Inaccuracy and information measures based on cumulative residual entropy are quite useful and have received considerable attention in many fields, such as statistics, probability, and reliability theory. In particular, many authors have studied cumulative residual inaccuracy between coherent systems based on system lifetimes. In a previous paper (Bueno and Balakrishnan, Prob. Eng. Inf. Sci. 36, 2022), we discussed a cumulative residual inaccuracy measure for coherent systems at component level, that is, based on the common, stochastically dependent component lifetimes observed under a non-homogeneous Poisson process. In this paper, using a point process martingale approach, we extend this concept to a cumulative residual inaccuracy measure between non-explosive point processes and then specialize the results to Markov occurrence times. If the processes satisfy the proportional risk hazard process property, then the measure determines the Markov chain uniquely. Several examples are presented, including birth-and-death processes and pure birth process, and then the results are applied to coherent systems at component level subject to Markov failure and repair processes.

This article examines the impact of the largest claims reinsurance treaties on loss reserve of the ceding company. The largest claims reinsurance, known as LCR, and ECOMOR reinsurance treaties are considered to be the two most appropriate reinsurance treaties for large or catastrophe claims. Then, it studies the impact of such treaties on loss reserves. Through a simulation study, it shown that, under a more general situation, the LCR treaty can be a more efficient (in some sense, see below) treaty than the ECOMOR treaty for the ceding company.

We consider fragmentation processes with values in the space of marked partitions of $\mathbb{N}$, i.e. partitions where each block is decorated with a nonnegative real number. Assuming that the marks on distinct blocks evolve as independent positive self-similar Markov processes and determine the speed at which their blocks fragment, we get a natural generalization of the self-similar fragmentations of Bertoin (Ann. Inst. H. Poincaré Prob. Statist.38, 2002). Our main result is the characterization of these generalized fragmentation processes: a Lévy–Khinchin representation is obtained, using techniques from positive self-similar Markov processes and from classical fragmentation processes. We then give sufficient conditions for their absorption in finite time to a frozen state, and for the genealogical tree of the process to have finite total length.

We propose a new approach to mortality prediction under survival energy hypothesis (SEH). We assume that a human is born with initial energy, which changes stochastically in time and the human dies when the energy vanishes. Then, the time of death is represented by the first hitting time of the survival energy (SE) process to zero. This study assumes that SE follows a time-inhomogeneous diffusion process and defines the mortality function, which is the first hitting time distribution function of the SE process. Although SEH is a fictitious construct, we illustrate that this assumption has the potential to yield a good parametric family of cumulative probability of death, and the parametric family yields surprisingly good predictions for future mortality rates.

Following the approach of standard filtering theory, we analyse investor valuation of firms, when these are modelled as geometric-Brownian state processes that are privately and partially observed, at random (Poisson) times, by agents. Tasked with disclosing forecast values, agents are able purposefully to withhold their observations; explicit filtering formulae are derived for downgrading the valuations in the absence of disclosures. The analysis is conducted for both a solitary firm and m co-dependent firms.

Let (Xt)0 ≤ t ≤ T be a one-dimensional stochastic process with independent and stationary increments, either in discrete or continuous time. In this paper we consider the problem of stopping the process (Xt) ‘as close as possible’ to its eventual supremum MT := sup0 ≤ t ≤ TXt, when the reward for stopping at time τ ≤ T is a nonincreasing convex function of MT - Xτ. Under fairly general conditions on the process (Xt), it is shown that the optimal stopping time τ takes a trivial form: it is either optimal to stop at time 0 or at time T. For the case of a random walk, the rule τ ≡ T is optimal if the steps of the walk stochastically dominate their opposites, and the rule τ ≡ 0 is optimal if the reverse relationship holds. An analogous result is proved for Lévy processes with finite Lévy measure. The result is then extended to some processes with nonfinite Lévy measure, including stable processes, CGMY processes, and processes whose jump component is of finite variation.

We study Szegö's theorem for a uniform algebra. In particular, we do it for the disc algebra or the bidisc algebra.

Let (Bt)0≤t≤T be either a Bernoulli random walk or a Brownian motion with drift, and let Mt := max{Bs: 0 ≤ s ≤ t}, 0 ≤ t ≤ T. In this paper we solve the general optimal prediction problem sup0≤τ≤TE[f(MT − Bτ], where the supremum is over all stopping times τ adapted to the natural filtration of (Bt) and f is a nonincreasing convex function. The optimal stopping time τ* is shown to be of ‘bang-bang’ type: τ* ≡ 0 if the drift of the underlying process (Bt) is negative and τ* ≡ T if the drift is positive. This result generalizes recent findings of Toit and Peskir (2009) and Yam, Yung and Zhou (2009), and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good stocks as long as possible.

We consider a Poisson cluster model, motivated by insurance applications. At each claim arrival time, modeled by the point of a homogeneous Poisson process, we start a cluster process which represents the number or amount of payments triggered by the arrival of a claim in a portfolio. The cluster process is a Lévy or truncated compound Poisson process. Given the observations of the process over a finite interval, we consider the expected value of the number and amount of payments in a future time interval. We also give bounds for the error encountered in this prediction procedure.

This paper studies limiting properties of discretely sampled Poisson shot noise processes. Versions of the law of large numbers and central limit theorem are derived under very general conditions. Examples demonstrating the utility of the results are included.

The entropy score of an observed outcome that has been given a probability forecast p is defined to be –log p. If p is derived from a probability model and there is a background model for which the same outcome has probability π, then the log ratio log(p/π) is the probability gain, and its expected value the information gain, for that outcome. Such concepts are closely related to the likelihood of the model and its entropy rate. The relationships between these concepts are explored in the case that the outcomes in question are the occurrence or nonoccurrence of events in a stochastic point process. It is shown that, in such a context, the mean information gain per unit time, based on forecasts made at arbitrary discrete time intervals, is bounded above by the entropy rate of the point process. Two examples illustrate how the information gain may be related to realizations with a range of values of ‘predictability'.

Lamperti's transformation, an isometry between self-similar and stationary processes, is used to solve some problems of linear estimation of continuous-time, self-similar processes. These problems include causal whitening and innovations representations on the positive real line, as well as prediction from certain finite and semi-infinite intervals. The method is applied to the specific case of fractional Brownian motion (FBM), yielding alternate derivations of known prediction results, along with some novel whitening and interpolation formulae. Some associated insights into the problem of discrete prediction are also explored. Closed-form expressions for the spectra and spectral factorization of the stationary processes associated with the FBM are obtained as part of this development.

In, Tory and Pickard show that a simple subclass of unilateral AR processes identifies with Gaussian Pickard random fields on Z2. First, we extend this result to the whole class of unilateral AR processes, by showing that they all satisfy a Pickard-type property, under which correlation matching and maximum entropy properties are assessed. Then, it is established that the Pickard property provides the ‘missing’ equations that complement the two-dimensional Yule-Walker equations, in the sense that the conjunction defines a one-to-one mapping between the set of AR parameters and a set of correlations. It also implies Markov chain conditions that allow exact evaluation of the likelihood and an exact sampling scheme on finite lattices.

The linear inverse problem of estimating the input random field in a first-kind stochastic integral equation relating two random fields is considered. For a wide class of integral operators, which includes the positive rational functions of a self-adjoint elliptic differential operator on L2(ℝd), the ill-posed nature of the problem disappears when such operators are defined between appropriate fractional Sobolev spaces. In this paper, we exploit this fact to reconstruct the input random field from the orthogonal expansion (i.e. with uncorrelated coefficients) derived for the output random field in terms of wavelet bases, transformed by a linear operator factorizing the output covariance operator. More specifically, conditions under which the direct orthogonal expansion of the output random field coincides with the integral transformation of the orthogonal expansion derived for the input random field, in terms of an orthonormal wavelet basis, are studied.

Prediction of future values of a shot noise process observed on a discrete lattice of points is considered. The shot magnitudes are assumed to be independent and identically distributed and to arrive via a Poisson process; the effect of each shot dissipates and/or accumulates according to a known shot function. Conditional mean and linear point predictors of future process values are developed. Distributional prediction, obtained through saddlepoint approximation of the conditional distributions of the process, is also explored.

We consider the inverse problem of estimating the input random field in a stochastic integral equation relating two random fields. The purpose of this paper is to present an approach to this problem using a Riesz-based or orthonormal-based series expansion of the input random field with uncorrelated random coefficients. We establish conditions under which the input series expansion induces (via the integral equation) a Riesz-based or orthonormal-based series expansion for the output random field. The estimation problem is studied considering two cases, depending on whether data are available from either the output random field alone, or from both the input and output random fields. Finally, we discuss this approach in the case of transmissivity estimation from piezometric head data, which was the original motivation of this work.

A random vibration model is investigated in this paper. The model is formulated as a cosine function with a constant frequency and a random walk phase. We show that this model is second-order stationary and can be rewritten as a vector-valued AR(1) model as well as a scalar ARMA(2, 1) model. The linear innovation sequence of the AR(1) model is shown to be a martingale difference sequence while the linear innovation sequence of the ARMA(2, 1) model is only an uncorrelated sequence. A non-linear predictor is derived from the AR(1) model while a linear predictor is derived from the ARMA(2, 1) model. We deduce that the non-linear predictor of this model has less mean square error than that of the linear predictor. This has significance, for example, for predicting seasonal phenomena with this model. In addition, the limit distributions of the sample mean, the finite Fourier transforms and the autocovariance functions are derived using a martingale approach. The limit distribution of autocovariance functions differs from the classical result given by Bartlett's formula.

Integration with respect to the fractional Brownian motion Z with Hurst parameter is discussed. The predictor is represented as an integral with respect to Z, solving a weakly singular integral equation for the prediction weight function.

Let C be a separable Hilbert Space, and let Λ be the halfplane {(m, n) ∈ Ζ2 : m ≥ 1} ∪ {(0, n) ∈ Ζ2 : n ≥ 0} of the integer lattice. Consider the subspace ℳc(Λ) of on the torus spanned by the C-valued trigonometric functions {Ceims+int : с ∈ C, (m, n) ∈ Λ}. The notion of a Λ-analytic operator on ℳc(Λ) is defined with respect to the family of shift operators {Smn}Λ on ℳC(Λ) given by (Smnƒ)(eis, eit) = eims+intƒ(eis, eit). The corresponding concepts of inner function, outer function and analytic range function are explored. These ideas are applied to the spectral factorization problem in prediction theory.

Our aim in this article is to derive an expression for the best linear predictor of a multivariate symmetric α stable process based on many past values. For this purpose we introduce a definition of dispersion for symmetric α stable random vectors and choose the linear predictor which minimizes the dispersion of the error vector.