Volume 11, Issue 1

Acyclic phase-type (PH) distributions have been a popular tool in survival analysis, thanks to their natural interpretation in terms of aging toward its inevitable absorption. In this article, we consider an extension to the bivariate setting for the modeling of joint lifetimes. In contrast to previous models in the literature that were based on a separate estimation of the marginal behavior and the dependence structure through a copula, we propose a new time-inhomogeneous version of a multivariate PH (mIPH) class that leads to a model for joint lifetimes without that separation. We study properties of mIPH class members and provide an adapted estimation procedure that allows for right-censoring and covariate information. We show that initial distribution vectors in our construction can be tailored to reflect the dependence of the random variables and use multinomial regression to determine the influence of covariates on starting probabilities. Moreover, we highlight the flexibility and parsimony, in terms of needed phases, introduced by the time inhomogeneity. Numerical illustrations are given for the data set of joint lifetimes of Frees et al., where 10 phases turn out to be sufficient for a reasonable fitting performance. As a by-product, the proposed approach enables a natural causal interpretation of the association in the aging mechanism of joint lifetimes that goes beyond a statistical fit.

This article establishes general conditions for posterior consistency of Bayesian finite mixture models with a prior on the number of components. That is, we provide sufficient conditions under which the posterior concentrates on neighborhoods of the true parameter values when the data are generated from a finite mixture over the assumed family of component distributions. Specifically, we establish almost sure consistency for the number of components, the mixture weights, and the component parameters, up to a permutation of the component labels. The approach taken here is based on Doob’s theorem, which has the advantage of holding under extraordinarily general conditions, and the disadvantage of only guaranteeing consistency at a set of parameter values that has probability one under the prior. However, we show that in fact, for commonly used choices of prior, this yields consistency at Lebesgue-almost all parameter values, which is satisfactory for most practical purposes. We aim to formulate the results in a way that maximizes clarity, generality, and ease of use.

This article proposes a framework to model the mutual volatility transmission between multiple assets and multiple trading places in different time zones. The model is estimated using a dataset containing daily returns of three stock indices – the MSCI Japan, the EuroStoxx 50, and the S&P 500 – each traded at three major trading places: the stock exchanges in Tokyo, London, and New York. Strong volatility transmission effects can be observed between New York and Tokyo, whereas current volatility in New York mostly depends on past volatility in New York. For the assets in consideration, spillovers are strong across trading zones, but weak across assets, suggesting a close connection between market places but only a loose volatility link between international stock indices. Volatility impulse response functions indicate a long-lasting and comparably large response of volatility in Tokyo, whereas they suggest a quicker volatility decay in London and New York.

Functions f f on [ 0 , 1 ] m {\left[0,1]}^{m} such that every composition f ∘ ( g 1 , … , g m ) f\circ \left({g}_{1},\ldots ,{g}_{m}) with d d -dimensional distribution functions g 1 , … , g m {g}_{1},\ldots ,{g}_{m} is again a distribution function, turn out to be characterized by a very natural monotonicity condition, which for d = 2 d=2 means ultramodularity. For m = 1 m=1 (and d = 2 d=2 ), this is equivalent with increasing convexity.

For probability measures μ , ν \mu ,\nu , and ρ \rho , define the cost functionals C ( μ , ρ ) ≔ sup π ∈ Π ( μ , ρ ) ∫ ⟨ x , y ⟩ π ( d x , d y ) and C ( ν , ρ ) ≔ sup π ∈ Π ( ν , ρ ) ∫ ⟨ x , y ⟩ π ( d x , d y ) , C\left(\mu ,\rho ):= \mathop{\sup }\limits_{\pi \in \Pi \left(\mu ,\rho )}\int \langle x,y\rangle \pi \left({\rm{d}}x,{\rm{d}}y)\hspace{1.0em}{\rm{and}}\hspace{1em}C\left(\nu ,\rho ):= \mathop{\sup }\limits_{\pi \in \Pi \left(\nu ,\rho )}\int \langle x,y\rangle \pi \left({\rm{d}}x,{\rm{d}}y), where ⟨ ⋅ , ⋅ ⟩ \langle \cdot ,\cdot \rangle denotes the scalar product and Π ( ⋅ , ⋅ ) \Pi \left(\cdot ,\cdot ) is the set of couplings. We show that two probability measures μ \mu and ν \nu on R d {{\mathbb{R}}}^{d} with finite first moments are in convex order (i.e., μ ≼ c ν \mu {\preccurlyeq }_{c}\nu ) iff C ( μ , ρ ) ≤ C ( ν , ρ ) C\left(\mu ,\rho )\le C\left(\nu ,\rho ) holds for all probability measures ρ \rho on R d {{\mathbb{R}}}^{d} with bounded support. This generalizes a result by Carlier. Our proof relies on a quantitative bound for the infimum of ∫ f d ν − ∫ f d μ \int f{\rm{d}}\nu -\int f{\rm{d}}\mu over all 1-Lipschitz functions f f , which is obtained through optimal transport (OT) duality and the characterization result of OT (couplings) by Rüschendorf, by Rachev, and by Brenier. Building on this result, we derive new proofs of well known one-dimensional characterizations of convex order. We also describe new computational methods for investigating convex order and applications to model-independent arbitrage strategies in mathematical finance.

The probability integral transform of a continuous random variable X X with distribution function F X {F}_{X} is a uniformly distributed random variable U = F X ( X ) U={F}_{X}\left(X) . We define the angular probability integral transform (APIT) as θ U = 2 π U = 2 π F X ( X ) {\theta }_{U}=2\pi U=2\pi {F}_{X}\left(X) , which corresponds to a uniformly distributed angle on the unit circle. For circular (angular) random variables, the sum modulus 2 π \pi of absolutely continuous independent circular uniform random variables is a circular uniform random variable, that is, the circular uniform distribution is closed under summation modulus 2 π \pi , and it is a stable continuous distribution on the unit circle. If we consider the sum (difference) of the APITs of two random variables, X 1 {X}_{1} and X 2 {X}_{2} , and test for the circular uniformity of their sum (difference) modulus 2 π \pi , this is equivalent to test of independence of the original variables. In this study, we used a flexible family of nonnegative trigonometric sums (NNTS) circular distributions, which include the uniform circular distribution as a member of the family, to evaluate the power of the proposed independence test by generating samples from NNTS alternative distributions that could be at a closer proximity with respect to the circular uniform null distribution.

Working with shuffles, we establish a close link between Kendall’s τ \tau , the so-called length measure, and the surface area of bivariate copulas and derive some consequences. While it is well known that Spearman’s ρ \rho of a bivariate copula A A is a rescaled version of the volume of the area under the graph of A A , in this contribution we show that the other famous concordance measure, Kendall’s τ \tau , allows for a simple geometric interpretation as well – it is inextricably linked to the surface area of A A .

This review discusses methods of testing for explosive bubbles in time series. A large number of recently developed testing methods under various assumptions about innovation of errors are covered. The review also considers the methods for dating explosive (bubble) regimes. Special attention is devoted to time-varying volatility in the errors. Moreover, the modelling of possible relationships between time series with explosive regimes is discussed.

A family of bivariate copulas given by: for v + 2 u < 2 v+2u\lt 2 , C ( u , v ) = F ( 2 F − 1 ( v ∕ 2 ) + F − 1 ( u ) ) C\left(u,v)=F\left(2{F}^{-1}\left(v/2)+{F}^{-1}\left(u)) , where F F is a strictly increasing cumulative distribution function of a symmetric, continuous random variable, and for v + 2 u ≥ 2 v+2u\ge 2 , C ( u , v ) = u + v − 1 C\left(u,v)=u+v-1 , is introduced. The basic properties and necessary conditions for absolute continuity of C C are discussed. Several examples are provided.

Copulas C C for which ( C t C ) 2 = C t C {({C}^{t}C)}^{2}={C}^{t}C are called pre-idempotent copulas, of which well-studied examples are idempotent copulas and complete dependence copulas. As such, we shall work mainly with the topology induced by the modified Sobolev norm, with respect to which the class ℛ {\mathcal{ {\mathcal R} }} of pre-idempotent copulas is closed and the class of factorizable copulas is a dense subset of ℛ {\mathcal{ {\mathcal R} }} . Identifying copulas with Markov operators on L 2 {L}^{2} , the one-to-one correspondence between pre-idempotent copulas and partial isometries is one of our main tools. In the same spirit as Darsow and Olsen’s work on idempotent copulas, we obtain an explicit characterization of pre-idempotent copulas, which is split into cases according to the atomicity of its associated σ \sigma -algebras, where the nonatomic case gives all factorizable copulas and the totally atomic case yields conjugates of ordinal sums of copies of the product copula.

The central mathematical tool discussed is a non-standard family of polynomials, univariate and bivariate, called Abel-Goncharoff polynomials. First, we briefly summarize the main properties of this family of polynomials obtained in the previous work. Then, we extend the remarkable links existing between these polynomials and the parking functions which are a classic object in combinatorics and computer science. Finally, we use the polynomials to determine the non-ruin probabilities over a finite horizon for a bivariate risk process, in discrete and continuous time, assuming that claim amounts are dependent via a partial Schur-constancy property.

In this article, we propose an omnibus test for comparing two survival functions under non-proportional hazards. The test statistic is based on a product-limit estimate of the restricted distance correlation, which is closely related to the L 2 {L}_{2} distance between survival curves. The strong consistency is established under mild regularity conditions. Our simulation studies show that the new test has satisfactory power under proportional hazard and various non-proportional hazards settings including delayed treatment effect, diminishing effect, and crossing survival curves; therefore, it can be a competitive alternative to the existing omnibus tests such as Kolmogorov-Smirnov test, Cramer-von Mises test, two-stage test, and the maxCombo test based on weighted log-rank statistics. Two extensions of the new test to one-sided alternatives and a Gaussian kernel are also discussed.

The article provides construction algorithms for consistent model classes of continuous spherical and elliptical distributions. The algorithms are based on characterization theorems for consistent families of generator functions. This characterization uses the term of complete monotonicity. The algorithms are applied to several examples of generators leading to consistent families of generators with explicit formulas for marginal densities of arbitrary dimension.