We define the p,q-Legendre-Stirling numbers PSp,q(n,k) of the second kind, which reduce to the Legendre-Stirling numbers PS(n,k) of the second kind discovered by Everitt, Littlejohn and Wellman when ...p=1 and q=1 and the q-Legendre-Stirling numbers PSq(n,k) of the second kind introduced by Mongelli when p=1. By introducing 0-1 tableaux of shape λ with even multiplicities, we give a combinatorial interpretation of the p,q-Legendre-Stirling numbers of the second kind. As an application of such a combinatorial interpretation, we derive three p,q-Legendre-Stirling identities, two of which can be viewed as p,q-analogues of two results due to Everitt-Littlejohn-Wellman and Andrews-Gawronski-Littlejohn, respectively. We also show that q-Legendre-Stirling numbers PSq(n,k) of the second kind are strongly q-log-concave by constructing a direct injection based on this combinatorial construction.
Let X₁,ellipsis (horizontal),Xn be independent and identically distributed random vectors with a (Lebesgue) density f. We first prove that, with probability 1, there is a unique log-concave maximum ...likelihood estimator graphic removed of f. The use of this estimator is attractive because, unlike kernel density estimation, the method is fully automatic, with no smoothing parameters to choose. Although the existence proof is non-constructive, we can reformulate the issue of computing graphic removed in terms of a non-differentiable convex optimization problem, and thus combine techniques of computational geometry with Shor's r-algorithm to produce a sequence that converges to graphic removed . An R version of the algorithm is available in the package LogConcDEAD--log-concave density estimation in arbitrary dimensions. We demonstrate that the estimator has attractive theoretical properties both when the true density is log-concave and when this model is misspecified. For the moderate or large sample sizes in our simulations, graphic removed is shown to have smaller mean integrated squared error compared with kernel-based methods, even when we allow the use of a theoretical, optimal fixed bandwidth for the kernel estimator that would not be available in practice. We also present a real data clustering example, which shows that our methodology can be used in conjunction with the expectation-maximization algorithm to fit finite mixtures of log-concave densities.
We consider an intensity control problem for a point process to maximize the expectation of a function of the time when the nth event occurs. We find the optimal control policy when the objective ...function is unimodal. Moreover, if the objective function is log-concave, so is the value function. As an application, we completely solve an intensity control problem that generalizes the problem studied by Brémaud (1976) and Defourny (2018). Also, we resolve the two conjectures made by Defourny (2018).
This paper extends the results of Prékopa (1973,1980) on strictly log-concave cumulative distributions to strictly log-concave probability distributions. It is shown that if a random variable follows ...a strictly logarithmic concave distribution, then the probability that the random variable is contained within a convex polytope is also strictly logarithmic concave. This formal result can be useful for identification and estimation of a general class of linear-index discrete response models, where the additively separable unobservable follows a strictly log-concave distribution.
Motivated by several recent studies on oligopolistic markets with consumer heterogeneity, we characterize the geometric properties of a class of inequalities involving the product of two integrals of ...the probability density quantile function and its reciprocal. Through three applications, we illustrate how our characterization can help enrich our understanding of the existing findings and shed some new insights, regarding the extent to which consumer welfare and social efficiency depend on the properties of the distributions of consumer valuation heterogeneity.
Recruiting candidates globally and across multiple sites in different geographic regions is necessary to speed up the enrollment of clinical trials. While patient enrollment can benefit from this ...globalization, initiating clinical trials has become much more complicated. In the start-up stage, the sites must be selected out of a set of potential candidates around the globe based on the specifics of those clinical trials, such as protocols, operational costs, and recruitment deadlines. Sites in one region can be very distinct from sites in another area. Yet, a common mistake in selecting sites is to rely on too little knowledge or subjective data. Poor selection decisions can lead to study delays and prolong the time to market for life-saving treatments. Thus, this paper proposes a novel framework to aid the decision-making in the global site selection problem (GSSP). To ensure that our framework accurately captures the uncertainty in recruitment time, we adopt a risk-based constraint that accounts for random patient enrollment. The extensive computational studies help quantify significant time-cost trade-offs as a potential solution to control the costs of conducting a trial.
•We aid managerial decision-making in clinical trials using data and business analytics.•Improved site selection minimizes study delays and optimizes clinical trial costs effectively.•A probabilistic constraint is incorporated to enhance decision-making under uncertainty.•Solution approach aids in selecting optimal sites globally, considering diverse geographic regions.
The Baxter numbers \(B_n\) enumerate a lot of algebraic and combinatorial objects such as the bases for subalgebras of the Malvenuto-Reutenauer Hopf algebra and the pairs of twin binary trees on ...\(n\) nodes.
The Turán inequalities and higher order Turán inequalities are related to the Laguerre-Pólya (\(\mathcal{L}\)-\(\mathcal{P}\)) class of real entire functions, and the \(\mathcal{L}\)-\(\mathcal{P}\) class has a close relation with the Riemann hypothesis. The Turán type inequalities have received much attention.
In this paper, we are mainly concerned with Turán type inequalities, or more precisely, the log-behavior, and the higher order Turán inequalities associated with the Baxter numbers. We prove the Turán inequalities (or equivalently, the log-concavity) of the sequences \(\{B_{n+1}/B_n\}_{n\geqslant 0}\) and \(\{\hspace{-2.5pt}\sqrtn{B_n}\}_{n\geqslant 1}\).
Monotonicity of the sequence \(\{\hspace{-2.5pt}\sqrtn{B_n}\}_{n\geqslant 1}\) is also obtained. Finally, we prove that the sequences \(\{B_n/n!\}_{n\geqslant 2}\) and \(\{B_{n+1}B_n^{-1}/n!\}_{n\geqslant 2}\) satisfy the higher order Turán inequalities.
We show that the number
of partitions with m even parts and largest hook length n is strongly unimodal with mode
for
. We establish this result by induction, using a 5-term recurrence due to Lin, ...Xiong and Yan, and two 4-term recurrences obtained by Zeilberger's algorithm. The sequence
is not log-concave. Using Möbius transformation and the method of interlacing zeros, we obtain that every zero of every generating polynomial
lies on the left half part of the circle
. Moreover, as an application of Wang and Zhang's characterization of root geometry of polynomial sequences that satisfy a recurrence, we confirm that the zeros are densely distributed on the half circle.
A number of papers have dealt with the preservation of log convexity and log concavity based on various operators. For instance, in Badía and Sangüesa 10, the preservation of log convexity and log ...concavity under Bernstein operators was discussed based on some characteristics of a stochastic process. However, regarding beta-type operators, the preservation of log concavity does not hold based on such a probabilistic method used in Badía and Sangüesa 10. In this study, we explore the preservation of log concavity for the beta operator using alternative probabilistic tools. Notably, we show results on the preservation of log concavity for monotone log concave functions. Further, some results of application to some specific functions, ageing classes of the deterioration Dirichlet process, related operators and order statistics are provided.
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