Strong Unimodality Research Articles

Signature Conditions for Distributional Properties of System Lifetimes if Component Lifetimes are I.I.D. Exponential

In this article, we consider coherent and mixed systems built of components with independent identically exponentially distributed lifetimes. We establish various sufficient and necessary conditions on the system signatures for asserting strong unimodality of the system lifetime as well as the monotonicity of its failure rate and mean residual life. Our investigations show that the precise description of the conditions characterizing the above distributional properties of the system lifetimes in the classic exponential model is a challenging task.

On firm size distribution: statistical models, mechanisms, and empirical evidence

In this work we explain the size distribution of business firms using a stochastic growth process that reproduces the main stylized facts documented in empirical studies. The steady state solution of this process is a three-parameter Dagum distribution, which possibly combines strong unimodality with a Paretian upper tail. Thanks to its flexibility, the proposed distribution is able to fit the whole range of firm size data, in contrast with traditional models that typically focus on large businesses only. An empirical application to Italian firms illustrates the practical merits of the Dagum distribution. Our findings go beyond goodness-of-fit per se, and shed light on possible connections between stochastic elements that influence firm growth and the meaning of parameters that appear in the steady state distribution of firm size. These results are ultimately relevant for studies into industrial organization and for policy interventions aimed at promoting sustainable growth and monitoring industrial concentration phenomena.

Strong unimodality of discrete order statistics

Alimohammadi et al. (2015) showed that the smallest and largest order statistics from a discrete strongly unimodal distribution, are strongly unimodal. In this note, we generalize their result to all order statistics, which proves their conjecture.

Kernel excess mass test for multimodality

SummaryIn this paper we propose a new statistical procedure for testing the multimodality of an underlying distribution. Peter Hall developed an innovative idea of calibrating the null distribution for the excess mass test statistic using the empirical distribution function. We find that the qualitative characteristics of a smooth underlying distribution function on the number of modes is barely preserved in the excess mass functional by the non‐smooth empirical distribution function. Instead of the empirical distribution function, we propose to use a kernel distribution function estimator. We derive the limiting distribution of the resulting test statistic under strong unimodality, based on which we apply the calibration idea to the proposed test statistic to obtain a cut‐off value. Our numerical study suggests that the calibrated kernel excess mass test has greater power than other existing methods. We also illustrate the use of the proposed method in a case study in astronomy which supports an assumption on a physical property of minor planets in the solar system.

Large time unimodality for classical and free Brownian motions with initial distributions

We prove that classical and free Brownian motions with initial distributions are unimodal for sufficiently large time, under some assumption on the initial distributions. The assumption is almost optimal in some sense. Similar results are shown for a symmetric stable process with index 1 and a positive stable process with index $1/2$. We also prove that free Brownian motion with initial symmetric unimodal distribution is unimodal, and discuss strong unimodality for free convolution.

New sufficient conditions for strong unimodality of multivariate discrete distributions

New sufficient conditions that ensure the strong unimodality of multivariate discrete distributions are obtained by the use of a special simplicial subdivision of multidimensional space. Strong unimodality of multivariate Pólya–Eggenberger distribution is shown.

Convolutions and generalization of logconcavity: Implications and applications

AbstractAdditive convolution of unimodal and α‐unimodal random variables are known as an old classic problem which has attracted the attention of many authors in theory and applied fields. Another type of convolution, called multiplicative convolution, is rather younger. In this article, we first focus on this newer concept and obtain several useful results in which the most important ones is that if is logconcave then so are and for some suitable increasing functions ϕ. This result contains and as two more important special cases. Furthermore, one table including more applied distributions comparing logconcavity of f(x) and and two comprehensive implications charts are provided. Then, these fundamental results are applied to aging properties, existence of moments and several kinds of ordered random variables. Multiplicative strong unimodality in the discrete case is also introduced and its properties are investigated. In the second part of the article, some refinements are made for additive convolutions. A remaining open problem is completed and a conjecture concerning convolution of discrete α‐unimodal distributions is settled. Then, we shall show that an existing result regarding convolution of symmetric discrete unimodal distributions is not correct and an easy alternative proof is presented. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 109–123, 2016

Discrete strong unimodality of order statistics

We show that discrete logconcavity of probability mass functions implies logconcavity of their distribution and survival functions. Applications are obtained and discrete strong unimodality of order statistics (OSs) is established. Illustrative examples are provided and finally discrete logconvexity of OSs is discussed.

SOME CONVEXITY PROPERTIES OF THE DISTRIBUTION OF LOWER k-RECORD VALUES WITH EXTENSIONS

Unimodality and strong unimodality of the distribution of ascendingly ordered random variables have been extensively studied in the literature, whereas these properties have not received much attention in the case of descendingly ordered random variates. In this paper, we show that log concavity of the reversed hazard rate implies that of the density function. Using this fundamental result, we establish some convexity properties of such random variables. To do this, we first provide a counterexample showing that a claim of Basak & Basak [7] about the lower record values is not valid. Then, we provide conditions under which unimodality properties of the distribution of lower k-record values would hold. Finally, some extensions to dual generalized order statistics in both univariate and multivariate cases are discussed.

Some new results on unimodality of generalized order statistics and their spacings

Because of the importance of ordered random variables and, in general, generalized order statistics (GOSs), in many branches of Statistics, a wide interest has been shown in investigating unimodality and strong unimodality of such random variables. Assuming certain restrictions on the model parameters and distributions, some authors have shown unimodality of GOSs. In this article, we shall provide some new results on unimodality of GOSs based on the population distribution function which contain and strengthen several known findings in this regard. A counterexample is also provided for the cases where the results are not valid in general. Unimodality of arbitrary spacings of GOSs based on exponential distributions is also discussed.

A multiplicative short proof for the unimodality of stable densities

Revisiting an article by Chernin and Ibragimov on unimodality of stable laws, we show that their approach to deduce the general case from the extremal ones, whose completion contained an error as discovered later by Kanter, can be carried out successfully in considering Bochner's subordination and multiplicative strong unimodality. This short proof of the unimodality of all stable densities yields also a multiplicative counterpart to Yamazato's additive ones.

Multiplicative strong unimodality for positive stable laws

It is known that real non-Gaussian stable laws are unimodal, not additive strongly unimodal, multiplicative strongly unimodal in the symmetric case, and that the only remaining relevant situation for the multiplicative strong unimodality is the one-sided case. It is shown here that positive α-stable distributions are multiplicative strongly unimodal if and only if α < 1/2.

On strong unimodality of multivariate discrete distributions

A discrete function f defined on Z n is said to be logconcave if f ( λ x + ( 1 − λ ) y ) ≥ [ f ( x ) ] λ [ f ( y ) ] 1 − λ for x , y , λ x + ( 1 − λ ) y ∈ Z n . A more restrictive notion is strong unimodality. Following Barndorff-Nielsen [O. Barndorff-Nielsen, Unimodality and exponential families, Commun. Statist. 1 (1973) 189–216] a discrete function p ( z ) , z ∈ Z n is called strongly unimodal if there exists a convex function f ( x ) , x ∈ R n such that f ( x ) = − log p ( x ) if x ∈ Z n . In this paper sufficient conditions that ensure the strong unimodality of a multivariate discrete distribution, are given. Examples of strongly unimodal multivariate discrete distributions are presented.

Neyman type A distribution revisited

The modality structure of the Neyman type A distribution is examined. The results obtained add to those previously stated in Barton (1957) (or in Johnsonet al. (1992, p. 370)) and in Shenton and Bowman (1967). Strong unimodality of this distribution is also considered. Problem 328 in Statistica Neerlandica (Theodorescu (1997)) and its solution (Borwein (2000)) play an important part.

Unimodality of the distribution of record statistics

Let X 1, X 2,…, be a sequence of independent and identically distributed random variables with absolutely continuous distribution function F. For n⩾1, we denote the order statistics of X 1, X 2,…, X n by X 1, n ⩽ X 2, n ⩽⋯⩽ X n, n . Define L(1)=1, L(n+1)= min{j : j>L(n), X j>X j−1,j−1}, andX(n)=X L(n),L(n), n⩾1. The sequence { X( n)} ({ L( n)}) is called upper record statistics (times). In this article, we deal with unimodality of record statistics. We also deal with the strong unimodality of record statistics.

Existence conditions and properties for the generalized Euler family of distributions

The paper gives existence conditions for the generalized Euler family of distributions and relates these to distributions already in the literature. The logconvexity, logconcavity, failure rate, infinite divisibility, strong unimodality, and over and under-dispersion properties are examined using the methods of Gupta et al. (J. Statist Plann. Inference 65 (1997) 225).

Multiplicative Strong Unimodality

Multiplicative strong unimodality is defined as the preservation of unimodality in products of independent random variables. An Ibragimov type theorem is proved. As an application, preserving unimodality for scale mixtures of gamma distributions is examined. It is also shown that multiplicative strong unimodal probability measures on R appear as images, by the exponential map, of classical strong unimodal ones. The connection to the star order is also established.

Unimodality of Probability Measures

Preface. 1. Prelude. 2. Khinchin Structures. 3. Concepts of Unimodality. 4. Khinchin's Classical Unimodality. 5. Discrete Unimodality. 6. Strong Unimodality. 7. Positivity of Functional Moments. Bibliography. Symbol Index. Name Index. Subject Index.

Two principal points of symmetric, strongly unimodal distributions

Examples illustrate that splitting a symmetric, unimodal distribution into two groups so as to minimize the sum of the within-group variances does not always occur at the mean. Strong unimodality is shown to be a sufficient condition for the optimal cutpoint to occur at the mean.

Mixtures with a Limited Number of Modal Intervals

We derive necessary and sufficient conditions in order that each mixture of a given family of probability densities have no more than $s$ modal intervals, with special attention to ordinary unimodality and strong unimodality of such mixtures.