Non-parametric estimation from simultaneous renewal–failure–degradation data with competing risks

A joint model for linear degradation and competing failure data with partial renewals is proposed. Non-parametric estimation procedures for failure intensities and failure probabilities as functions of degradation level are given. Asymptotic properties of the estimators are investigated. To cite this article: V. Bagdonavi?ius et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Approximations of some hazard rate estimators in a competing risks model

A general model involving k competing risks is studied and the hazard rates of these risks are simultaneously estimated. The estimators are strongly approximated by Gaussian processes and the limiting distribution of certain statistics are obtained.     

Likelihood-based inference for bivariate latent failure time models with competing risks under the generalized FGM copula

Many existing latent failure time models for competing risks do not provide closed form expressions of sub-distribution functions. This paper suggests a generalized FGM copula models with the Burr III failure time distribution such that the sub-distribution functions have closed form expressions. Under the suggested model, we develop a likelihood-based inference method along with its computational tools and asymptotic theory. Based on the expressions of the sub-distribution functions, we propose goodness-of-fit tests. Simulations are conducted to examine the performance of the proposed methods. A real data from the reliability analysis of the radio transmitter-receivers are analyzed to illustrate the proposed methods. The computational programs are made available in the R package GFGM.copula.     

Dependent competing risks: Cause elimination and its impact on survival

The dependent competing risks model of human mortality is considered, assuming that the dependence between lifetimes is modelled by a multivariate copula function. The effect on the overall survival of removing one or more causes of death is explored under two alternative definitions of removal, ignoring the causes and eliminating them. Under the two definitions of removal, expressions for the overall survival functions in terms of the specified copula (density) and the net (marginal) survival functions are given. The net survival functions are obtained as a solution to a system of non-linear differential equations, which relates them through the specified copula (derivatives) to the crude (sub-) survival functions, estimated from data. The overall survival functions in a model with four competing risks, cancer, cardiovascular diseases, respiratory diseases and all other causes grouped together, have been implemented and evaluated, based on cause-specific mortality data for England and Wales published by the Office for National Statistics, for the year 2007. We show that the two alternative definitions of removal of a cause of death have different effects on the overall survival and in particular on the life expectancy at birth and at age 65, when one, two or three of the competing causes are removed. An important conclusion is that the eliminating definition is better suited for practical use in competing risks’ applications, since it is more intuitive, and it suffices to consider only positive dependence between the lifetimes which is not the case under the alternative ignoring definition.     

Maximum likelihood estimation of a survival function under the koziol-green proportional hazards model

Let X and Y be two independent competing lifetimes wih survival functions S_F(t) and S_G(t). In this paper we, study a proportional-hazards competing risks model characterized by Armitage (1959): S_G(t) = S_G(t)]^β for certain positive constant β. The maximum likelihood estimator for S_F is proposed, examined and compared with the product limit estimator of Kaplan and Meier (1958). The MLE is shown to be asymptotically more efficient that the PLE under the specified model.     

The concept lattice functors

This paper is concerned with the relationship between contexts, closure spaces, and complete lattices. It is shown that, for a unital quantale L, both formal concept lattices and property oriented concept lattices are functorial from the category L-Ctx of L-contexts and infomorphisms to the category L-Sup of complete L-lattices and suprema-preserving maps. Moreover, the formal concept lattice functor can be written as the composition of a right adjoint functor from L-Ctx to the category L-Cls of L-closure spaces and continuous functions and a left adjoint functor from L-Cls to L-Sup.     

Error-free computer solution of certain systems of linear equations

In this article we propose a procedure which generates the exact solution for the system Ax = b, where A is an integral nonsingular matrix and b is an integral vector, by improving the initial floating-point approximation to the solution. This procedure, based on an easily programmed method proposed by Aberth [1], first computes the approximate floating-point solution x^* by using an available linear equation solving algorithm. Then it extracts the exact solution x from x^* if the error in the approximation x^* is sufficiently small. An a posteriori upper bound for the error of x^* is derived when Gaussian Elimination with partial pivoting is used. Also, a computable upper bound for |det(A)|, which is an alternative to using Hadamard's inequality, is obtained as a byproduct of the Gaussian Elimination process.     

Some inverse problems involving conditional expectations

Let (Ω, F, P) be a probability space, let H be a sub-σ-algebra of F, and let Y be positive and H-measurable with E[Y] = 1. We discuss the structure of the convex set CE(Y; H) = {X ∈ pF: Y = E[X|H]} of random variables whose conditional expectation given H is the prescribed Y. Several characterizations of extreme points of CE(Y; H) are obtained. A necessary and sufficient condition is given in order that CE(Y; H) be the closed, convex hull of its extreme points. For the case of finite F we explicitly calculate the extreme points of CE(Y; H), identify pairs of adjacent extreme points, and characterize extreme points of CE(Y; H) ? CE(Z; G), where G is a second sub-σ-algebra of F and Z ∈ pG. When H = σ(Y) and appropriate topological hypotheses hold, extreme points of CE(Y; H) are shown to be in explicit one-to-one correspondence with certain left inverses of Y. Finally, it is shown how the same approach can be applied to the problem of extremal random measures on $\text{R}$₊ with a prescribed compensator, to deduce that the number of extreme points is zero or one.     

Local times for stochastic processes which are subordinate to Gaussian processes

Let X and Y be random vectors of the same dimension such that Y has a normal distribution with mean vector O and covariance matrix R. Let g(x), x≥0, be a bounded nonincreasing function. X is said to be g-subordinate to Y if |Ee^iu′X| ≤ g(u′Ru) for all real vectors u of the same dimension as X. This is used to define the g-subordination of a real stochastic process X(t), 0 ≤ t ≤ 1, to a Gaussian process Y(t), 0 ≤ t ≤ 1. It is shown that the basic local time properties of a given Gaussian process are shared by all the processes that age g-subordinate to it. It is shown in particular that certain random series, including some random Fourier series, are g-subordinate to Gaussian processes, and so have their local time properties.     

A computational method for the indefinite quadratic programming problem

We present an algorithm for the quadratic programming problem of determining a local minimum of ?(x)=$\text{1}\text{2}$x^TQx+c^Tx such that A^Tx?b where Q ymmetric matrix which may not be positive definite. Our method combines the active constraint strategy of Murray with the Bunch-Kaufman algorithm for the stable decomposition of a symmetric matrix. Under the active constraint strategy one solves a sequence of equality constrained problems, the equality constraints being chosen from the inequality constraints defining the original problem. The sequence is chosen so that ?(x) continues to decrease and x remains feasible. Each equality constrained subproblem requires the solution of a linear system with the projected Hessian matrix, which is symmetric but not necessarily positive definite. The Bunch-Kaufman algorithm computes a decomposition which facilitates the stable determination of the solution to the linear system. The heart of this paper is a set of algorithms for updating the decomposition as the method progresses through the sequence of equality constrained problems. The algorithm has been implemented in a FORTRAN program, and a numerical example is given.     

Non-parametric estimation of lifetime and repair time criteria for a semi-Markov process

In this Note, we model an industrial system by a semi-Markov process where failure and repair phenomena are in mutual competition. A non-parametric estimation method for system component lifetime and repair time distributions and for associated hazard rate functions is proposed. The lifetime and repair time empirical distributions are reduced to two Kaplan–Meier estimators. A numerical example from an industrial system with three components and one repair man modeled by a birth and death process is provided to illustrate the previous results. To cite this article: A.-L. Afchain, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

A topological property enjoyed by near points but not by large points

Let H denote the halfline [0,∞). A point p?βH?H is called a near point if p is in the closure of some countable discrete closed subspace of H. In addition, a point p?βH?H is called a large point if p is not in the closure of a closed subset of H of finite Lebesgue measure. We will show that for every autohomeomorphism ? of βH?H and for each near point p we have that ?(p) is not large. In addition, we establish, under CH, the existence of a point x?βH?H such that for each autohomeomorphism ? of βH?H the point ?(x) is neither large nor near.     

Properties of Euclidean and non-Euclidean distance matrices

A distance matrix D of order n is symmetric with elements $\text{?}\text{1}\text{2}\text{d}{}_{\mathrm{ij}}{}^2$, where d_ii=0. D is Euclidean when the $\text{1}\text{2}\text{n(n?1)}$ quantities d_ij can be generated as the distances between a set of n points, X (n×p), in a Euclidean space of dimension p. The dimensionality of D is defined as the least value of p=rank(X) of any generating X; in general p+1 and p+2 are also acceptable but may include imaginary coordinates, even when D is Euclidean. Basic properties of Euclidean distance matrices are established; in particular, when ρ=rank(D) it is shown that, depending on whether e^TD^?e is not or is zero, the generating points lie in either p=ρ?1 dimensions, in which case they lie on a hypersphere, or in p=ρ?2 dimensions, in which case they do not. (The notation e is used for a vector all of whose values are one.) When D is non-Euclidean its dimensionality p=r+s will comprise r real and s imaginary columns of X, and (r, s) are invariant for all generating X of minimal rank. Higher-ranking representations can arise only from p+1=(r+1)+s or p+1=r+ (s+1) or p+2=(r+1)+(s+1), so that not only are r, s invariant, but they are both minimal for all admissible representations X.     

On the existence of free topological groups

Given a Tychonoff space X and classes U and V of topological groups, we say that a topological group G = G(X, U, V) is a free (U,V)-group over X if (a) X is a subspace of G, (b) G ϵ U, and (c) every continuous f: X → H with H ϵ V extends uniquely to a continuous homomorphism f̄: G→H. For certain classes U and V, we consider the question of the existence of free (U,V)- groups. Our principal results are the following. Let PA and CA denote, respectively, the class ofpseudocompact Abelian groups and the class of compact Abelian groups. Then

• 1.(a) there is a free (PA,PA)-group over X iff; X=Ø and
• 2.(b) there is for each X a free (PA,CA)-group over X in which X is closed.

     

Matricial logarithmic derivatives

If θ is a norm on Cⁿ, then the mapping $\text{A→}\text{lim}{}_{\mathrm{h\downarrow 0}}\text{6I+hA6}{}_{\mathrm{\theta }}\text{?1}\text{/h}$ from M_n(C) (=C^{n × n}) into R is called the logarithmic derivative induced by the vector norm θ. In this paper we generalize this concept to a mapping γ from M_n(C) into M_k(R), where k ? n. Denoting by α(B) the spectral abscissa of a square matrix B (the largest of the real parts of the eigenvalues), we show, in particular, that α(A) ?α(γ(A)). As a byproduct we obtain simple sufficient conditions for the stability of a matrix.     

Borch’s theorem,equal margins,and efficient allocation

The economic concept of margin guides or justifies the sharing of risks and resources. Broadly, by Borch’s theorem, competing claimants, ends or users ought see equal margins along any efficient allocation.However helpful this maxim, its application is often hampered, and occasionally misguided, by concerns with the differentiability of objectives—or with the interiority of solutions. Circumventing such concerns, this paper introduces a quite applicable, generalized notion, called essential margin.Presuming transferable or quasi-linear utility, the coincidence of such margins supports efficiency, competitive equilibria, and core solutions. The said coincidence also defines deductibles and prioritized claims, seen in finance and insurance.     

Construction d’entrelacs hyperpoliques et arithmétiques

In this Note, we continue the study started in [4] about arithmetic hyperbolic links L such that S³\L is homeomorphic to H³/Γ, where Γ is not conjugate to a subgroup of any Bianchi group. One describes the steps of the construction, different from the construction in [4], which permits us to determine the first known examples in M₂(Q (i√39)) and M₂(Q (i√6 respectively.     

Non-parametric estimation from simultaneous degradation and failure time dataEstimation non-paramétrique à partir de données de dégradation et de défaillances

The paper considers degradation and failure time models with multiple failure modes. Dependence of traumatic failure intensities on the degradation level are included into the models. Non-parametric estimators of various reliability characteristics are proposed. Theorems on simultaneous asymptotic distribution of random functions characterizing degradation and intensities of traumatic events are proposed. To cite this article: V. Bagdonavi?ius et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 183–188.     

ε-Forced adaptive policy in age replacement

We consider the problem of minimizing the long-run average cost per unit time in age replacement with an unknown failure distribution. Utilizing the idea of ε-forced choices, which may be thought of as a modification of forced choice circles given by Fox and Rolph (Adaptive policies for Markov renewal programs, Ann. Statist.50 (1973), 334–341), we construct the adaptive policy which has a non-parametrically good property and is Bayes against a Dirichlet process prior.     

Separation properties in algebraic categories of topological spaces

Full subcategories C ? Top of the category of topological spaces, which are algebraic over Set in the sense of Herrlich [2], have pleasant separation properties, mostly subject to additional closedness assumptions. For instance, every C-object is a T₁-space, if the two-element discrete space belongs to C. Moreover, if C is closed under the formation of finite powers in Top and even varietal [2], then every C-object is Hausdorff. Hence, the T₂-axiom turns out to be (nearly) superfluous in Herrlich's and Strecker's characterization of the category of compact Hausdorff spaces [1], although it is essential for the proof.If we think of C-objects X as universal algebras (with possibly infinite operations), then the subalgebras of X form the closed sets of a compact topology on X, provided that the ordinal spaces [0, β] belong to C. This generalizes a result in [3]. The subalgebra topology is used to prove criterions for the Hausdorffness of every space in C, if C is only algebraic.