Complete Monotone Research Articles

Systematic and Durable Analysis of the Rule based Cauchy’s Inequality in solving the n-order Differential Equations

This paper instigates the multivariate generalization of the widely used Cauchy inequality 1 + x ≤ ex where x can be any non-negative real number. The results of this study can be the solution for the Cauchy’s problem for particular Ordinary Differential Equation (ODE). This is also related to study of the complete monotone function and the divided differences theory. The proof is based on the empty product convention notion and the Beppo Levi theorem of Monotone convergence. This study is also extended to multivariate generalization of the simultaneous inequalities. Key Words: ODEs, Inequalities, Population Dynamics, Simultaneous Inequalities, Divided differences.

Strong solution of a hydrodinamics problem with memory

In the last few years, the concepts of fractional calculus were frequently applied to other disciplines. Recently, this subject has been extended in various directions such as signal processing, applied mathematics, bio-engineering, viscoelasticity, fluid mechanics, and fluid dynamics. In fluid dynamics, the fractional derivative models were used widely in the past for the study of viscoelastic materials such as polymers in the glass transition and in the glassy state. Recently, it has increasingly been seen as an efficient tool through which a useful generalization of physical concepts can be obtained. The fractional derivatives used most are the Riemann--Liouville fractional derivative and the Caputo fractional derivative. It is well known that these operators exhibit difficulties in applications. For example, the Riemann--Liouville derivative of a constant is not zero. We deal with so called temporal fractional derivative as a prototype of general fractional derivative. We prove the global strong solvability of a linear and quasilinear initial-boundary value problems with a singular complete monotone kernels. Our main tool is a theory of evolutionary integral equations. An abstract fractional order differential equation is studied, which contains as particular case the Rayleigh–Stokes problem for a generalized second-grade fluid with a fractional derivative model. This paper concerns with an initial-boundary value problem for the Navier--Stokes--Voigt equations describing unsteady flows of an incompressible viscoelastic fluid. We give the strong formulation of this problem as a nonlinear evolutionary equation in Sobolev spaces. Using the Galerkin method with a special basis of eigenfunctions of the Stokes operator, we construct a global-in-time strong solution in two-dimensional domain. We also establish an $L_2$ decay estimate for the velocity field under the assumption that the external forces field is conservative.

Least squares estimation of a completely monotone pmf: From Analysis to Statistics

We consider the class of completely monotone probability mass functions (pmf) from a statistical perspective. An element in this class is known to be a mixture of geometric pmfs, a consequence of the celebrated Hausdorff Theorem. We show that the complete monotone least squares estimator exists, is strongly consistent and converges weakly to the truth at the n-rate. Furthermore, we fully describe its limit distribution as the unique solution of a well-posed minimization problem. Through a simulation study we assess the performance of the method under different scenarios.

Multiple monotonicity of discrete distributions: The case of the Poisson model

Shape constrained estimation in discrete settings has received increasing attention in statistics. Among the most important shape constrained models is multiple monotonicity, including $k$-monotonicity, for a given integer $k\in [1,\infty )$, and complete monotonicity. Multiple monotonicity provides a nice generalization of monotonicity and convexity and has been successfully used in applications related to estimation of species richness. Although fully nonparametric, it is of great interest to determine some of the well-known parametric distributions which belong to this model. Among the most important examples are the family of Poisson distributions and mixtures thereof. In Giguelay (2017) $k$-monotonicity of Poisson distributions was connected to the roots of a certain polynomial, but a typographical error occurred while writing its expression. In this note, we correct that typographical error and give a detailed proof that a Poisson distribution with rate $\lambda \in [0,\infty )$ is $k$-monotone if and only if $\lambda \le \lambda _{k}$, where $\lambda _{k}$ is the smallest zero of the $k$-th degree Laguerre polynomial $L_{k}(x)=\sum _{j=0}^{k}(-1)^{j}\binom{k}{j}x^{j}/j!$, $x\ge 0$. This result yields the sufficient condition that a mixture of Poisson distributions is $k$-monotone if the support of the mixing distribution is included in $[0,\lambda _{k}]$. Furthermore, we show that the only complete monotone Poisson distribution is the Dirac distribution at $0$.

Many disjoint edges in topological graphs

A monotone cylindrical graph is a topological graph drawn on an open cylinder with an infinite vertical axis satisfying the condition that every vertical line intersects every edge at most once. It is called simple if any pair of its edges have at most one point in common: an endpoint or a point at which they properly cross. We say that two edges are disjoint if they do not intersect. We show that every simple complete monotone cylindrical graph on n vertices contains Ω(n1−ϵ) pairwise disjoint edges for any ϵ>0. As a consequence, we show that every simple complete topological graph (drawn in the plane) with n vertices contains Ω(n12−ϵ) pairwise disjoint edges for any ϵ>0. This improves the previous lower bound of Ω(n13) by Suk which was reproved by Fulek and Ruiz-Vargas. We remark that our proof implies a polynomial time algorithm for finding this set of pairwise disjoint edges.

Many disjoint edges in topological graphs

A monotone cylindrical graph is a topological graph drawn on an open cylinder with an infinite vertical axis satisfying the condition that every vertical line intersects every edge at most once. It is called simple if any pair of its edges have at most one point in common: an endpoint or a point at which they properly cross. We say that two edges are disjoint if they do not intersect. We show that every simple complete monotone cylindrical graph on n vertices contains Ω(n1−ϵ) pairwise disjoint edges for any ϵ>0. As a consequence, we show that every simple complete topological graph (drawn in the plane) with n vertices contains Ω(n12−ϵ) pairwise disjoint edges for any ϵ>0. This improves the previous lower bound of Ω(n13) by Suk which was reproved by Fulek and Ruiz-Vargas. We remark that our proof implies a polynomial time algorithm for finding this set of pairwise disjoint edges.

Complete Monotone Quasiconcave Duality

We introduce a notion of complete monotone quasiconcave duality, motivated by some economic applications. We show that this duality holds for important classes of quasiconcave functions.

Computing sets of expected utility maximizing distributions for common utility functions

The set of distribution functions that maximize expected utility for some utility function in a given class is the optimal set. This paper presents an algorithm for determining the optimal set of distributions for an important class of preferences and general finite sets of alternatives. The class of preferences considered, which includes the commonly used log, exponential and power functions, is the set of utility functions with complete monotone marginal utility (all derivatives alternate in sign). The algorithm is based on the necessary and sufficient conditions for infinite degree convex stochastic dominance, and is implemented using the solutions to a parametric family of linear programming problems. The algorithm is intended to be applied to sample data, and nonparametric statistical inference procedures are provided.

Efficient ML Estimation of the Multivariate Normal Distribution from Incomplete Data

It is well known that the maximum likelihood estimates (MLEs) of a multivariate normal distribution from incomplete data with a monotone pattern have closed-form expressions and that the MLEs from incomplete data with a general missing-data pattern can be obtained using the Expectation-Maximization (EM) algorithm. This article gives closed-form expressions, analogous to the extension of the Bartlett decomposition, for both the MLEs of the parameters and the associated Fisher information matrix from incomplete data with a monotone missing-data pattern. For MLEs of the parameters from incomplete data with a general missing-data pattern, we implement EM and Expectation-Constrained-Maximization-Either (ECME), by augmenting the observed data into a complete monotone sample. We also provide a numerical example, which shows that the monotone EM (MEM) and monotone ECME (MECME) algorithms converge much faster than the EM algorithm.

On the Mittag–Leffler distributions

We first prove that the Mittag–Leffler distributions belong to the class of distributions with complete monotone derivative. Then we investigate the fundamental properties of the Mittag–Leffler distributions and of their extensions, including the tail behavior of distribution, the explicit expressions for moments of all orders and for the density functions. The latter has been used to correct some inverse Laplace transforms given in the literature. As a by-product, the moments of negative (integral) orders are used to characterize the positive stable distributions with exponent α∈[ 1 3 ,1] .

Ein neues Zusammenhangsmaß für ordinalskalierte Merkmale / A New Measure of Association for Ordinal Variables

Zusammenfassung In dieser Arbeit wird das Zusammenhangsmaß T) vorgeschlagen, das die Stärke und die Richtung monotoner Beziehungen zwischen ordinalskalierten Merkmalen mißt. Dieser Zusammenhang wird unter der Bedingung gegebener Randverteilungen dieser Merkmale gemessen, weil diese Randverteilungen die Zusammenhangsstruktur der betrachteten Merkmale entscheidend determinieren. Die Konstruktion des Maßes beruht daher auf geeignet modifizierten Abständen bzw. der Position der gegebenen zweidimensionalen Häufigkeitsverteilung zu den aus ihren Randverteilungen ableitbaren zweidimensionalen Häufigkeitsverteilungen, die durch einen vollständigen monoton positiven bzw. negativen Zusammenhang gekennzeichnet sind. Ein Nachteil von T) ist sicherlich seine komplizierte Konstruktion und Berechnung, aber mit Hilfe eines Computerprogramms kann r) in Bruchteilen von Sekunden berechnet werden. Demgegenüber steht der Vorteil seiner anschaulichen Interpretierbarkeit auf der Grundlage der Position der gegebenen zweidimensionalen Häufigkeitsverteilung zwischen den beiden aus ihren Randverteilungen ableitbaren Extremen.

Distributions with complete monotone derivative and geometric infinite divisibility

It is shown that a distribution with complete monotone derivative is geometrically infinitely divisible and that the class of distributions with complete monotone derivative is a proper subclass of the class of geometrically infinitely divisible distributions.

Distributions with complete monotone derivative and geometric infinite divisibility

It is shown that a distribution with complete monotone derivative is geometrically infinitely divisible and that the class of distributions with complete monotone derivative is a proper subclass of the class of geometrically infinitely divisible distributions.