Stochastic ordering of classical discrete distributions

We provide a lower bound on the probability that a binomial random variable is exceeding its mean. Our proof employs estimates on the mean absolute deviation and the tail conditional expectation of binomial random variables.Keywords: lower bounds; binomial tails; tail conditional expectation; mean absolute deviation; hazard rate order 1 Prologue, related work and main result Given a positive integer n and a real number p ∈ (0, 1), we denote by Bin(n, p) a binomial random variable of parameters n and p. Here and later, given two random variables X, Y , the notation X ∼ Y will indicate that X and Y have the same distribution. The main purpose of this note is to illustrate that estimates on the mean absolute deviation of a binomial random variable yield a lower bound on P [X ≥ np], where X ∼ Bin(n, p). It should come as no surprise that there exists general machinery that can be employed to such a problem. For example, using Cauchy-Schwartz inequality one can show that, for any random variable Z whose mean equals zero, it holdsand the bound can be improved further under information on higher moments (see Veraar [11]). If we now let Z = X − np, where X ∼ Bin(n, p), then (1) provides a lower bound

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“…Proof. This is a well known result (see [2,7]). We include some details of the proof for the sake of completeness.…”

Section: Proof Of Theorem 11supporting

confidence: 54%

A lower bound on the probability that a binomial random variable is exceeding its mean

Pelekis

Ramon

2016

Statistics & Probability Letters

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“…So, from (6) and (7), we can conclude that only the left-hand side of (5) is bounded from above by something greater than 1 − δ and, hence, (4) need not imply (5). Therefore, some pursuit learning specific considerations are needed and, to the authors' knowledge, there is no obvious way to fill this gap.…”

Section: Existing Proofs Of ε-Optimalitymentioning

confidence: 96%

“…The proof in [15] implicitly assumes that (4) implies (5). A slightly different oversight is made in [7], [12], and [19].…”

Section: Existing Proofs Of ε-Optimalitymentioning

confidence: 99%

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On ε-Optimality of the Pursuit Learning Algorithm

Martin

Tilak

2012

J. Appl. Probab.

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Estimator algorithms in learning automata are useful tools for adaptive, real-time optimization in computer science and engineering applications. In this paper we investigate theoretical convergence properties for a special case of estimator algorithmsthe pursuit learning algorithm. We identify and fill a gap in existing proofs of probabilistic convergence for pursuit learning. It is tradition to take the pursuit learning tuning parameter to be fixed in practical applications, but our proof sheds light on the importance of a vanishing sequence of tuning parameters in a theoretical convergence analysis.

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“…Second, note that D h and D h+1 are (shifted) binomial random variables and satisfy the conditions of Theorem (1.a) in Klenke and Mattner (2010), which implies that D h+1 ≤ st D h or that for all nondecreasing functions γ :…”

Section: Appendixmentioning

confidence: 99%

Dynamically optimizing the administration of vaccines from multi-dose vials

et al. 2014

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Many vaccines are manufactured in large, multi-dose vials that once opened, must be used within a matter of hours. As a result, clinicians (especially those in remote locations) face difficult tradeoffs between opening a vial to satisfy a potentially small immediate demand versus retaining the vial to satisfy a potentially large future demand. This article formulates a Markov decision process model that determines when to conserve vials as a function of time of day, the current vial inventory, and the remaining clinic-days until the next replenishment. The objective is to minimize open-vial waste while administering as many vaccinations as possible. It is analytically established that the optimal policy is of a threshold type. Furthermore, an extensive sensitivity analysis is conducted that speaks to the benefits of consolidating demand, investing in buffer stock, and adopting different vial sizes. Lastly, a practical heuristic is evaluated and shown to perform competitively with the optimal policy.Keywords: Markov decision process, perishable inventory model, multi-dose vial, vaccine wastage * Corresponding author gitis, etc. A key step in achieving this goal is to reduce the large amount of vaccine "wastage", i.e., doses that are manufactured and shipped but then not administered while still viable, in developing countries. WHO (2005) estimates the overall global vaccine wastage rate to be a staggering 50% which varies with vial size, vaccine type, location, etc. More specifically, a recent study conducted in Bangladesh (Guichard et al., 2010) reports average wastage rates of 85.1% for BCG (tuberculosis), 71.2% for measles, 36.6% for TT (tetanus), and 44.2% for DPT (diphtheria, pertussis, and tetanus combination). Reducing wastage has been mandated by WHO (2005) and identified as a key factor in maintaining the financial sustainability of immunization programs in the world's poorest countries (Kamara et al., 2008) and in increasing the delivery of vaccines to patients. In the overall effort to improve vaccine coverage rates, it is critical that clinics manage vaccine supplies efficiently in the field to minimize wastage and maximize the number of immunizations administered.The primary driver of vaccine wastage is the fact that many vaccines are manufactured in large, multi-dose vials, which are more economical to produce than single-dose vials (Lee et al., 2010) but result in so-called open-vial waste (Guichard et al., 2010). Although unopened vials can be stored in reliable temperature-controlled environments for relatively long periods of time, for many vaccine types once a vial is reconstituted (i.e., "opened"), all of its doses must be used within a relatively short period of time

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Stochastic ordering of classical discrete distributions

Cited by 26 publications

References 11 publications

A lower bound on the probability that a binomial random variable is exceeding its mean

A lower bound on the probability that a binomial random variable is exceeding its mean

On ε-Optimality of the Pursuit Learning Algorithm

Dynamically optimizing the administration of vaccines from multi-dose vials

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