(s):

Katsunori Ano ◽

Hideo Kakinuma ◽

Naoto Miyoshi

Keyword(s):

Finite Number ◽

Closed Form ◽

Selection Rule ◽

Random Variables ◽

Secretary Problem ◽

Optimal Selection ◽

Maximum Probability ◽

Optimal Rule ◽

Interesting Implication

We study the multi-selection version of the so-called odds theorem by Bruss (2000). We observe a finite number of independent 0/1 (failure/success) random variables sequentially and want to select the last success. We derive the optimal selection rule when m (≥ 1) selection chances are given and find that the optimal rule has the form of a combination of multiple odds-sums. We provide a formula for computing the maximum probability of selecting the last success when we have m selection chances and also provide closed-form formulae for m = 2 and 3. For m = 2, we further give the bounds for the maximum probability of selecting the last success and derive its limit as the number of observations goes to ∞. An interesting implication of our result is that the limit of the maximum probability of selecting the last success for m = 2 is consistent with the corresponding limit for the classical secretary problem with two selection chances.

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Maximizing the probability of stopping on any of the last m successes in independent Bernoulli trials with random horizon

Advances in Applied Probability ◽

10.1017/s0001867800005139 ◽

2011 ◽

Vol 43 (03) ◽

pp. 760-781 ◽

Cited By ~ 3

Author(s):

Mitsushi Tamaki

Keyword(s):

Process Model ◽

Arrival Time ◽

Random Number ◽

Secretary Problem ◽

Main Concern ◽

Bernoulli Trials ◽

Asymptotic Results ◽

Optimal Rule ◽

Random Horizon ◽

Threshold Rule

We consider the problem of maximizing the probability of stopping on any of the last m successes in independent Bernoulli trials with random horizon of length N, where m is a predetermined integer. A prior is given for N. It is known that, when N is degenerate, i.e. P{N = n} = 1 for a given n > m, the sum-the-multiplicative-odds theorem gives the solution and shows that the optimal rule is a threshold rule, i.e. it stops on the first success appearing after a given stage. However, when N is nondegenerate, the optimal rule is not necessarily a threshold rule. So our main concern in Section 2 is to give a sufficient condition for the optimal rule to be a threshold rule when N is a bounded random variable such that P{N ≤ n} = 1. Application will be made to the usual (discrete arrival time) secretary problem with a random number N of applicants in Section 3. When N is uniform or curtailed geometric, the optimal rules are shown to be threshold rules and their asymptotic results are obtained. We also examine, as a nonhomogeneous Poisson process model, an intermediate prior that allows N to be uniform or degenerate. In Section 4 we consider a continuous arrival time version of the secretary problem with a random number M of applicants. It is shown that, whatever the distribution of M, we can win with probability greater than or equal to u m *, where u m * is, as given in (1.4), the asymptotic win probability of the usual secretary problem when N degenerates to n and n → ∞.

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Maximizing the probability of stopping on any of the last m successes in independent Bernoulli trials with random horizon

Advances in Applied Probability ◽

10.1239/aap/1316792669 ◽

2011 ◽

Vol 43 (3) ◽

pp. 760-781 ◽

Cited By ~ 1

Author(s):

Mitsushi Tamaki

Keyword(s):

Process Model ◽

Arrival Time ◽

Random Number ◽

Secretary Problem ◽

Main Concern ◽

Bernoulli Trials ◽

Asymptotic Results ◽

Optimal Rule ◽

Random Horizon ◽

Threshold Rule

We consider the problem of maximizing the probability of stopping on any of the last m successes in independent Bernoulli trials with random horizon of length N, where m is a predetermined integer. A prior is given for N. It is known that, when N is degenerate, i.e. P{N = n} = 1 for a given n > m, the sum-the-multiplicative-odds theorem gives the solution and shows that the optimal rule is a threshold rule, i.e. it stops on the first success appearing after a given stage. However, when N is nondegenerate, the optimal rule is not necessarily a threshold rule. So our main concern in Section 2 is to give a sufficient condition for the optimal rule to be a threshold rule when N is a bounded random variable such that P{N ≤ n} = 1. Application will be made to the usual (discrete arrival time) secretary problem with a random number N of applicants in Section 3. When N is uniform or curtailed geometric, the optimal rules are shown to be threshold rules and their asymptotic results are obtained. We also examine, as a nonhomogeneous Poisson process model, an intermediate prior that allows N to be uniform or degenerate. In Section 4 we consider a continuous arrival time version of the secretary problem with a random number M of applicants. It is shown that, whatever the distribution of M, we can win with probability greater than or equal to um*, where um* is, as given in (1.4), the asymptotic win probability of the usual secretary problem when N degenerates to n and n → ∞.

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Odds Theorem with Multiple Selection Chances

Journal of Applied Probability ◽

10.1239/jap/1294170522 ◽

2010 ◽

Vol 47 (4) ◽

pp. 1093-1104 ◽

Cited By ~ 4

Author(s):

Katsunori Ano ◽

Hideo Kakinuma ◽

Naoto Miyoshi

Keyword(s):

Finite Number ◽

Closed Form ◽

Selection Rule ◽

Random Variables ◽

Secretary Problem ◽

Optimal Selection ◽

Maximum Probability ◽

Optimal Rule ◽

Interesting Implication

We study the multi-selection version of the so-called odds theorem by Bruss (2000). We observe a finite number of independent 0/1 (failure/success) random variables sequentially and want to select the last success. We derive the optimal selection rule when m (≥ 1) selection chances are given and find that the optimal rule has the form of a combination of multiple odds-sums. We provide a formula for computing the maximum probability of selecting the last success when we have m selection chances and also provide closed-form formulae for m = 2 and 3. For m = 2, we further give the bounds for the maximum probability of selecting the last success and derive its limit as the number of observations goes to ∞. An interesting implication of our result is that the limit of the maximum probability of selecting the last success for m = 2 is consistent with the corresponding limit for the classical secretary problem with two selection chances.

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Prophet Inequalities for Independent and Identically Distributed Random Variables from an Unknown Distribution

Mathematics of Operations Research ◽

10.1287/moor.2021.1167 ◽

2021 ◽

Author(s):

José Correa ◽

Paul Dütting ◽

Felix Fischer ◽

Kevin Schewior

Keyword(s):

Stopping Time ◽

Random Variables ◽

Optimal Solution ◽

Random Order ◽

Secretary Problem ◽

Maximum Value ◽

Optimal Stopping Theory ◽

Long Time ◽

Object Of Study ◽

Independent And Identically Distributed

A central object of study in optimal stopping theory is the single-choice prophet inequality for independent and identically distributed random variables: given a sequence of random variables [Formula: see text] drawn independently from the same distribution, the goal is to choose a stopping time τ such that for the maximum value of α and for all distributions, [Formula: see text]. What makes this problem challenging is that the decision whether [Formula: see text] may only depend on the values of the random variables [Formula: see text] and on the distribution F. For a long time, the best known bound for the problem had been [Formula: see text], but recently a tight bound of [Formula: see text] was obtained. The case where F is unknown, such that the decision whether [Formula: see text] may depend only on the values of the random variables [Formula: see text], is equally well motivated but has received much less attention. A straightforward guarantee for this case of [Formula: see text] can be derived from the well-known optimal solution to the secretary problem, where an arbitrary set of values arrive in random order and the goal is to maximize the probability of selecting the largest value. We show that this bound is in fact tight. We then investigate the case where the stopping time may additionally depend on a limited number of samples from F, and we show that, even with o(n) samples, [Formula: see text]. On the other hand, n samples allow for a significant improvement, whereas [Formula: see text] samples are equivalent to knowledge of the distribution: specifically, with n samples, [Formula: see text] and [Formula: see text], and with [Formula: see text] samples, [Formula: see text] for any [Formula: see text].

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Optimal selection of stochastic intervals under a sum constraint

Advances in Applied Probability ◽

10.2307/1427427 ◽

1987 ◽

Vol 19 (2) ◽

pp. 454-473 ◽

Cited By ~ 16

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.

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Probability maximizing approach to a secretary problem by random change-point of the distribution law of the observed process

Journal of Applied Probability ◽

10.2307/3213668 ◽

1984 ◽

Vol 21 (1) ◽

pp. 98-107 ◽

Cited By ~ 1

Author(s):

Minoru Yoshida

Keyword(s):

Distribution Function ◽

Stopping Time ◽

Random Variables ◽

Secretary Problem ◽

Independent Random Variables ◽

Distribution Law ◽

Common Distribution ◽

Common Distribution Function ◽

Random Moment ◽

The Moment

Before some random moment θ, independent identically distributed random variables x1, · ··, xθ–1 with common distribution function μ (dx) appear consecutively. After the moment θ, independent random variables xθ, xθ+1, · ·· have another common distribution function f (x)μ (dx). Our information about θ can be constructed only by successively observed values of the x's.In this paper we find an optimal stopping policy by which we can maximize the probability that the quantity associated with the stopping time is the largest of all θ + m – 1 quantities for a given integer m.

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Dependent Random Variables with Independent Subsets - II

Canadian Mathematical Bulletin ◽

10.4153/cmb-1990-004-6 ◽

1990 ◽

Vol 33 (1) ◽

pp. 24-28 ◽

Cited By ~ 6

Author(s):

Y. H. Wang

Keyword(s):

Joint Distribution ◽

Random Variables ◽

Marginal Distributions ◽

Dependent Random Variables ◽

One Dimensional ◽

The One

AbstractIn this paper, we consolidate into one two separate problems - dependent random variables with independent subsets and construction of a joint distribution with given marginals. Let N = {1,2,3,...} and X = {Xn; n ∊ N} be a sequence of random variables with nondegenerate one-dimensional marginal distributions {Fn; n ∊ N}. An example is constructed to show that there exists a sequence of random variables Y = {Yn; n ∊ N} such that the components of a subset of Y are independent if and only if its size is ≦ k, where k ≧ 2 is a prefixed integer. Furthermore, the one-dimensional marginal distributions of Y are those of X.

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On the Sequential Detection Problem

Calcutta Statistical Association Bulletin ◽

10.1177/0008068319890301 ◽

1989 ◽

Vol 38 (3-4) ◽

pp. 129-146

Author(s):

Uttam Bandyopadhyay

Keyword(s):

Infinite Sequence ◽

Random Variables ◽

Stopping Rule ◽

Independent Random Variables ◽

Sequential Detection ◽

Detection Problem ◽

Asymptotic Results ◽

Rule Based ◽

Unknown Point ◽

Cumulative Sums

In this paper, for an infinite sequence of independent random variables, we have considered the problem of estimation of an unknown point ( q) where a change in the distribution of the random variables occurs. Attaching suitable scores for the observed values. of the random variables, a stopping rule based on the cumulative sums of these scores has been proposed. Some asymptotic results useful for studying the performance of the proposed procedure havo beon obtained.

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