Semi-parametric estimation for heavy tailed distributions

In this paper, we generalize several studies in the area of extreme value theory for the estimation of the extreme value index and the second order parameter. Weak consistency and asymptotic normality are proven under classical assumptions. Some numerical simulations and computations are also performed to illustrate the finite-sample and the limiting behavior of the estimators.     

Robust and asymptotically unbiased estimation of extreme quantiles for heavy tailed distributions

A robust and asymptotically unbiased extreme quantile estimator is derived from a second order Pareto-type model and its asymptotic properties are studied under suitable regularity conditions. The finite sample properties of the proposed estimator are investigated with a small simulation experiment.     

On the tail index of a heavy tailed distribution

This paper proposes some new estimators for the tail index of a heavy tailed distribution when only a few largest values are observed within blocks. These estimators are proved to be asymptotically normal under suitable conditions, and their Edgeworth expansions are obtained. Empirical likelihood method is also employed to construct confidence intervals for the tail index. The comparison for the confidence intervals based on the normal approximation and the empirical likelihood method is made in terms of coverage probability and length of the confidence intervals. The simulation study shows that the empirical likelihood method outperforms the normal approximation method.     

On complexity of multistage stochastic programs under heavy tailed distributions

In this paper, the complexity of sample average approximation (SAA) of multistage stochastic programs under heavy tailed distributions is investigated. Specifically, we estimate confidence levels when the accuracy parameter and sample size are given under independently and identically distributed (iid) and non-iid conditional samples, respectively. Different from the existing works, we emphasize the impact of heavy tailed distributions, non-iid conditional sampling and stages dependence of the random process in multistage stochastic programs.     

On the extremes of a class of non-linear processes with heavy tailed innovations

The aim of this paper is to look at the limiting form of certain empirical point processes induced by a particular class of non-linear processes generated by heavy tailed innovations. Such asymptotic results will be highly useful in obtaining the weak limiting behavior of various functionals of the underlying process including the asymptotic distribution of upper and lower order statistics. In particular, we investigate the maximum limiting distribution and its corresponding extremal index. The results are applied to the study of the extremal properties of bilinear processes.     

Convexification for data fitting

The main results reported in this paper are two theorems concerning the use of a newtype of risk-averting error criterion for data fitting. The first states that the convexity region of the risk-averting error criterion expands monotonically as its risk-sensitivity index increases. The risk-averting error criterion is easily seen to converge to the mean squared error criterion as its risk-sensitivity index goes to zero. Therefore, the risk-averting error criterion can be used to convexify the mean squared error criterion to avoid local minima. The second main theorem shows that as the risk-sensitivity index increases to infinity, the risk-averting error criterion approaches the minimax error criterion, which is widely used for robustifying system controllers and filters.     

Circulant type matrices with heavy tailed entries

We study the limiting spectral distribution for a class of circulant type random matrices with heavy tailed input sequence. Unlike the light tailed case where the limit is nonrandom, here the limit is a random probability distribution. We provide an explicit representation of the limit.     

高负荷下带重尾服务强占优先排队的扩散逼近

考虑的排队系统是单服务台,顾客的初始到来是依泊松过程来到服务台,顾客的服务时间是重尾分布,服务的原则是强占优先服务.在高负荷条件下对此模型进行研究,获得了系统中的负荷过程,离去过程和队长过程的扩散逼近.     

Estimation of limiting conditional distributions for the heavy tailed long memory stochastic volatility process

We consider Stochastic Volatility processes with heavy tails and possible long memory in volatility. We study the limiting conditional distribution of future events given that some present or past event was extreme (i.e. above a level which tends to infinity). Even though extremes of stochastic volatility processes are asymptotically independent (in the sense of extreme value theory), these limiting conditional distributions differ from the i.i.d. case. We introduce estimators of these limiting conditional distributions and study their asymptotic properties. If volatility has long memory, then the rate of convergence and the limiting distribution of the centered estimators can depend on the long memory parameter (Hurst index).     

On robust estimation of location for arbitrarily right-censored data

Extension to censored data of classes of robust location estimators are discussed. Efficient estimators no longer exist, but linear functions of order statistics have a stability property, and variances can be lower than that of the Kaplan-Meier median.     

Stochastic approximation with long range dependent and heavy tailed noise

Stability and convergence properties of stochastic approximation algorithms are analyzed when the noise includes a long range dependent component (modeled by a fractional Brownian motion) and a heavy tailed component (modeled by a symmetric stable process), in addition to the usual ‘martingale noise’. This is motivated by the emergent applications in communications. The proofs are based on comparing suitably interpolated iterates with a limiting ordinary differential equation. Related issues such as asynchronous implementations, Markov noise, etc. are briefly discussed.     

Steady-state asymptotics for tandem,split-match and other feedforward queues with heavy tailed service

Consider a stable FIFO GI/GI/1 → /GI/1 tandem queue in which the equilibrium distribution of service time at the second node S(2) is subexponential. It is shown that when the service time at the first node has a lighter tail, the tail of steady-state delay at the second node, D(2), has the same asymptotics as if it were a GI/GI/1 queue: $$x \to \infty $$ where S _e(2) has equilibrium (integrated tail) density P(S(2) > $x$ )/E[S(2)], and ρ₂ = λE[S(2)] (λ is the arrival rate of customers). The same result holds for tandem queues with more than two stations. For split-match (fork-join) queues with subexponential service times, we derive the asymptotics for both the sojourn time and the queue length. Finally, more generally, we consider feedforward generalized Jackson networks and obtain similar results.     

Least squares and Chebyshev fitting for parameter estimation in ODEs

We discuss the problem of determining parameters in mathematical models described by ordinary differential equations. This problem is normally treated by least squares fitting. Here some results from nonlinear mean square approximation theory are outlined which highlight the problems associated with nonuniqueness of global and local minima in this fitting procedure. Alternatively, for Chebyshev fitting and for the case of a single differential equation, we extend and apply the theory of [17, 18] which ensures a unique global best approximation. The theory is applied to two numerical examples which show how typical difficulties associated with mean square fitting can be avoided in Chebyshev fitting.This paper is presented as an outcome of the LMS Durham Symposium convened by Professor C.T.H. Baker on 4th-14th July 1992 with support from the SERC under Grant reference number GR/H03964.     

A weighted least squares method for scattered data fitting

In this paper, we present a weighted least squares method to fit scattered data with noise. Existence and uniqueness of a solution are proved and an error bound is derived. The numerical experiments illustrate that our weighted least squares method has better performance than the traditional least squares method in case of noisy data.     

Maximum consistency method for data fitting under interval uncertainty

The work is devoted to application of global optimization in data fitting problem under interval uncertainty. Parameters of the linear function that best fits intervally defined data are taken as the maximum point for a special (“recognizing”) functional which is shown to characterize consistency between the data and parameters. The new data fitting technique is therefore called “maximum consistency method”. We investigate properties of the recognizing functional and present interpretation of the parameter estimates produced by the maximum consistency method.     

Maximum likelihood estimation for stochastic volatility in mean models with heavy‐tailed distributions

In this article, we introduce a likelihood‐based estimation method for the stochastic volatility in mean (SVM) model with scale mixtures of normal (SMN) distributions. Our estimation method is based on the fact that the powerful hidden Markov model (HMM) machinery can be applied in order to evaluate an arbitrarily accurate approximation of the likelihood of an SVM model with SMN distributions. Likelihood‐based estimation of the parameters of stochastic volatility models, in general, and SVM models with SMN distributions, in particular, is usually regarded as challenging as the likelihood is a high‐dimensional multiple integral. However, the HMM approximation, which is very easy to implement, makes numerical maximum of the likelihood feasible and leads to simple formulae for forecast distributions, for computing appropriately defined residuals, and for decoding, that is, estimating the volatility of the process. Copyright © 2017 John Wiley & Sons, Ltd.     

Estimating tail probabilities of heavy tailed distributions with asymptotically zero relative error

Efficient estimation of tail probabilities involving heavy tailed random variables is amongst the most challenging problems in Monte-Carlo simulation. In the last few years, applied probabilists have achieved considerable success in developing efficient algorithms for some such simple but fundamental tail probabilities. Usually, unbiased importance sampling estimators of such tail probabilities are developed and it is proved that these estimators are asymptotically efficient or even possess the desirable bounded relative error property. In this paper, as an illustration, we consider a simple tail probability involving geometric sums of heavy tailed random variables. This is useful in estimating the probability of large delays in M/G/1 queues. In this setting we develop an unbiased estimator whose relative error decreases to zero asymptotically. The key idea is to decompose the probability of interest into a known dominant component and an unknown small component. Simulation then focuses on estimating the latter ‘residual’ probability. Here we show that the existing conditioning methods or importance sampling methods are not effective in estimating the residual probability while an appropriate combination of the two estimates it with bounded relative error. As a further illustration of the proposed ideas, we apply them to develop an estimator for the probability of large delays in stochastic activity networks that has an asymptotically zero relative error.      

Revisiting fitting monotone polynomials to data

We revisit Hawkins’ (Comput Stat 9(3):233–247, 1994) algorithm for fitting monotonic polynomials and discuss some practical issues that we encountered using this algorithm, for example when fitting high degree polynomials or situations with a sparse design matrix but multiple observations per $x$ -value. As an alternative, we describe a new approach to fitting monotone polynomials to data, based on different characterisations of monotone polynomials and using a Levenberg–Marquardt type algorithm. We consider different parameterisations, examine effective starting values for the non-linear algorithms, and discuss some limitations. We illustrate our methodology with examples of simulated and real world data. All algorithms discussed in this paper are available in the R Development Core Team (A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, 2011) package MonoPoly.