A robust prediction error criterion for pareto modelling of upper tails

Estimation of the Pareto tail index from extreme order statistics is an important problem in many settings. The upper tail of the distribution, where data are sparse, is typically fitted with a model, such as the Pareto model, from which quantities such as probabilities associated with extreme events are deduced. The success of this procedure relies heavily not only on the choice of the estimator for the Pareto tail index but also on the procedure used to determine the number k of extreme order statistics that are used for the estimation. The authors develop a robust prediction error criterion for choosing k and estimating the Pareto index. A Monte Carlo study shows the good performance of the new estimator and the analysis of real data sets illustrates that a robust procedure for selection, and not just for estimation, is needed.     

Generalized order statistic processes and Pfeifer records

We introduce a uniform generalized order statistic process. It is a simple Markov process whose initial segment can be identified with a set of uniform generalized order statistics. A standard marginal transformation leads to a generalized order statistic process related to non-uniform generalized order statistics. It is then demonstrated that the nth variable in such a process has the same distribution as an nth Pfeifer record value. This process representation of Pfeifer records facilitates discussion of the possible limit laws for Pfeifer records and, in some cases, of sums thereof. Because of the close relationship between Pfeifer records and generalized order statistics, the results shed some light on the problem of determining the nature of the possible limiting distributions of the largest generalized order statistic.     

OPTIMAL SEMIPARAMETRIC INFERENCE FOR THE TAIL INDEX BASED ON RATIOS OF THE LARGEST EXTREMES

The k largest order statistics in a random sample from a common heavy‐tailed parent distribution with a regularly varying tail can be characterized as Fréchet extremes. This paper establishes that consecutive ratios of such Fréchet extremes are mutually independent and distributed as functions of beta random variables. The maximum likelihood estimator of the tail index based on these ratios is derived, and the exact distribution of the maximum likelihood estimator is determined for fixed k, and the asymptotic distribution as k →∞ . Inferential procedures based upon the maximum likelihood estimator are shown to be optimal. The Fréchet extremes are not directly observable, but a feasible version of the maximum likelihood estimator is equivalent to Hill's statistic. A simple diagnostic is presented that can be used to decide on the largest value of k for which an assumption of Fréchet extremes is sustainable. The results are illustrated using data on commercial insurance claims arising from fires and explosions, and from hurricanes.     

On tail index estimation based on multivariate data

This article is devoted to the study of tail index estimation based on i.i.d. multivariate observations, drawn from a standard heavy-tailed distribution, that is, of which Pareto-like marginals share the same tail index. A multivariate central limit theorem for a random vector, whose components correspond to (possibly dependent) Hill estimators of the common tail index α, is established under mild conditions. We introduce the concept of (standard) heavy-tailed random vector of tail index α and show how this limit result can be used in order to build an estimator of α with small asymptotic mean squared error, through a proper convex linear combination of the coordinates. Beyond asymptotic results, simulation experiments illustrating the relevance of the approach promoted are also presented.     

Asymptotic tail behavior of a random sum with conditionally dependent subexponential summands

In this paper, we obtain some results for the asymptotic behavior of the tail probability of a random sum S_τ = ∑^τ_{k = 1}X_k, where the summands X_k, k = 1, 2, …, are conditionally dependent random variables with a common subexponential distribution F, and the random number τ is a non negative integer-valued random variable, independent of {X_k: k ? 1}.     

Small-sample comparison of the exact and asymptotic upper tail probabilities of chi-squared goodness-of-fit statistics: the binomila and the mixture binomial

The exact and asymptotic upper tail probabilities (α = .10, .05, .01, .001) of the three chi-squared goodness-of-fit statistics Pearson's X ², likelihood ratioG ², and powerdivergence statisticD ²(λ), with λ= 2/3 are compared by complete enumeration for the binomial and the mixture binomial. For the two-component mixture binomial, three cases have been distinguished. 1. Both success probabilities and the mixing weights are unknwon. 2. One of the two success probabilities is known. And 3., the mixing weights are known. The binomial was investigated for the number of cellsk, being between 3 and 6 with sample sizes between 5 and 100, for k = 7 with sample sizes between 5 and 45, and for k = 10 with sample sizes ranging from 5 to 20. For the mixture binomial, solely k = 5 cells were considered with sample sizes from 5 to 100 and k = 8 cells with sample sizes between 4 and 20. Rating the relative accuracy of the chi-squared approximation in terms of ±10% and ±20% intervals around α led to the following conclusions for the binomial: 1. Using G2 is not recommendable. 2. At the significance levels α=.10 and α=.05X ² should be preferred over D ²; D ² is the best choice at α = .01. 3. Cochran's (1954; Biometrics, 10, 417-451) rule for the minimum expectation when using X ² seems to generalize to the binomial for G ² and D ² ; as a compromise, it gives a rather strong lower limit for the expected cell frequencies in some circumstances, but a rather liberal in others. To draw similar conclusions concerning the mixture binomial was not possible, because in that case, the accuracy of the chi-squared approximation is not only a function of the chosen test statistic and of the significance level, but also heavily depends on the numerical value of theinvolved unknown parameters and on the hypothesis to be tested. Thereto, the present study may give rise only to warnings against the application of mixture models to small samples.     

On the asymptotic independence of numbers of observations near order statistics

In this paper, we consider the numbers of observations in two-sided neighbourhoods of the kth and (n?r)th order statistics from a sample of size n and show that they are asymptotically independent as n→∞. We also establish a result that generalizes all the existing results regarding the asymptotic independence of numbers of observations in the left and right neighbourhoods of order statistics. Finally, we consider the limiting joint behaviour of numbers of observations in the neighbourhoods of s central order statistics and establish that they are asymptotically independent.     

Some characterizations of geometric tail distributions based on record values

Let R_j be the jth upper record value from an infinite sequence of independent identically distributed positive integer valued random variables. We show that their common distribution must have geometric tail if R_j+k?R_j and R_j are partially independent for some j≥1 and k≥1 or if E(R_j+2?R_j+1| R_j) is a constant. Three versions of partial independence, each of which provides a characterization of the geometric tail are presented.     

On residual lifetimes in sequential (n ? k?+?1)-out-of-n systems

Sequential order statistics is an extension of ordinary order statistics. They model the successive failure times in sequential k-out-of-n systems, where the failures of components possibly affect the residual lifetimes of the remaining ones. In this paper, we consider the residual lifetime of the components after the kth failure in the sequential (n − k + 1)-out-of-n system. We extend some results on the joint distribution of the residual lifetimes of the remaining components in an ordinary (n − k + 1)-out-of-n system presented in Bairamov and Arnold (Stat Probab Lett 78(8):945–952, 2008) to the case of the sequential (n − k + 1)-out-of-n system.     

On Pitman's measure of closeness of k-records

Pitman closeness of both the upper and lower k-record statistics to the population quantiles of a location–scale family of distributions is studied. For the population median, the Pitman-closest k-record is also determined. In the case of symmetric distributions, the Pitman closeness probabilities of k-record statistics are shown to be distribution-free, and explicit expressions are also derived for these probabilities. Exact expressions are derived for the required probabilities for uniform and exponential distributions. Numerical results are given for these families and also the Pitman-closest k-record is determined.     

Non‐parametric Estimation of Extreme Risk Measures from Conditional Heavy‐tailed Distributions

In this paper, we introduce a new risk measure, the so‐called conditional tail moment. It is defined as the moment of order a ≥ 0 of the loss distribution above the upper α‐quantile where α ∈ (0,1). Estimating the conditional tail moment permits us to estimate all risk measures based on conditional moments such as conditional tail expectation, conditional value at risk or conditional tail variance. Here, we focus on the estimation of these risk measures in case of extreme losses (where α ↓0 is no longer fixed). It is moreover assumed that the loss distribution is heavy tailed and depends on a covariate. The estimation method thus combines non‐parametric kernel methods with extreme‐value statistics. The asymptotic distribution of the estimators is established, and their finite‐sample behaviour is illustrated both on simulated data and on a real data set of daily rainfalls.     

Pfeifer records process

In this article, we study the limit distribution of sums of Pfeifer records. Motivated by the results obtained by Arnold and Villaseñor [Generalized order statistics process and Pfeifer records, Statistics 46(3) (2012), pp. 373–385], we show that the partial sum process of Pfeifer records converge to a function of the Brownian motion. The normalization is either a sequence of appropriate constants or a sequence of functions, depending on the tail behaviour of the underlying variables. These results, in particular, prove stronger version of results obtained in Villaseñor and Arnold [On limit laws for sums of Pfeifer records, Extremes 10 (2007), pp. 235–248] and Bose and Gangopadhyay [Convergence of linear functions of Pfeifer records, Extremes 13 (2010), pp. 89–106] and extends results of Bose et al. [Partial sum process for records, Extremes 8 (2005), pp. 43–56] from classical records to Pfeifer records.     

Small -sample comparison of the exact and symptotic upper tail probabilitiesof chi-squared goodness -of-fit statistics: pearson's x 2,likelihood ratio,and power-divergence statistic(λ = 2/3)

The exact and asymptotic upper tail probabilities ( α= .l0, .05, .01, .001) of the three chi-squared goodness-of-fit statistics Pearson's X ², likelihood ratioG ², and power-divergence statisticD ² (λ ) , with λ = 2/3, are compared numerically for simple null hypotheses not involving parameter estimation. Three types of such hypotheses were investigated (equal cell probabilities, proportional cell probabilities, some fixed small expectations together with some increasing large expectations) for the number of cells being between 3 and 15, and for sample sizes from 10 to 40, increasing by steps of one. Rating the relative accuracy of the chi-squared approximation in terms of ±10% and ±20% intervals around α led to the following conclusions: 1. Using G ² is not recommended. 2 . At the more relevant significance levels α = .10 and α = .05X ² should be preferred over D ². Solely in case of unequal cell probabilitiesD ² is the better choice at α = .O1 and α = .001. 3 . Yarnold's (1970; Journal of the Amerin Statistical Association, 65, 864-886) rule for the minimum expectation when using X ² ("If the number of cells k is 3 or more, and if r denotes the number of expectations less than 5, then the minimum expectation may be as small as 5r/k.") generalizes to D ²; it gives a good lower limit for the expected cell frequencies, however, when the number of cells is greater than 3. For k = 3 , even sample sizes over 15 may be insufficient.     

Small-Sample Quantile Estimators in a Large Nonparametric Model

The large nonparametric model in this note is a statistical model with the family ? of all continuous and strictly increasing distribution functions. In the abundant literature of the subject, there are many proposals for nonparametric estimators that are applicable in the model. Typically the kth order statistic X _k:n is taken as a simplest estimator, with k = [nq], or k = [(n + 1)q], or k = [nq] + 1, etc. Often a linear combination of two consecutive order statistics is considered. In more sophisticated constructions, different L-statistics (e.g., Harrel–Davis, Kaigh–Lachenbruch, Bernstein, kernel estimators) are proposed. Asymptotically the estimators do not differ substantially, but if the sample size n is fixed, which is the case of our concern, differences may be serious. A unified treatment of quantile estimators in the large, nonparametric statistical model is developed.     

Distribution-free confidence intervals for quantiles and tolerance intervals in terms of k-records

In this paper, we consider the problem of determining non-parametric confidence intervals for quantiles when available data are in the form of k-records. Distribution-free confidence intervals as well as lower and upper confidence limits are derived for fixed quantiles of an arbitrary unknown distribution based on k-records of an independent and identically distributed sequence from that distribution. The construction of tolerance intervals and limits based on k-records is also discussed. An exact expression for the confidence coefficient of these intervals are derived. Some tables are also provided to assist in choosing the appropriate k-records for the construction of these confidence intervals and tolerance intervals. Some simulation results are presented to point out some of the features and properties of these intervals. Finally, the data, representing the records of the amount of annual rainfall in inches recorded at Los Angeles Civic Center, are used to illustrate all the results developed in this paper and also to demonstrate the improvements that they provide on those based on either the usual records or the current records.     

Bias reduction of a tail index estimator through an external estimation of the second-order parameter

In this paper, we first consider a class of consistent semi-parametric estimators of a positive tail index γ, parameterised in a tuning or control parameter α. Such a control parameter enables us to have access, for any available sample, to an estimator of the tail index γ with a null dominant component of asymptotic bias, and consequently with a reasonably flat mean squared error pattern, as a function of k, the number of top-order statistics considered. Such a control parameter depends on a second-order parameter ρ, which will be adequately estimated so that we may achieve a high efficiency relative to the classical Hill estimator, provided we use a number of top-order statistics larger than the one usually required for the estimation through the Hill estimator. An illustration of the behaviour of the estimators is provided, through the analysis of the daily log-returns on the Euro–US$ exchange rates.     

Kernel estimators for the second order parameter in extreme value statistics

We develop and study in the framework of Pareto-type distributions a general class of kernel estimators for the second order parameter ρ$\rho$, a parameter related to the rate of convergence of a sequence of linearly normalized maximum values towards its limit. Inspired by the kernel goodness-of-fit statistics introduced in Goegebeur et al. (2008), for which the mean of the normal limiting distribution is a function of ρ$\rho$, we construct estimators for ρ$\rho$ using ratios of ratios of differences of such goodness-of-fit statistics, involving different kernel functions as well as power transformations. The consistency of this class of ρ$\rho$ estimators is established under some mild regularity conditions on the kernel function, a second order condition on the tail function 1−F of the underlying model, and for suitably chosen intermediate order statistics. Asymptotic normality is achieved under a further condition on the tail function, the so-called third order condition. Two specific examples of kernel statistics are studied in greater depth, and their asymptotic behavior illustrated numerically. The finite sample properties are examined by means of a simulation study.     

Improved reduced-bias tail index and quantile estimators

In this paper, we deal with bias reduction techniques for heavy tails, trying to improve mainly upon the performance of classical high quantile estimators. High quantiles depend strongly on the tail index γ$\gamma$, for which new classes of reduced-bias estimators have recently been introduced, where the second-order parameters in the bias are estimated at a level _k1$k_1$ of a larger order than the level k at which the tail index is estimated. Doing this, it was seen that the asymptotic variance of the new estimators could be kept equal to the one of the popular Hill estimators. In a similar way, we now introduce new classes of tail index and associated high quantile estimators, with an asymptotic mean squared error smaller than that of the classical ones for all k in a large class of heavy-tailed models. We derive their asymptotic distributional properties and compare them with those of alternative estimators. Next to that, an illustration of the finite sample behavior of the estimators is also provided through a Monte Carlo simulation study and the application to a set of real data in the field of insurance.     

Tail Probability Ratios of Normal and Student’s t Distributions

The ratio of normal tail probabilities and the ratio of Student’s t tail probabilities have gained an increased attention in statistics and related areas. However, they are not well studied in the literature. In this paper, we systematically study the functional behaviors of these two ratios. Meanwhile, we explore their difference as well as their relationship. It is surprising that the two ratios behave very different to each other. Finally, we conclude the paper by conducting some lower and upper bounds for the two ratios.