A nonparametric estimation procedure for bivariate extreme value copulas

• Proposes the Multimodal Bivariate Extreme Value (MMBEV) distribution. • Extends bivariate extreme value theory to capture multimodal dependencies. • Applies MMBEV to model extreme climate data with complex dependencies. • Demonstrates MMBEV’s flexibility through simulations and real-world analysis. • Establishes links between MMBEV and bivariate Weibull distributions. Modeling multivariate extreme values in complex systems increasingly demands flexible distributions capable of capturing multimodal behavior. Classical bivariate extreme value (BEV) distributions, though well established, are inherently unimodal and may inadequately represent data with heterogeneity or regime-switching—features commonly observed in environmental and financial applications. This paper introduces the Multimodal Bivariate Extreme Value (MBEV) distribution, a new class that extends the traditional BEV framework to accommodate multimodal structures through additional shape parameters. Unlike previous generalizations restricted to specific cases such as the bivariate Gumbel model, the MBEV encompasses the entire BEV family. We explore theoretical properties of the MBEV model, including its stochastic representation and its connection to the bivariate Weibull distribution. Through Monte Carlo simulations, we assess the performance of maximum likelihood estimators and demonstrate their robustness. An application to climate data from Brasília, Brazil, illustrates the practical value of the MBEV model in capturing complex dependencies among extreme variables such as wind gust speed, relative humidity, and dew point temperature. Model selection criteria (AIC and BIC) confirm the superiority of the MBEV model over classical BEV distributions. Overall, the MBEV model offers a flexible and interpretable framework for modeling bivariate extremes, with potential applications in climatology, survival analysis, reliability, and finance. This work advances the frontier of extreme value modeling by addressing multimodal dependence in heterogeneous multivariate contexts.

A test when the Fisher information may be infinite, exemplified by a test for marginal independence in extreme value distributions

A polynomial model for bivariate extreme value distributions

Maximum likelihood estimation of dependence parameter using ranked set sampling

SUMMARY Bivariate extreme value distributions arise as the limiting distributions of renormalized componentwise maxima. No natural parametric family exists for the dependence between the marginal distributions, but there are considerable restrictions on the dependence structure. We consider modelling the dependence function with parametric models, for which two new models are presented. Tests for independence, and discriminating between models, are also given. The estimation procedure, and the flexibility of the new models, are illustrated with an application to sea level data. Extreme value theory has recently been an area of much theoretical and practical work. Univariate theory is a well documented area, whereas bivariate/multivariate extreme value theory has, until recently, received surprisingly little attention. In the multivariate case, no natural parametric family exists for the dependence structure, so this must be modelled in some way. In the analysis of environmental extreme value data, there is a need for models of dependence between extremes from different sources: for example at various sea ports, or at various -points of a river. In this paper we consider bivariate extreme value distributions. We assume, without loss of generality, that we have exponential marginal distributions with unit means. The class of bivariate exponential distributions in which we are interested, satisfy a strong stability relation. Exponential variables (X, Y) satisfy the stability relation if and only if W = min (aX, bY) is also exponentially distributed for all a, b > 0 (Pickands, 1981). Therefore, the models we will consider have particular application in reliability and survival analysis. One approach to modelling the dependence structure is via parametric models. This requires a flexible family of models which satisfy certain constraints. Models are of two kinds: either differentiable, or nondifferentiable. All nondifferentiable models give distri- butions which are singular, with nonzero probability concentrated on a certain subspace. The differentiable models have densities, but the existing models are symmetric which leads to the variables being exchangeable. Here, we present two new asymmetric differenti- able models, which have increased flexibility. Properties of the differentiable models are examined. Estimation of the parametric models has previously been by ad hoc methods, because there is a nonregular estimation problem when the margins are independent. For the

We approximate the distribution of the TCP-flow rate by deriving it from the joint bivariate distribution of the flow sizes and flow durations of a given access network. The latter distribution is represented by a bivariate extreme value distribution using the Pickand’s dependence A-function. We estimate the A-function to measure the dependencies of random pairs: TCP-flow size and duration, the rate of TCP-flow and size, as well as the rate and duration. We provide a method to test that the achieved estimate of A-function is good and perform the analysis with one concrete data example.

This paper addresses two central problems in simulation methodology: the generation of confidence intervals for the steady state means of the output sequences and the sequential use of these confidence intervals to control the run length. The variance of the sample mean of a covariance stationary process is given approximately by p(0)/N, where p(f) is the spectral density at frequency f and N is the sample size. In an earlier paper we developed a method of confidence interval generation based on the estimation of p(0) through the least squares fit of a quadratic to the logarithm of the periodogram. This method was applied in a run length control procedure to a sequence of batched means. As the run length increased the batch means were rebatched into larger batch sizes so as to limit storage requirements. In this rebatching the shape of the spectral density changes, gradually becoming flat as N increases. Quadratics were chosen as a compromise between small sample bias and large sample stability. In this paper we consider smoothing techniques which adapt to the changing spectral shape in an attempt to improve both the small and large sample behavior of the method. The techniques considered are polynomial smoothing with the degree selected sequentially using standard regression statistics, polynomial smoothing with the degree selected by cross validation, and smoothing splines with the amount of smoothing determined by cross validation. These techniques were empirically evaluated both for fixed sample sizes and when incorporated into the sequential run length control procedure. For fixed sample sizes they did not improve the small sample behavior and only marginally improved the large sample behavior when compared with the quadratic method. Their performance in the sequential procedure was unsatisfactory. Hence, the straightforward quadratic technique recommended in the earlier paper is still recommended as an effective, practical technique for simulation confidence interval generation and run length control.

Examples for the coefficient of tail dependence and the domain of attraction of a bivariate extreme value distribution

Performance measures of phylogenetic estimation methods such as accuracy, consistency, and power are an attempt at summarizing an ensemble of a given estimator's behavior. These summaries characterize an ensemble behavior with a single number, leading to a variety of definitions. In particular, the relationships between different performance measures such as accuracy and consistency or accuracy and error depend on the exact definition of these measures. In addition, it is relatively common to use large-sample behavior to infer similar behavior for small samples. In fact, large-sample results such as the claimed asymptotic efficiency of the maximum-likelihood estimator are often uninformative for small samples. Conversely, small-sample behavior using simulations is sometimes used to imply large-sample behavior such as consistency. However, such extrapolation is often difficult. How the performance of a phylogenetic estimator scales with the addition of taxa must be qualified with respect to whether the whole tree is being estimated or a fixed subset of taxa is being estimated. It must also be qualified with respect to how tree models are sampled. Over the ensemble of all possible trees of a given size, the performance of the estimators for the whole tree estimate suffers when the tree size becomes larger. However, under certain models of cladogenesis, the estimate can improve with the addition of taxa. In fact, at all numbers of taxa there are subsets of tree models that are easier to estimate than others. This suggests that with judicious addition or subtraction of taxa we can move from tree models that are more difficult to estimate at one number of taxa to those that are easier to estimate at another number of taxa.

The events of interest in any survival analysis study are regularly subject to censoring. There are various censoring schemes, including right or left censoring, and interval censoring. The most frequent censoring scheme is the right censoring, where subjects might drop out of the study or simply because not all events of interest occur before the end of the study. Moreover, for each subject, additional information referred to as covariates is registered at the beginning or throughout the study, such as age, sex, undergoing treatment, etc. The classical model to study the distribution of the events of interest, while accounting for additional information, is the Cox model. The Cox model expresses the hazard function of a subject given a set of covariates in terms of a baseline hazard, for which all covariates are zero, and an exponential function of the covariates and corresponding regression parameters. The baseline hazard can be left completely unspecified while estimating the regression parameters. Nonetheless, in practice, there are numerous studies in which the baseline hazard appears to be monotone. Time to death or to the onset of a disease are observed to have a nondecreasing baseline hazard, while the survival or recovery time after a successful medical treatment usually exhibit a nonincreasing baseline hazard. The aim of this thesis is to study the behavior of nonparametric baseline hazard and baseline density estimators in the Cox model under monotonicity constraints. The event times are assumed to be right censored and the censoring mechanism is assumed to be independent of the event of interest and non-informative. The covariates are assumed to be time-independent, usually recorded at the beginning of the study. In addition to point estimates, interval estimates of a monotone baseline hazard will be provided, based on a likelihood ratio method, along with testing at a fixed point. Furthermore, kernel smoothed estimates of a monotone baseline hazard will be defined and their behavior will be investigated. In Chapter 2, we propose several nonparametric monotone estimators of a baseline hazard or a baseline density within the Cox model. We derive the nonparametric maximum likelihood estimator of a nondecreasing baseline hazard and we consider a Grenander-type estimator, defined as the left-hand slope of the greatest convex minorant of the Breslow estimator. The two estimators are then shown to be strongly consistent and asymptotically equivalent. Moreover, we derive their common limit distribution at a fixed point. The two equivalent estimators of a nonincreasing baseline hazard and their asymptotic properties are acquired similarly. Furthermore, we introduce a Grenander-type estimator of a nonincreasing baseline density, defined as the left-hand slope of the least concave majorant of an estimator of the baseline cumulative distribution function derived from the Breslow estimator. This estimator is proven to be strongly consistent and its asymptotic distribution at a fixed point is derived. Chapter 3 provides an asymptotic linear representation of the Breslow estimator of the baseline cumulative hazard function in the Cox model. This representation can be used to derive the asymptotic distribution of the Grenander type estimator of a monotone baseline hazard estimator. The representation consists of an average of independent random variables and a term involving the difference between the maximum partial likelihood estimator and the underlying regression parameter. The order of the remainder term is arbitrarily close to n^-1. Chapter 4 focuses on interval estimation and on testing whether a monotone baseline hazard function in the Cox model has a particular value at a fixed point, via a likelihood ratio method. Nonparametric maximum likelihood estimators under the null hypothesis are defined for both nondecreasing and nonincreasing baseline hazard functions. These characterizations, along with those of the monotone nonparametric maximum likelihood estimators provide the asymptotic distribution of the likelihood ratio test. This asymptotic distribution enables, via inversion, the construction of pointwise confidence intervals. This method of constructing confidence intervals avoids the issue of estimating the nuisance parameters, as in the case of confidence intervals based on the asymptotic distribution of the estimators. Simulations indicate that the two methods yield confidence intervals with comparable coverage probabilities. Nonetheless, the confidence intervals based on the likelihood ratio are smaller, on average. Finally, in chapter 5 we consider smooth baseline hazard estimators. The estimators are obtained by kernel smoothing the maximum likelihood and Grenander-type estimators of a monotone baseline hazard function. Three different estimators are proposed for a nondecreasing baseline hazard, which are provided by the interchange of the smoothing and isotonization step. With this respect, we define a smoothed maximum likelihood estimator (SMLE), as well as a smoothed Grenander type (SG) estimator and a Grenander type smoothed (GS) estimator. All estimators are shown to be strongly pointwise or uniformly consistent.

The aim of our study was to test the robustness and efficiency of maximum likelihood with respect to different long branch effects on multiple-taxon trees. We simulated data of different alignment lengths under two different 11-taxon trees and a broad range of different branch length conditions. The data were analyzed with the true model parameters as well as with estimated and incorrect assumptions about among-site rate variation. If length differences between connected branches strongly increase, tree inference with the correct likelihood model assumptions can fail. We found that incorporating invariant sites together with distributed site rates in the tree reconstruction (+I) increases the robustness of maximum likelihood in comparison with models using only . The results show that for some topologies and branch lengths the reconstruction success of maximum likelihood under the correct model is still low for alignments with a length of 100,000 base positions. Altogether, the high confidence that is put in maximum likelihood trees is not always justified under certain tree shapes even if alignment lengths reach 100,000 base positions.

Estimating the basic reproduction number from surveillance data on past epidemics

In this paper, two bivariate models based on the proposed methods of Marshall and Olkin are introduced. In the first model, the new bivariate distribution is presented based on the proposed method of Marshall and Olkin (1967) which has natural interpretations, and it can be applied in fatal shock models or in competing risks models. In the second model, the proposed method of Marshall and Olkin (1997) is generalized to bivariate case and a new bivariate distribution is introduced. We call these new distributions as the bivariate Gompertz (BGP) distribution and bivariate Gompertz-geometric (BGPG) distribution, respectively. Moreover, the BGP model can be obtained as a special case of the BGPG model. Then, we present various properties of the new bivariate models. In this regard, the joint and conditional density functions, the joint cumulative distribution function can be obtained in compact forms. Also, the aging properties and the bivariate hazard gradient are discussed. This model has five unknown parameters and the maximum likelihood estimators cannot be obtained in explicit form. We propose to use the EM algorithm to compute the maximum likelihood estimators of the unknown parameters, and it is computationally quite tractable. Also, Monte Carlo simulations are performed to investigate the effectiveness of the proposed algorithm. Finally, we analyze three real data sets for illustrative purposes.

Model averaging generally provides better predictions than model selection, but the existing model averaging methods cannot lead to parsimonious models. Parsimony is an especially important property when the number of parameters is large. To achieve a parsimonious model averaging coefficient estimator, we suggest a novel criterion for choosing weights. Asymptotic properties are derived in two practical scenarios: (i) one or more correct models exist in the candidate model set and (ii) all candidate models are misspecified. Under the former scenario, it is proved that our method can put the weight one to the smallest correct model and the resulting model averaging estimators of coefficients have many zeros and thus lead to a parsimonious model. The asymptotic distribution of the estimators is also provided. Under the latter scenario, prediction is mainly focused on and we prove that the proposed procedure is asymptotically optimal in the sense that its squared prediction loss and risk are asymptotically identical to those of the best—but infeasible—model averaging estimator. Numerical analysis shows the promise of the proposed procedure over existing model averaging and selection methods.

A systematic approach is given for constructing continuous bivariate distributions with specified marginals and ixed dependence measures. This approach is based on linear combinations of independent random variables and results in bivariate distributions that can attain the Fréchet bounds. The dependence measures considered are Spearman's rho and Kendall's tau. Applications to testing for sensitivity in simulation models are discussed.