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 (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 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 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.

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.

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

scholarly journals Almost conservation laws for stochastic nonlinear Schrödinger equations

2021 ◽  
Author(s):  
Kelvin Cheung ◽  
Guopeng Li ◽  
Tadahiro Oh
Keyword(s):  
Conservation Laws ◽  
Stopping Time ◽  
Energy Space ◽  
Stochastic Forcing ◽  
Time Dynamics ◽  
Nonlinear Schrödinger ◽  
Time Argument ◽  
Cubic Nonlinear Schrödinger Equation ◽  
Stochastic Setting ◽  
Ito's Lemma

AbstractIn this paper, we present a globalization argument for stochastic nonlinear dispersive PDEs with additive noises by adapting the I-method (= the method of almost conservation laws) to the stochastic setting. As a model example, we consider the defocusing stochastic cubic nonlinear Schrödinger equation (SNLS) on $${\mathbb {R}}^3$$ R 3 with additive stochastic forcing, white in time and correlated in space, such that the noise lies below the energy space. By combining the I-method with Ito’s lemma and a stopping time argument, we construct global-in-time dynamics for SNLS below the energy space.

General Schema Theory for Genetic Programming with Subtree-Swapping Crossover: Part II

2003 ◽  
Vol 11 (2) ◽  
pp. 169-206 ◽  
Author(s):  
Riccardo Poli ◽  
Nicholas Freitag McPhee
Keyword(s):  
Genetic Programming ◽  
Schema Theory ◽  
Exact Formulation ◽  
Expected Number ◽  
Size Evolution ◽  
General Schema ◽  
Definition Of ◽  
Effective Fitness ◽  
Selection Of ◽  
Exact Definition

This paper is the second part of a two-part paper which introduces a general schema theory for genetic programming (GP) with subtree-swapping crossover (Part I (Poli and McPhee, 2003)). Like other recent GP schema theory results, the theory gives an exact formulation (rather than a lower bound) for the expected number of instances of a schema at the next generation. The theory is based on a Cartesian node reference system, introduced in Part I, and on the notion of a variable-arity hyperschema, introduced here, which generalises previous definitions of a schema. The theory includes two main theorems describing the propagation of GP schemata: a microscopic and a macroscopic schema theorem. The microscopic version is applicable to crossover operators which replace a subtree in one parent with a subtree from the other parent to produce the offspring. Therefore, this theorem is applicable to Koza's GP crossover with and without uniform selection of the crossover points, as well as one-point crossover, size-fair crossover, strongly-typed GP crossover, context-preserving crossover and many others. The macroscopic version is applicable to crossover operators in which the probability of selecting any two crossover points in the parents depends only on the parents' size and shape. In the paper we provide examples, we show how the theory can be specialised to specific crossover operators and we illustrate how it can be used to derive other general results. These include an exact definition of effective fitness and a size-evolution equation for GP with subtree-swapping crossover.

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

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].

scholarly journals Real Zeros of a Class of Hyperbolic Polynomials with Random Coefficients

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Mina Ketan Mahanti ◽  
Amandeep Singh ◽  
Lokanath Sahoo
Keyword(s):  
Random Variables ◽  
Random Coefficients ◽  
Expected Number ◽  
Hyperbolic Polynomial ◽  
Gaussian Random Variables ◽  
Hyperbolic Polynomials ◽  
Asymptotic Value ◽  
Real Zeros ◽  
Number Of Real Zeros

We have proved here that the expected number of real zeros of a random hyperbolic polynomial of the formy=Pnt=n1a1cosh⁡t+n2a2cosh⁡2t+⋯+nnancosh⁡nt, wherea1,…,anis a sequence of standard Gaussian random variables, isn/2+op(1). It is shown that the asymptotic value of expected number of times the polynomial crosses the levely=Kis alson/2as long asKdoes not exceed2neμ(n), whereμ(n)=o(n). The number of oscillations ofPn(t)abouty=Kwill be less thann/2asymptotically only ifK=2neμ(n), whereμ(n)=O(n)orn-1μ(n)→∞. In the former case the number of oscillations continues to be a fraction ofnand decreases with the increase in value ofμ(n). In the latter case, the number of oscillations reduces toop(n)and almost no trace of the curve is expected to be present above the levely=Kifμ(n)/(nlogn)→∞.

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