On an inverse problem in mixture failure rates modelling

Mixtures of decreasing failure rate (DFR) distributions are always DFR. It turns out that very often mixtures of increasing failure rate distributions can decrease or show even more complicated patterns of dependence on time. For studying this and other relevant effects two simple models of mixing with additive and multiplicative failure rates are considered. It is shown that for these models an inverse problem can be solved, which means that given an arbitrary shape of the mixture failure rate and a mixing distribution, the failure rate for a governing distribution can be uniquely obtained. Some examples are considered where this operation can be performed explicitly. Possible generalizations are discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

On extreme points inl 1

It is proved that every bounded closed and convex subset ofl ₁ is the closed convex hull of its extreme points. The research reported in this document has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18 through the European Office of Aerospace Research (OAR) United States Air Force.     

Strict convexity of some subsets of Hankel operators

We find some extreme points in the unit ball of the set of Hankel operators and show that the unit ball of the set of compact Hankel operators is strictly convex. We use this result to show that the collection of lower triangular Toeplitz contractions is strictly convex. We also find some extreme points in certain reduced Cowen sets and discuss cases in which they are or are not strictly convex.

     

Exchangeable Random Orders and Almost Uniform Distributions

Random orders on invariant under permutations are called exchangeable. The compact and convex set of all random total orders is shown to be a Bauer simplex whose set of extreme points, the socalled totally ordered paintbox processes, is homeomorphically parametrized by almost uniform distributions on the unit interval, i.e. by probability measures w on [0, 1] whose distribution functions are w-almost surely the identity.     

Regularity and Integration of Set-Valued Maps Represented by Generalized Steiner Points

A family of probability measures on the unit ball in generates a family of generalized Steiner (GS-)points for every convex compact set in . Such a “rich” family of probability measures determines a representation of a convex compact set by GS-points. In this way, a representation of a set-valued map with convex compact images is constructed by GS-selections (which are defined by the GS-points of its images). The properties of the GS-points allow to represent Minkowski sum, Demyanov difference and Demyanov distance between sets in terms of their GS-points, as well as the Aumann integral of a set-valued map is represented by the integrals of its GS-selections. Regularity properties of set-valued maps (measurability, Lipschitz continuity, bounded variation) are reduced to the corresponding uniform properties of its GS-selections. This theory is applied to formulate regularity conditions for the first-order of convergence of iterated set-valued quadrature formulae approximating the Aumann integral.      

The facialQ-topology for compact convex sets

The structure of the set of closed two-sided ideals in aC*-algebraU with identity is described by means of a topology on the set _e K of extreme points of the state spaceK ofU. Recent results of Alfsen, Andersen, Combes, Perdrizet, Wils, and others have shown that such a topology can be defined on the set _e K of extreme points of an arbitrary compact convex subset of a locally convex Hausdorff topological vector space.The structure of the set of closed left ideals in aC*-algebraU with identity can also be described by means of a set of subsets of the set _e K of extreme points of its state spaceK. Akemann, Giles, and Kummer showed that this formed a more general structure than a topology which was called aq-topology. In this paper it is shown that for a reasonably wide class of compact convex subsetsK of locally convex Hausdorff topological vector spaces such aq-topology can also be defined on _e K and that it shares many of the properties of theq-topology defined forC*-algebras. The methods used depend strongly upon recent results of Alfsen and Shultz on the spectral theory of affine functions on compact convex sets.     

Exponential distributions on semigroups

Three fundamental characterizations of the standard exponential distribution on [0, ) are the remaining life, memoryless and constant failure properties. Analogs of these properties are studied for distributions on a class of semigroups in which the semigroup operation replaces addition, a compatible partial order replaces the ordinary order, and a left-invariant measure replaces Lebesgue measure. Partial characterizations of exponential distributions on such semigroups are obtained and the semigroup formulation provides new characterizations of certain aging properties studied in reliability-increasing failure rate, new better than used, and increasing failure rate average.     

On compact multipliers of topological algebras

It is shown that if the maximal ideal space (A) of a semisimple commutative complete metrizable locally convex algebra contains no isolated points, then every compact multiplies is trivial. In particular, compact multipliers on semisimple commutative Fréchet algebras whose maximal ideal space has no isolated points are identically zero.     

Some characterizing properties of the simplex

We shall prove that a convex body in ^d (d2) is a simplex if, and only if, each of its Steiner symmetrals is a convex double cone over the symmetrization space or, equivalently, has exactly two extreme points outside of this hyperplane. In [3] it is shown that every Steiner symmetral of an arbitrary d-simplex is such a double cone, more precisely a bipyramid. Therefore our main aim is to prove that a convex body which is not a simplex has Steiner symmetrals with more than two extreme points not in the symmetrization space. Some equivalent properties of simplices will also be given.     

Distributionen mit Werten in topologischen Vektorräumen I

In this article, distributions with values in a (not necessarily locally convex) topological vector space E are defined to be the elements of. Operations on such distributions can be introduced as usual. In general, not every continuous E-valued function defines a distribution, but the elements of do. A large part of the theory is reduced to the locally convex case by use of the so-called locally convex subspaces of E (cf. [2]). We prove that the presheaf of E-valued distributions on open subsets of ^N is a topological sheaf. We give integral representation theorems for bounded mappings in and, and we show that, on bounded subsets of ^N, each distribution defined on ^N with values in a (p)-space is a derivative of a function in.

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Während der Fertigstellung eines Teils dieser Arbeit Research Associate der University of Maryland, Md. 20742, USA.     

Planar Point Sets With Large Minimum Convex Decompositions

We show the existence of sets with $n$ points ( $n\ge 4$ ) for which every convex decomposition contains more than $\frac{35}{32}n-\frac{3}{2}$ polygons, which refutes the conjecture that for every set of $n$ points there is a convex decomposition with at most $n+C$ polygons. For sets having exactly three extreme points we show that more than $n+\sqrt{2(n-3)}-4$ polygons may be necessary to form a convex decomposition.     

Concave functions,blaschke products,and polygonal mappings

We consider the class Co(p) of all conformal maps of the unit disk onto the exterior of a bounded convex set. We prove that the triangle mappings, i.e., the functions that map the unit disk onto the exterior of a triangle, are among the extreme points of the closed convex hull of Co(p). Moreover, we prove a conjecture on the closed convex hull of Co(p) for all p ∈ (0, 1) which had partially been proved by the authors for some values of p ∈ (0, 1).     

Entropy splitting for antiblocking corners and perfect graphs

We characterize pairs of convex setsA, B in thek-dimensional space with the property that every probability distribution (p ₁,...,p _k ) has a repsesentationp _i =a _l .b _i , aA, bB.Minimal pairs with this property are antiblocking pairs of convex corners. This result is closely related to a new entropy concept. The main application is an information theoretic characterization of perfect graphs.Research was partially sponsored by the Hungarian National Foundation, Scientific Research Grants No 1806 and 1812.     

Understanding the shape of the mixture failure rate (with engineering and demographic applications)

Mixtures of distributions are usually effectively used for modelling heterogeneity. It is well known that mixtures of DFR distributions are always DFR. On the other hand, mixtures of IFR distributions can decrease, at least in some intervals of time. As IFR distributions often model lifetimes governed by ageing processes, the operation of mixing can dramatically change the pattern of ageing. Therefore, the study of the shape of the observed (mixture) failure rate in a heterogeneous setting is important in many applications. We study discrete and continuous mixtures, obtain conditions for the mixture failure rate to tend to the failure rate of the strongest populations and describe asymptotic behaviour as t→∞. Some demographic and engineering examples are considered. The corresponding inverse problem is discussed. Copyright © 2009 John Wiley & Sons, Ltd.     

AnF-space with trivial dual where the Krein-Milman theorem holds

We show that in certain non-locally convex Orlicz function spacesL _ϕ with trivial dual every compact convex set is locally convex and hence the Krein-Milman theorem holds. This complements the example constructed by Roberts of a compact convex set without extreme points inL _p (0<p<1) and answers a question raised by Shapiro.     

Characterizations of continuous distributions through inequalities involving the expected values of selected functions

Nanda (2010) and Bhattacharjee et al. (2013) characterized a few distributions with help of the failure rate, mean residual, log-odds rate and aging intensity functions. In this paper, we generalize their results and characterize some distributions through functions used by them and Glaser’s function. Kundu and Ghosh (2016) obtained similar results using reversed hazard rate, expected inactivity time and reversed aging intensity functions. We also, via w(·)-function defined by Cacoullos and Papathanasiou (1989), characterize exponential and logistic distributions, as well as Type 3 extreme value distribution and obtain bounds for the expected values of selected functions in reliability theory. Moreover, a bound for the varentropy of random variable X is provided.     

Norm or numerical radius attaining polynomials on C(K)

Let be the Banach space of all complex-valued continuous functions on a compact Hausdorff space K. We study when the following statement holds: every norm attaining n-homogeneous complex polynomial on attains its norm at extreme points. We prove that this property is true whenever K is a compact Hausdorff space of dimension less than or equal to one. In the case of a compact metric space a characterization is obtained. As a consequence we show that, for a scattered compact Hausdorff space K, every continuous n-homogeneous complex polynomial on can be approximated by norm attaining ones at extreme points and also that the set of all extreme points of the unit ball of is a norming set for every continuous complex polynomial. Similar results can be obtained if “norm” is replaced by “numerical radius.”     

On representations of the feasible set in convex optimization

We consider the convex optimization problem \({\min_{\mathbf{x}} \{f(\mathbf{x}): g_j(\mathbf{x})\leq 0, j=1,\ldots,m\}}\) where f is convex, the feasible set \({\mathbf{K}}\) is convex and Slater’s condition holds, but the functions g _j ’s are not necessarily convex. We show that for any representation of \({\mathbf{K}}\) that satisfies a mild nondegeneracy assumption, every minimizer is a Karush-Kuhn-Tucker (KKT) point and conversely every KKT point is a minimizer. That is, the KKT optimality conditions are necessary and sufficient as in convex programming where one assumes that the g _j ’s are convex. So in convex optimization, and as far as one is concerned with KKT points, what really matters is the geometry of \({\mathbf{K}}\) and not so much its representation.