Extreme value copula estimation based on block maxima of a multivariate stationary time series

The core of the classical block maxima method consists of fitting an extreme value distribution to a sample of maxima over blocks extracted from an underlying series. In asymptotic theory, it is usually postulated that the block maxima are an independent random sample of an extreme value distribution. In practice however, block sizes are finite, so that the extreme value postulate will only hold approximately. A more accurate asymptotic framework is that of a triangular array of block maxima, the block size depending on the size of the underlying sample in such a way that both the block size and the number of blocks within that sample tend to infinity. The copula of the vector of componentwise maxima in a block is assumed to converge to a limit, which, under mild conditions, is then necessarily an extreme value copula. Under this setting and for absolutely regular stationary sequences, the empirical copula of the sample of vectors of block maxima is shown to be a consistent and asymptotically normal estimator for the limiting extreme value copula. Moreover, the empirical copula serves as a basis for rank-based, nonparametric estimation of the Pickands dependence function of the extreme value copula. The results are illustrated by theoretical examples and a Monte Carlo simulation study.     

On the limiting joint distribution of the extreme order statistics

The connection between extreme values and record-low values is exploited to derive simply the limiting joint distribution of the r largest order statistics. The use of this distribution in the modelling of corrosion phenomena is considered, and the extrapolation of maxima in space and time is described in this context. There has been recent emphasis on movement away from classical extreme value theory to more efficient estimation procedures. This shift is continued with the illustration of the extra precision of predicted maxima obtained from a model based on extreme order statistics over the classical extreme value approach.     

Detecting distributional changes in samples of independent block maxima using probability weighted moments

The analysis of seasonal or annual block maxima is of interest in fields such as hydrology, climatology or meteorology. In connection with the celebrated method of block maxima, we study several tests that can be used to assess whether the available series of maxima is identically distributed. It is assumed that block maxima are independent but not necessarily generalized extreme value distributed. The asymptotic null distributions of the test statistics are investigated and the practical computation of approximate p-values is addressed. Extensive Monte-Carlo simulations show the adequate finite-sample behavior of the studied tests for a large number of realistic data generating scenarios. Illustrations on several environmental datasets conclude the work.     

Equivalence classes of regularly varying functions

A useful method to derive limit results for partial maxima and record values of independent, identically distributed random variables is to start from one specific probability distribution and to extend the result for this distribution to a class of distributions.This method involves an extended theory of regularly varying functions. In this paper, equivalence classes of regularly varying functions (in the extended sense) are studied, which is relevant to the problems mentioned above.     

Practical Extreme Value Modelling of Hydrological Floods and Droughts: A Case Study

Estimation of flood and drought frequencies is important for reservoir design and management, river pollution, ecology and drinking water supply. Through an example based on daily streamflow observations, we introduce a stepwise procedure for estimating quantiles of the hydrological extremes floods and droughts. We fit the generalised extreme value (GEV) distribution by the method of block maxima and the generalised Pareto (GP) distribution by applying the peak over threshold method. Maximum likelihood, penalized maximum likelihood and probability weighted moments are used for parameter estimation. We incorporate trends and seasonal variation in the models instead of splitting the data, and investigate how the observed number of extreme events, the chosen statistical model, and the parameter estimation method effect parameter estimates and quantiles. We find that a seasonal variation should be included in the GEV distribution fitting for floods using block sizes less than one year. When modelling droughts, block sizes of one year or less are not recommended as significant model bias becomes visible. We conclude that the different characteristics of floods and droughts influence the choices made in the extreme value modelling within a common inferential strategy.This revised version was published online in March 2005 with corrections to the cover date.     

Exploration of probabilistic mould growth assessment

A sufficiently long continuous duration of favourable conditions, particularly the availability of water and nutrients, is necessary for the mould growth on the surface of a building component. The availability of water is commonly quantified by the relative humidity on the building component surface. Some parameters, substantially affecting the relative humidity randomly fluctuate in time. Therefore, considering continuous duration of favourable conditions as a basic characteristic for mould growth assessment, its occurrence in time is represented by a stochastic process. Exploration and comparison of current methods for probabilistic assessment of mould growth resulted in emphasising the application of the theory of extreme values. Exceedance probabilities of practically important duration levels are estimated by the classical block maxima model for extremes and the point process model. As a quantification of potential mould risk the mean return period of a considered duration of favourable mould growth conditions is suggested. This quantification may be used for comparison of different designs, assessment of retrofitting effectiveness or as an input for cost–benefit analysis.     

Estimated conditional score function for missing mechanism model with nonignorable nonresponse

Missing data mechanism often depends on the values of the responses, which leads to nonignorable nonresponses. In such a situation, inference based on approaches that ignore the missing data mechanism could not be valid. A crucial step is to model the nature of missingness. We specify a parametric model for missingness mechanism, and then propose a conditional score function approach for estimation. This approach imputes the score function by taking the conditional expectation of the score function for the missing data given the available information. Inference procedure is then followed by replacing unknown terms with the related nonparametric estimators based on the observed data. The proposed score function does not suffer from the non-identifiability problem, and the proposed estimator is shown to be consistent and asymptotically normal. We also construct a confidence region for the parameter of interest using empirical likelihood method. Simulation studies demonstrate that the proposed inference procedure performs well in many settings. We apply the proposed method to a data set from research in a growth hormone and exercise intervention study.     

The root–unroot algorithm for density estimation as implemented via wavelet block thresholding

We propose and implement a density estimation procedure which begins by turning density estimation into a nonparametric regression problem. This regression problem is created by binning the original observations into many small size bins, and by then applying a suitable form of root transformation to the binned data counts. In principle many common nonparametric regression estimators could then be applied to the transformed data. We propose use of a wavelet block thresholding estimator in this paper. Finally, the estimated regression function is un-rooted by squaring and normalizing. The density estimation procedure achieves simultaneously three objectives: computational efficiency, adaptivity, and spatial adaptivity. A numerical example and a practical data example are discussed to illustrate and explain the use of this procedure. Theoretically it is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the Besov classes. The estimator also automatically adapts to the local smoothness of the underlying function, and attains the local adaptive minimax rate for estimating functions at a point. There are three key steps in the technical argument: Poissonization, quantile coupling, and oracle risk bound for block thresholding in the non-Gaussian setting. Some of the technical results may be of independent interest.     

On approximating max-stable processes and constructing extremal copula functions

There is an infinite number of parameters in the definition of multivariate maxima of moving maxima (M4) processes, which poses challenges in statistical applications where workable models are preferred. This paper establishes sufficient conditions under which an M4 process with infinite number of parameters may be approximated by an M4 process with finite number of parameters. In statistical inferences, the paper focuses on a family of sectional multivariate extreme value copula (SMEVC) functions which is derived from the joint distribution functions of M4 processes. A new non-standard parameter estimation procedure is introduced, which is based on order statistics of ratios of (transformed) marginal unit Fréchet random variables, and is shown via simulation to be more efficient than a semi-parametric estimation procedure. In real data analysis, empirical results show that SMEVCs are more flexible for modeling various dependence structures, and perform better than the widely used Gumbel-Hougaard copulas.     

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Dealing with the missing values is an important object in the field of data mining. Besides, the properties of compositional data lead to that traditional imputation methods may get undesirable result if they are directly used in this type of data. As a result, the management of missing values in compositional data is of great significant. To solve this problem, this paper uses the relationship between compositional data and Euclidean data, and proposes a new method based on Random Forest for missing values in compositional data. This method has been implemented and evaluated using both simulated and real-world databases, then the experimental results reveal that the new imputation method can be widely used in various types of data sets and has good performance than other methods.     

Allocation of resources according to a fractional objective

The problem considered is that of the allocation of resources to activities according to a fractional measure given by the ratio of “return” to “cost”. The return is the sum of returns from the activities, each activity being described by a concave return function. There is a positive fixed cost and a variable cost that depend linearly on the allocations. Properties related to the uniqueness of optimal solutions and the number of non-zero allocations are derived. A method is given by which any set of feasible allocations can be used to derive an upper bound of the optimal value of the objective function: optimal and almost-optimal allocations can be recognized. Allocations can be generated by a fast incremental method that is described. The method utilizes data in sequential order and can be used to solve large problems.     

A Law of the Iterated Logarithm for Extreme Queue Length in Multiphase Queues

The object of this research in queueing theory is the Law of the Iterated Logarithm (LIL) under the conditions of heavy traffic in Multiphase Queueing Systems (MQS). In this paper, the LIL is proved for extreme values of important probabilistic characteristics of the MQS investigated as well as maxima and minima of the summary queue length of customers and maxima and minima of the queue length of customers. Also, the paper presents a survey on the works for extreme values in queues and the queues in heavy traffic.      

More accurate breakdown voltage estimation for the new step-up test method in the gumbel distribution model

The estimation problems for the conventional step-up method (the observed breakdown voltages are not given at all) and the new step-up method (some of the observed breakdown voltages are given) are analyzed when the underlying probability distribution (of breakdown voltage level) is assumed to be gumbel distributions for minima and maxima. The new step-up test method has advantages compared to the conventional method: (1) the confidence intervals of the estimates become smaller and (2) the estimates can be obtained with higher probability. In some case of real step-up breakdown voltage test, a fit of the gumbel distribution to the data case is found to be superior to that of the normal distribution, which suggests the usefulness of the gumbel distribution for the underlying distribution in the step-up breakdown voltage test.     

The Law of the Iterated Logarithm for Extreme Waiting Time in Multiphase Queueing Systems

The objective of this research in the queueing theory is the law of the iterated logarithm (LIL) under the conditions of heavy traffic in multiphase queueing systems (MQS). In this paper, the LIL is proved for the extreme values of some important probabilistic characteristics of the MQS, namely, maxima and minima of the summary waiting time of a customer, and maxima and minima of the waiting time of a customer.     

Probabilistic clustering via Pareto solutions and significance tests

The present paper proposes a new strategy for probabilistic (often called model-based) clustering. It is well known that local maxima of mixture likelihoods can be used to partition an underlying data set. However, local maxima are rarely unique. Therefore, it remains to select the reasonable solutions, and in particular the desired one. Credible partitions are usually recognized by separation (and cohesion) of their clusters. We use here the p values provided by the classical tests of Wilks, Hotelling, and Behrens–Fisher to single out those solutions that are well separated by location. It has been shown that reasonable solutions to a clustering problem are related to Pareto points in a plot of scale balance vs. model fit of all local maxima. We briefly review this theory and propose as solutions all well-fitting Pareto points in the set of local maxima separated by location in the above sense. We also design a new iterative, parameter-free cutting plane algorithm for the multivariate Behrens–Fisher problem.     

Maximum likelihood estimation from fuzzy data using the EM algorithm

A method is proposed for estimating the parameters in a parametric statistical model when the observations are fuzzy and are assumed to be related to underlying crisp realizations of a random sample. This method is based on maximizing the observed-data likelihood defined as the probability of the fuzzy data. It is shown that the EM algorithm may be used for that purpose, which makes it possible to solve a wide range of statistical problems involving fuzzy data. This approach, called the fuzzy EM (FEM) method, is illustrated using three classical problems: normal mean and variance estimation from a fuzzy sample, multiple linear regression with crisp inputs and fuzzy outputs, and univariate finite normal mixture estimation from fuzzy data.     

Nonparametric long term prediction of stock returns with generated bond yields

Recent empirical approaches in forecasting equity returns or premiums found that dynamic interactions among the stock and bond are relevant for long term pension products. Automatic procedures to upgrade or downgrade risk exposure could potentially improve long term performance for such products. The risk and return of bonds is more easy to predict than the risk and return of stocks. This and the well known stock-bond correlation motivates the inclusion of the current bond yield in a model for the prediction of excess stock returns. Here, we take the actuarial long term view using yearly data, and focus on nonlinear relationships between a set of covariates. We employ fully nonparametric models and apply for estimation a local-linear kernel smoother. Since the current bond yield is not known, it is predicted in a prior step. The structure imposed this way in the final estimation process helps to circumvent the curse of dimensionality and reduces bias in the estimation of excess stock returns. Our validated stock prediction results show that predicted bond returns improve stock prediction significantly.     

CAPM with fuzzy returns and hypothesis testing

Over the last four decades, several estimation issues of the beta have been discussed extensively in many articles. An emerging consensus is that the betas are time-dependent and their estimates are impacted by the return interval and the length of the estimation period. These findings lead to the prominence of the practical implementation of the Capital Asset Pricing Model. Our goal in this paper is two-fold. After studying the impact of the return interval on the beta estimates, we analyze the sample size effects on the preceding estimation. Working in the framework of fuzzy set theory, we first associate the returns based on closing prices with the intraperiod volatility for the representation by the means of a fuzzy random variable in order to incorporate the effect of the interval period over which the returns are measured in the analysis. Next, we use these fuzzy returns to estimate the beta via fuzzy least square method in order to deal efficiently with outliers in returns, often caused by structural breaks and regime switches in the asset prices. A bootstrap test is carried out to investigate whether there is a linear relationship between the market portfolio fuzzy return and the given asset fuzzy return. Finally, the empirical results on French stocks suggest that our beta estimates seem to be more stable than the ordinary least square (OLS) estimates when the return intervals and the sample size change.     

Regression based reserving models and partial information

We consider a class of distribution-free regression models only defined in terms of moments, which can be used to model separate reported but not settled reserves, and incurred but not reported reserves. These regression models can be estimated using standard least squares and method of moments techniques, similar to those used in the distribution-free chain-ladder model. Further, these regression models are closely related to double chain-ladder type models, and the suggested estimation techniques could serve as alternative estimation procedures for these models. Due to the simple structure of the models it is possible to obtain Mack-type mean squared error of prediction estimators. Moreover, the analysed regression models can be used on different levels of detailed data, and by using the least squares estimation techniques it is possible to show that the precision in the reserve predictor is improved by using more detailed data. These regression models can be seen as a sequence of linear models, and are therefore also easy to bootstrap non-parametrically.     

Specifying measurement errors for required lifetime estimation performance

Lifetime estimation based on the measured health monitoring data has long been investigated and applied in reliability and operational management communities and practices, such as planning maintenance schedules, logistic supports, and production planning. It is known that measurement error (ME) is a source of uncertainty in the measured data considerably affecting the performance of data driven lifetime estimation. While the effect of ME on the performance of data driven lifetime estimation models has been studied recently, a reversed problem—“the specification of the ME range to achieve a desirable lifetime estimation performance” has not been addressed. This problem is related to the usability of the measured health monitoring data for estimating the lifetime. In this paper, we deal with this problem and develop guidelines regarding the formulation of specification limits to the distribution-related ME characteristics. By referring to one widely applied Wiener process-based degradation model, permissible values for the ME bias and standard deviation can be given under a specified lifetime estimation requirement. If the performance of ME does not satisfy the permissible values, the desirable performance for lifetime estimation cannot be ensured by the measured health monitoring data. We further analyze the effect of ME on an age based replacement decision, which is one of the most common and popular maintenance policies in maintenance scheduling. Numerical examples and a case study are provided to illustrate the implementation procedure and usefulness of theoretical results.