Further Results on the Bivariate Semi-parametric Singular Family of Distributions

General classes of bivariate distributions are well studied in literature. Most of these classes are proposed via a copula formulation or extensions of some characterisation properties in the univariate case. In Kundu(2022) we see one such semi-parametric family useful to model bivariate data with ties. This model is a general semi-parametric model with a baseline. In this paper we present a characterisation property of this class of distributions in terms of a functional equation. The general solution to this equation is explored. Necessary and sufficient conditions under which the solution becomes a bivariate distribution is investigated

Teleconnected warm and cold extremes of North American wintertime temperatures

Current models for spatial extremes are concerned with the joint upper (or lower) tail of the distribution at two or more locations. Such models cannot account for teleconnection patterns of two-meter surface air temperature ($T_{2m}$) in North America, where very low temperatures in the contiguous Unites States (CONUS) may coincide with very high temperatures in Alaska in the wintertime. This dependence between warm and cold extremes motivates the need for a model with opposite-tail dependence in spatial extremes. This work develops a statistical modeling framework which has flexible behavior in all four pairings of high and low extremes at pairs of locations. In particular, we use a mixture of rotations of common Archimedean copulas to capture various combinations of four-corner tail dependence. We study teleconnected $T_{2m}$ extremes using ERA5 reanalysis of daily average two-meter temperature during the boreal winter. The estimated mixture model quantifies the strength of opposite-tail dependence between warm temperatures in Alaska and cold temperatures in the midlatitudes of North America, as well as the reverse pattern. These dependence patterns are shown to correspond to blocked and zonal patterns of mid-tropospheric flow. This analysis extends the classical notion of correlation-based teleconnections to considering dependence in higher quantiles

Multivariate peaks-over-threshold with latent variable representations of generalized Pareto vectors

Generalized Pareto distributions with positive tail index arise from embedding a Gamma random variable for the rate of an exponential distribution. In this paper, we exploit this property to define a flexible and statistically tractable modeling framework for multivariate extremes based on componentwise ratios between any two random vectors with exponential and Gamma marginal distributions. To model multivariate threshold exceedances, we propose hierarchical constructions using a latent random vector with Gamma margins, whose Laplace transform is key to obtaining the multivariate distribution function. The extremal dependence properties of such constructions, covering asymptotic independence and asymptotic dependence, are studied. We detail two useful parametric model classes: the latent Gamma vectors are sums of independent Gamma components in the first construction (called the convolution model), whereas they correspond to chi-squared random vectors in the second construction. Both of these constructions exhibit asymptotic independence, and we further propose a parametric extension (called beta-scaling) to obtain asymptotic dependence. We demonstrate good performance of likelihood-based estimation of extremal dependence summaries for several scenarios through a simulation study for bivariate and trivariate Gamma convolution models, including a hybrid model mixing bivariate subvectors with asymptotic dependence and independence

Hierarchical space-time modeling of asymptotically independent exceedances with an application to precipitation data

<p>The statistical modeling of space-time extremes in environmental applications is key to understanding complex dependence structures in original event data and to generating realistic scenarios for impact models. In this context of high-dimensional data, we propose a novel hierarchical model for high threshold exceedances defined over continuous space and time by embedding a space-time Gamma process convolution for the rate of an exponential variable, leading to asymptotic independence in space and time. Its physically motivated anisotropic dependence structure is based on geometric objects moving through space-time according to a velocity vector. We demonstrate that inference based on weighted pairwise likelihood is fast and accurate. The usefulness of our model is illustrated by an application to hourly precipitation data from a study region in Southern France, where it clearly improves on an alternative censored Gaussian space-time random field model. While classical limit models based on threshold-stability fail to appropriately capture relatively fast joint tail decay rates between asymptotic dependence and classical independence, strong empirical evidence from our application and other recent case studies motivates the use of more realistic asymptotic independence models such as ours.</p

Innovations in Quantitative Risk Management

Quantitative Finance; Game Theory, Economics, Social and Behav. Sciences; Finance/Investment/Banking; Actuarial Science

Some aspects of stochastic modelling

This thesis is mainly concerned with methods for the comparison of distribution functions which depend on the whole of the distribution functions, as against those methods which depend on the comparison of derived statistics. The main methods considered are partial and pre-orderings (Chapter 2) and probability metrics (Chapter 4). Chapter 0 contains a general discussion of mathematical and stochastic modelling. The aim of this chapter is to introduce the concept of the structure of a stochastic model, which is fundamental to the notion of comparability of stochastic models. Chapter 1 is introductory. Chapter 2 considers the comparison of distribution functions via partial orderings. Various partial and pre-orderings are described and their properties studied. In particular, a survey of comparisons for certain common distribution functions is given. In Chapter 3 it is shown how the orderings introduced in Chapter 2 lead to definitions of monotonicity and comparability for stochastic models. Examples are given for Markov chains, martingales, renewal processes and queueing processes. Proofs of results of Daley (1968), Kalmykov (1962) and others on the monotonicity of Markov chains are shown to be elementary. Chapter 4 deals with the use of metrics to compare distribution functions. Some of the more common probability metrics are listed. New bounds on the supremum (uniform) metric are derived for non-negative integer-valued random variables. Probability metrics are fundamental to the notion of the stability of stochastic models, and this is considered in Chapter 5. In Chapter 6 monotonicity properties are used to investigate the behaviour of a class of branching processes which allow for interaction between male and female in the production of offspring. Sufficient conditions for certain extinction and non-certain extinction are given for the general model, whilst for models with superadditive mating functions necessary and sufficient conditions are given for almost sure extinction. Comparison of models which allow for sexual reproduction with ordinary Galton-Watson branching processes shows that, at least for small populations, there is a significant difference in both the probabilities of extinction and the rates of growth. These are investigated numerically using a truncation technique. Theoretical bounds are given for the error in the calculated values of the extinction probabilities