Nonparametric estimation of compound distributions with applications in insurance

A nonparametric estimator of the distribution functionG of a random sum of independent identically distributed random variables, with distribution functionF, is proposed in the case where the distribution of the number of summands is known and a random sample fromF is available. This estimator is found by evaluating the functional that mapsF ontoG at the empirical distribution function based on the random sample. Strong consistency and asymptotic normality of the resulting estimator in a suitable function space are established using appropriate continuity and differentiability results for the functional. Bootstrap confidence bands are also obtained. Applications to the aggregate claims distribution function and to the probability of ruin in the Poisson risk model are presented.     

A Note on Equilibrium Problems with Properly Quasimonotone Bifunctions

In this paper, we consider some well-known equilibrium problems and their duals in a topological Hausdorff vector space X for a bifunction F defined on K x K,where K is a convex subset of X. Some necessary conditions are investigated, proving different results depending on the behaviour of F on the diagonal set. The concept of proper quasimonotonicity for bifunctions is defined, and the relationship with generalized monotonicity is investigated. The main result proves that the condition of proper quasimonotonicity is sharp in order to solve the dual equilibrium problem on every convex set.     

Inference on a distribution function from ranked set samples

Consider independent observations \((X_i,R_i)\) with random or fixed ranks \(R_i\), while conditional on \(R_i\), the random variable \(X_i\) has the same distribution as the \(R_i\)-th order statistic within a random sample of size k from an unknown distribution function F. Such observation schemes are well known from ranked set sampling and judgment post-stratification. Within a general, not necessarily balanced setting we derive and compare the asymptotic distributions of three different estimators of the distribution function F: a stratified estimator, a nonparametric maximum-likelihood estimator and a moment-based estimator. Our functional central limit theorems generalize and refine previous asymptotic analyses. In addition, we discuss briefly pointwise and simultaneous confidence intervals for the distribution function with guaranteed coverage probability for finite sample sizes. The methods are illustrated with a real data example, and the potential impact of imperfect rankings is investigated in a small simulation experiment.

     

Iterated Boolean random varieties and application to fracture statistics models

Models of random sets and of point processes are introduced to simulate some specific clustering of points, namely on random lines in R² and R³ and on random planes in R³. The corresponding point processes are special cases of Cox processes. The generating distribution function of the probability distribution of the number of points in a convex set K and the Choquet capacity T (K) are given. A possible application is to model point defects in materials with some degree of alignment. Theoretical results on the probability of fracture of convex specimens in the framework of the weakest link assumption are derived, and used to compare geometrical effects on the sensitivity of materials to fracture.     

Restricted Isometry Property of Matrices with?Independent Columns and Neighborly Polytopes by?Random Sampling

This paper considers compressed sensing matrices and neighborliness of a centrally symmetric convex polytope generated by vectors ±X ₁,…,±X _N ∈ℝ ⁿ , (N≥n). We introduce a class of random sampling matrices and show that they satisfy a restricted isometry property with overwhelming probability. In particular, we prove that matrices with i.i.d. centered and variance 1 entries that satisfy uniformly a subexponential tail inequality possess the restricted isometry property with overwhelming probability. We show that such “sensing” matrices are valid for the exact reconstruction process of m-sparse vectors via ℓ ₁ minimization with m≤Cn/log ²(cN/n). The class of sampling matrices we study includes the case of matrices with columns that are independent isotropic vectors with log-concave densities. We deduce that if K⊂ℝ ⁿ is a convex body and X ₁,…,X _N ∈K are i.i.d. random vectors uniformly distributed on K, then, with overwhelming probability, the symmetric convex hull of these points is an m-centrally-neighborly polytope with m∼n/log ²(cN/n).     

Regularized minimization under weaker hypotheses

LetV andW be two Banach spaces, withV reflexive, a bounded convex set ofV, A a linear mapping fromV intoW, and letF be a convex functional onW. We minimizeJ(v)=F(Av) on using hypotheses about particular sequences in IfV is uniformly convex, we prove existence and uniqueness of a solution of minimal norm minimizingJ. In the Hilbert space case, withF defined byF(w)=w–f ²,f given inW, we get existence and uniqueness of the projection off on A(), which generalizes the case where A() is a closed set ofW (taking closed andA continuous). Finally, we give examples, and we study an unbounded operator case.     

An L0（F,R）-valued Function＇s Intermediate Value Theorem and Its Applications to Random Uniform Convexity

Let (Ω , F , P ) be a probability space and L0 ( F, R ) the algebra of equivalence classes of real- valued random variables on (Ω , F , P ). When L0 ( F, R ) is endowed with the topology of convergence in probability, we prove an intermediate value theorem for a continuous local function from L0 ( F, R ) to L0 ( F, R ). As applications of this theorem, we first give several useful expressions for modulus of random convexity, then we prove that a complete random normed module ( S,|| · ||) is random uniformly convex iff Lp ( S ) is uniformly convex for each fixed positive number p such that 1 p + ∞ .     

Algorithm for finding a solution to the inclusion 0∈F(x)

Using the steepest-descent method combined with the Armijo stepsize rule, we give an algorithm for finding a solution to the inclusion 0F(x), whereF is a set-valued map with smooth support function. As an example, we consider the special caseF(x)=g(x)+K, withK being a convex cone andg a single-valued function. The relation between the present algorithm and that given by Burke and Han is also discussed.The valuable comments and helpful suggestions of the referee are gratefully acknowledged. Sincere thanks are due to Dr. J. Burke for submitting the necessary material and to Dr. C. Lemarechal for advices and encouragement.     

Testing linear operators—An average case study

Given a specification linear operatorS, we want to test an implementation linear operatorA and determine whether it conforms to the specification operator according to an error criterion. In an earlier paper [3], we studied a worst case error in which we test whether the error is no more than a given bound ε>0 for all elements in a given setF, i.e., sup _{fεf∥Sf—Af∥≤ε}. In this work, we study the average error instead, i. e., ∫ _F ∥Sf-Af∥²μ(df)ɛ≤², where μ is a probability measure onF. We assume that an upper boundK on the norm of the difference ofS andA is given a priori. It turns out that any finite number of tests is in general inconclusive with the average error. Therefore, as in the worst case, we allow a relaxation parameter α>0 and test for weak conformance with an error bound (1+α)ε. Then a finite number of tests from an arbitrary orthogonal complete sequence is conclusive. Furthermore, the eigenvectors of the covariance operatorC _μ of the probability measure μ provide an almost optimal test sequence. This implies that the test set isuniversal; it only depends on the set of valid inputsF and the measure μ, and is independent ofS, A, and the other parameters of the problem. However, the minimal number of tests does depend on all the parameters of the testing problem, i.e., ε, α,K, and the eigenvalues ofC _μ. In contrast to the worst case setting, it also depends on the dimensiond of the range space ofS andA. This work was done while consulting at Bell Laboratories, and is partially supported by the National Science Foundation and the Air Force Office of Scientific Research.     

A functional characterization of certain mixed volumes

A function over the convex coneK{inn}, of convex bodiesK in Euclideann-space (where addition is vector addition, positive scalar multiplication is dilatation), which is linear overK{inn}, increasing with respect to set inclusion, and zero at point bodies must be a mixed volumeV(K; đ, p−1;σ ₁, …,σ _n−p). Heređ, takenp−1 times, is inK{inn} andσ ₁, …,σ _n−pare pairwise orthogonal unit segments spanning the orthogonal complement of the affine hull ofđ.     

On the area and perimeter of a random convex hull in a bounded convex set

Suppose K is a compact convex set in ℝ² and X _i , 1≤i≤n, is a random sample of points in the interior of K. Under general assumptions on K and the distribution of the X _i we study the asymptotic properties of certain statistics of the convex hull of the sample. Received: 24 July 1996/Revised version: 24 February 1998     

On Sequential Fixed-Size Confidence Regions for the Mean Vector

In order to construct a fixed-size confidence region for the mean vector of an unknown distribution functionF, a new purely sequential sampling strategy is proposed first. For this new procedure, under some regularity conditions onF, the coverage probability is shown (Theorem 2.1) to be at least (1−α)−Bα²d²+o(d²) asd→0, where (1−α) is the preassigned level of confidence,Bis an appropriate functional ofF, and 2dis the preassigned diameter of the proposed spherical confidence region for the mean vector ofF. An accelerated version of the stopping rule is also provided with the analogous second-order characteristics (Theorem 3.1). In the special case of ap-dimensional normal random variable, analogous purely sequential and accelerated sequential procedures as well as a three-stage procedure are briefly introduced together with their asymptotic second-order characteristics.     

<Emphasis Type="Italic">α</Emphasis>-Conservative approximation for probabilistically constrained convex programs

In this paper, we address an approximate solution of a probabilistically constrained convex program (PCCP), where a convex objective function is minimized over solutions satisfying, with a given probability, convex constraints that are parameterized by random variables. In order to approach to a solution, we set forth a conservative approximation problem by introducing a parameter α which indicates an approximate accuracy, and formulate it as a D.C. optimization problem.     

Semicontinuity and Supremal Representation in the Calculus of Variations

We study the weak* lower semicontinuity properties of functionals of the form

Asymptotics of random contractions

In this paper we discuss the asymptotic behaviour of random contractions X=RS, where R, with distribution function F, is a positive random variable independent of S∈(0,1). Random contractions appear naturally in insurance and finance. Our principal contribution is the derivation of the tail asymptotics of X assuming that F is in the max-domain of attraction of an extreme value distribution and the distribution function of S satisfies a regular variation property. We apply our result to derive the asymptotics of the probability of ruin for a particular discrete-time risk model. Further we quantify in our asymptotic setting the effect of the random scaling on the Conditional Tail Expectations, risk aggregation, and derive the joint asymptotic distribution of linear combinations of random contractions.     

Random points in halfspheres

A random spherical polytope P_n in a spherically convex set as considered here is the spherical convex hull of n independent, uniformly distributed random points in K. The behaviour of P_n for a spherically convex set K contained in an open halfsphere is quite similar to that of a similarly generated random convex polytope in a Euclidean space, but the case when K is a halfsphere is different. This is what we investigate here, establishing the asymptotic behaviour, as n tends to infinity, of the expectation of several characteristics of P_n, such as facet and vertex number, volume and surface area. For the Hausdorff distance from the halfsphere, we obtain also some almost sure asymptotic estimates. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 3–22, 2017     

An extremal property of convex hexagons

The following conjecture is discussed: if K is a plane convex figure and T is a triangle of maximal area contained in K, then K is contained in ?5 \sqrt {5} T. It is shown that it suffices to check the conjecture in the case where K is a convex hexagon, but the conjecture is proved only in the case where K is a pentagon. Bibliography: 2 titles.     

Further results on increasing paths in the one-skeleton of a convex body

Various problems are considered in an attempt to generalize the simplex algorithm of linear programming to a much wider class of convex bodies than the class of convex polytopes. A conjecture of D.G. Larman and C.A. Rogers is disproved by constructing a three-dimensional convex body K with an extreme point e, so that for a certain linear functional f, there are no paths in the one-skeleton of K leading from e, along which f strictly increases. Their conjectured generalization is, however, proved for the large class of three-dimensional convex bodies, all of whose extreme points are exposed.A strong generalization of the simplex algorithm is obtained for the class of all finite-dimensional convex bodies, where, for a given exposed point e of a convex body K, it is possible to find f-strictly-increasing paths in the one-skeleton of K, leading from e, for almost all linear functionals f.Research sponsored by the British Science Research Council.     

Minimizing and Stationary Sequences of Convex Constrained Minimization Problems

In the asymptotic analysis of the minimization problem for a nonsmooth convex function on a closed convex set X in ⁿ, one can consider the corresponding problem of minimizing a smooth convex function F on ⁿ, where F denotes the Moreau–Yosida regularization of f. We study the interrelationship between the minimizing/stationary sequence for f and that for F. An algorithm is given to generate iteratively a possibly unbounded sequence, which is shown to be a minimizing sequence of f under certain regularity and uniform continuity assumptions.     

Products and factors of Banach function spaces

Given two Banach function spaces we study the pointwise product space E · F, especially for the case that the pointwise product of their unit balls is again convex. We then give conditions on when the pointwise product E · M(E, F) = F, where M(E, F) denotes the space of multiplication operators from E into F.