arXiv subjects

Mogens Bladt

Publications and source records attributed to Mogens Bladt.

At least 19 recordsLinked to original sources

arXiv2024

Likelihood estimation for stochastic differential equations with mixed effects

Stochastic differential equations provide a powerful tool for modelling dynamic phenomena affected by random noise. In case of repeated observations of time series for several experimental units, it is often the case that some of the parameters vary between the individual experimental units, which has motivated a considerable interest in stochastic differential equations with mixed effects, where a subset of the parameters are random. These models enable simultaneous representation of randomness in the dynamics and variability between experimental units. When the data are observations at discrete time points, the likelihood function is only rarely explicitly available, so for likelihood-based inference to be feasible, numerical methods are needed. We present Gibbs samplers and stochastic EM-algorithms based on augmented data obtained by the simple method for simulation of diffusion bridges in Bladt and Sørensen (2014). This method is easy to implement and has no tuning parameters. The method is, moreover, computationally efficient at low sampling frequencies because the computing time increases linearly with the time between observations. The algorithms can be extended to models with measurement errors. The Gibbs sampler as well as the EM-algorithm are shown to simplify considerably for exponential families of diffusion processes, including many models used in practice. In a simulation study, the estimation methods are shown to work well for Ornstein-Uhlenbeck processes and t-diffusions with mixed effects. Finally, the Gibbs sampler is applied to neuronal data.

stat.ME↗

arXiv2024

Aggregate Markov models in life insurance: properties and valuation

In multi-state life insurance, an adequate balance between analytic tractability, computational efficiency, and statistical flexibility is of great importance. This might explain the popularity of Markov chain modelling, where matrix analytic methods allow for a comprehensive treatment. Unfortunately, Markov chain modelling is unable to capture duration effects, so this paper presents aggregate Markov models as an alternative. Aggregate Markov models retain most of the analytical tractability of Markov chains, yet are non-Markovian and thus more flexible. Based on an explicit characterization of the fundamental martingales, matrix representations of the expected accumulated cash flows and corresponding prospective reserves are derived for duration-dependent payments with and without incidental policyholder behaviour. Throughout, special attention is given to a semi-Markovian case. Finally, the methods and results are illustrated in a numerical example.

math.PR↗

arXiv2023

Aggregate Markov models in life insurance: estimation via the EM algorithm

In this paper, we consider statistical estimation of time-inhomogeneous aggregate Markov models. Unaggregated models, which corresponds to Markov chains, are commonly used in multi-state life insurance to model the biometric states of an insured. By aggregating microstates to each biometric state, we are able to model dependencies between transitions of the biometric states as well as the distribution of occupancy in these. This allows for non--Markovian modelling in general. Since only paths of the macrostates are observed, we develop an expectation-maximization (EM) algorithm to obtain maximum likelihood estimates of transition intensities on the micro level. Special attention is given to a semi-Markovian case, known as the reset property, which leads to simplified estimation procedures where EM algorithms for inhomogeneous phase-type distributions can be used as building blocks. We provide a numerical example of the latter in combination with piecewise constant transition rates in a three-state disability model with data simulated from a time-inhomogeneous semi-Markov model. Comparisons of our fits with more classic GLM-based fits as well as true and empirical distributions are provided to relate our model with existing models and their tools.

math.ST↗

arXiv2022

Phase-type representations of stochastic interest rates with applications to life insurance

The purpose of the present paper is to incorporate stochastic interest rates into a matrix-approach to multi-state life insurance, where formulas for reserves, moments of future payments and equivalence premiums can be obtained as explicit formulas in terms of product integrals or matrix exponentials. To this end we consider the Markovian interest model, where the rates are piecewise deterministic (or even constant) in the different states of a Markov jump process, and which is shown to integrate naturally into the matrix framework. The discounting factor then becomes the price of a zero-coupon bond which may or may not be correlated with the biometric insurance process. Another nice feature about the Markovian interest model is that the price of the bond coincides with the survival function of a phase-type distributed random variable. This, in particular, allows for calibrating the Markovian interest rate models using a maximum likelihood approach to observed data (prices) or to theoretical models like e.g. a Vasicek model. Due to the denseness of phase-type distributions, we can approximate the price behaviour of any zero-coupon bond with interest rates bounded from below by choosing the number of possible interest rate values sufficiently large. For observed data models with few data points, lower dimensions will usually suffice, while for theoretical models the dimensionality is only a computational issue.

q-fin.RM↗

arXiv2022

Mortality modeling and regression with matrix distributions

In this paper we investigate the flexibility of matrix distributions for the modeling of mortality. Starting from a simple Gompertz law, we show how the introduction of matrix-valued parameters via inhomogeneous phase-type distributions can lead to reasonably accurate and relatively parsimonious models for mortality curves across the entire lifespan. A particular feature of the proposed model framework is that it allows for a more direct interpretation of the implied underlying aging process than some previous approaches. Subsequently, towards applications of the approach for multi-population mortality modeling, we introduce regression via the concept of proportional intensities, which are more flexible than proportional hazard models, and we show that the two classes are asymptotically equivalent. We illustrate how the model parameters can be estimated from data by providing an adapted EM algorithm for which the likelihood increases at each iteration. The practical feasibility and competitiveness of the proposed approach, including the right-censored case, are illustrated by several sets of mortality and survival data.

stat.ME↗

arXiv2022

Estimating absorption time distributions of general Markov jump processes

The estimation of absorption time distributions of Markov jump processes is an important task in various branches of statistics and applied probability. While the time-homogeneous case is classic, the time-inhomogeneous case has recently received increased attention due to its added flexibility and advances in computational power. However, commuting sub-intensity matrices are assumed, which in various cases limits the parsimonious properties of the resulting representation. This paper develops the theory required to solve the general case through maximum likelihood estimation, and in particular, using the expectation-maximization algorithm. A reduction to a piecewise constant intensity matrix function is proposed in order to provide succinct representations, where a parametric linear model binds the intensities together. Practical aspects are discussed and illustrated through the estimation of notoriously demanding theoretical distributions and real data, from the perspective of matrix analytic methods.

math.ST↗

arXiv2021

Continuous scaled phase-type distributions

Products between phase-type distributed random variables and any independent, positive and continuous random variable are studied. Their asymptotic properties are established, and an expectation-maximization algorithm for their effective statistical inference is derived and implemented using real-world datasets. In contrast to discrete scaling studied in earlier literature, in the present continuous case closed-form formulas for various functionals of the resulting distributions are obtained, which facilitates both their analysis and implementation. The resulting mixture distributions are very often heavy-tailed and yet retain various properties of phase-type distributions, such as being dense (in weak convergence) on the set of distributions with positive support.

math.PR↗

arXiv2021

Fluctuation theory for one-sided Lévy processes with a matrix-exponential time horizon

There is an abundance of useful fluctuation identities for one-sided Lévy processes observed up to an independent exponentially distributed time horizon. We show that all the fundamental formulas generalize to time horizons having matrix exponential distributions, and the structure is preserved. Essentially, the positive killing rate is replaced by a matrix with eigenvalues in the right half of the complex plane which, in particular, applies to the positive root of the Laplace exponent and the scale function. Various fundamental properties of thus obtained matrices and functions are established, resulting in an easy to use toolkit. An important application concerns deterministic time horizons which can be well approximated by concentrated matrix exponential distributions. Numerical illustrations are also provided.

math.PR↗

arXiv2021

Multivariate phase-type theory for the site frequency spectrum

Linear functions of the site frequency spectrum (SFS) play a major role for understanding and investigating genetic diversity. Estimators of the mutation rate (e.g. based on the total number of segregating sites or average of the pairwise differences) and tests for neutrality (e.g. Tajima's D) are perhaps the most well-known examples. The distribution of linear functions of the SFS is important for constructing confidence intervals for the estimators, and to determine significance thresholds for neutrality tests. These distributions are often approximated using simulation procedures. In this paper we use multivariate phase-type theory to specify, characterize and calculate the distribution of linear functions of the site frequency spectrum. In particular, we show that many of the classical estimators of the mutation rate are distributed according to a discrete phase-type distribution. Neutrality tests, however, are generally not discrete phase-type distributed. For neutrality tests we derive the probability generating function using continuous multivariate phase-type theory, and numerically invert the function to obtain the distribution. A main result is an analytically tractable formula for the probability generating function of the SFS. Software implementation of the phase-type methodology is available in the R package phasty, and R code for the reproduction of our results is available as an accompanying vignette.

stat.ME↗

arXiv2020

Fitting inhomogeneous phase-type distributions to data: the univariate and the multivariate case

The class of inhomogeneous phase-type distributions (IPH) was recently introduced in Albrecher and Bladt (2019) as an extension of the classical phase-type (PH) distributions. Like PH distributions, the class of IPH is dense in the class of distributions on the positive halfline, but leads to more parsimonious models in the presence of heavy tails. In this paper we propose a fitting procedure for this class to given data. We furthermore consider an analogous extension of Kulkarni's multivariate phase-type class (Kulkarni, 1989) to the inhomogeneous framework and study parameter estimation for the resulting new and flexible class of multivariate distributions. As a by-product, we amend a previously suggested fitting procedure for the homogeneous multivariate phase-type case and provide appropriate adaptations for censored data. The performance of the algorithms is illustrated in several numerical examples, both for simulated and real-life insurance data.

math.ST↗

arXiv2020

Matrix Mittag--Leffler distributions and modeling heavy-tailed risks

In this paper we define the class of matrix Mittag-Leffler distributions and study some of its properties. We show that it can be interpreted as a particular case of an inhomogeneous phase-type distribution with random scaling factor, and alternatively also as the absorption time of a semi-Markov process with Mittag-Leffler distributed interarrival times. We then identify this class and its power transforms as a remarkably parsimonious and versatile family for the modelling of heavy-tailed risks, which overcomes some disadvantages of other approaches like the problem of threshold selection in extreme value theory. We illustrate this point both on simulated data as well as on a set of real-life MTPL insurance data that were modeled differently in the past.

math.ST↗

arXiv2020

Multivariate fractional phase--type distributions

We extend the Kulkarni class of multivariate phase--type distributions in a natural time--fractional way to construct a new class of multivariate distributions with heavy-tailed Mittag-Leffler(ML)-distributed marginals. The approach relies on assigning rewards to a non--Mar\-ko\-vi\-an jump process with ML sojourn times. This new class complements an earlier multivariate ML construction \cite{multiml} and in contrast to the former also allows for tail dependence. We derive properties and characterizations of this class, and work out some special cases that lead to explicit density representations.

math.PR↗

arXiv2020

Multivariate Matrix Mittag--Leffler distributions

We extend the construction principle of multivariate phase-type distributions to establish an analytically tractable class of heavy-tailed multivariate random variables whose marginal distributions are of Mittag-Leffler type with arbitrary index of regular variation. The construction can essentially be seen as allowing a scalar parameter to become matrix-valued. The class of distributions is shown to be dense among all multivariate positive random variables and hence provides a versatile candidate for the modelling of heavy-tailed, but tail-independent, risks in various fields of application.

math.PR↗

arXiv2019

Inhomogeneous phase--type distributions and heavy tails

We extend the construction principle of phase-type (PH) distributions to allow for inhomogeneous transition rates and show that this naturally leads to direct probabilistic descriptions of certain transformations of PH distributions. In particular, the resulting matrix distributions enable to carry over fitting properties of PH distributions to distributions with heavy tails, providing a general modelling framework for heavy-tail phenomena. We also illustrate the versatility and parsimony of the proposed approach for the modelling of a real-world heavy-tailed fire insurance dataset.

math.PR↗

arXiv2019

Matrix calculations for inhomogeneous Markov reward processes, with applications to life insurance and point processes

A multi--state life insurance model is naturally described in terms of the intensity matrix of an underlying (time--inhomogeneous) Markov process which describes the dynamics for the states of an insured person. Between and at transitions, benefits and premiums are paid, defining a payment process, and the technical reserve is defined as the present value of all future payments of the contract. Classical methods for finding the reserve and higher order moments involve the solution of certain differential equations (Thiele and Hattendorf, respectively). In this paper we present an alternative matrix--oriented approach based on general reward considerations for Markov jump processes. The matrix approach provides a general framework for effortlessly setting up general and even complex multi--state models, where moments of all orders are then expressed explicitly in terms of so--called product integrals (matrix--exponentials) of certain matrices. As Thiele and Hattendorf type of theorems can be retrieved immediately from the matrix formulae, this methods also provides a quick and transparent approach to proving these classical results. Methods for obtaining distributions and related properties of interest (e.g. quantiles or survival functions) of the future payments are presented from both a theoretical and practical point of view (via Laplace transforms and methods involving orthogonal polynomials).

math.PR↗

arXiv2018

Phase-type distributions in population genetics

Probability modelling for DNA sequence evolution is well established and provides a rich framework for understanding genetic variation between samples of individuals from one or more populations. We show that both classical and more recent models for coalescence (with or without recombination) can be described in terms of the so-called phase-type theory, where complicated and tedious calculations are circumvented by the use of matrices. The application of phase-type theory consists of describing the stochastic model as a Markov model by appropriately setting up a state space and calculating the corresponding intensity and reward matrices. Formulae of interest are then expressed in terms of these aforementioned matrices. We illustrate this by a few examples calculating the mean, variance and even higher order moments of the site frequency spectrum in the multiple merger coalescent models, and by analysing the mean and variance for the number of segregating sites for multiple samples in the two-locus ancestral recombination graph. We believe that phase-type theory has great potential as a tool for analysing probability models in population genetics. The compact matrix notation is useful for clarification of current models, in particular their formal manipulation (calculation), but also for further development or extensions.

q-bio.PE↗

arXiv2017

Fitting phase--type scale mixtures to heavy--tailed data and distributions

We consider the fitting of heavy tailed data and distribution with a special attention to distributions with a non--standard shape in the "body" of the distribution. To this end we consider a dense class of heavy tailed distributions introduced recently, employing an EM algorithm for the the maximum likelihood estimates of its parameters. We present methods for fitting to observed data, histograms, censored data, as well as to theoretical distributions. Numerical examples are provided with simulated data and a benchmark reinsurance dataset. We empirically demonstrate that our model can provide excellent fits to heavy--tailed data/distributions with minimal assumptions

math.ST↗

arXiv2014

Simulation of multivariate diffusion bridge

We propose simple methods for multivariate diffusion bridge simulation, which plays a fundamental role in simulation-based likelihood and Bayesian inference for stochastic differential equations. By a novel application of classical coupling methods, the new approach generalizes a previously proposed simulation method for one-dimensional bridges to the multi-variate setting. First a method of simulating approximate, but often very accurate, diffusion bridges is proposed. These approximate bridges are used as proposal for easily implementable MCMC algorithms that produce exact diffusion bridges. The new method is much more generally applicable than previous methods. Another advantage is that the new method works well for diffusion bridges in long intervals because the computational complexity of the method is linear in the length of the interval. In a simulation study the new method performs well, and its usefulness is illustrated by an application to Bayesian estimation for the multivariate hyperbolic diffusion model.

math.ST↗