Asymptotic ruin probabilities of the renewalmodel with constant interest force and dependent heavy-tailed claims

In this paper,we investigate the asymptotic behavior for the finite- and infinite-time ruin probabilities of a nonstandard renewal model in which the claims are identically distributed but not necessarily independent. Under the assumptions that the identical distribution of the claims belongs to the class of extended regular variation(ERV) and that the tails of joint distributions of every two claims are negligible compared to the tails of their margins,we obtain the precise approximations for the finite- and infinite-time ruin probabilities.     

一类拟平稳序列极值的极限律

Ｋ．Ｆ．Ｔｕｒｋｍａｎ讨论了一类拟平稳序列最大值的渐近分布。本文利用点过程收全党一理得到水平超出点过程的收敛定理和第ｒ个最大值的渐近分布及前ｒ个最大值的联合渐近分布。     

Renewal-theoretic considerations in evaluation of forecast monitoring schemes

To evaluate the performance of a scheme for monitoring forecasts that are generated by exponential smoothing, forecasters have traditionally used a simulation-based estimator of some characteristic of the associated run-length distribution. The most frequently cited performance measures are the average run length and the probability that the run length does not exceed a user-specified cut-off point. However, there is disagreement about the definition of run length that is appropriate in the context of forecasting. In this paper we use fundamental results from renewal theory to establish the precise relationships between conflicting formulations both of the average run length and of the probability density function for the run length. More generally we derive the asymptotic mean and the asymptotic distribution function of the forward recurrence time for both ordinary and delayed renewal processes whose interoccurrence distributions are arithmetic with a given span. We discuss the practical significance of these results in the context of forecasting. Our findings bear directly on the way in which simulation experiments should be designed and executed to compare alternative forecast monitoring schemes.     

Asymptotics in a time-dependent renewal risk model with stochastic return

In this paper we study the asymptotic tail behavior for a non-standard renewal risk model with a dependence structure and stochastic return. An insurance company is allowed to invest in financial assets such as risk-free bonds and risky stocks, and the price process of its portfolio is described by a geometric Lévy process. By restricting the claim-size distribution to the class of extended regular variation (ERV) and imposing a constraint on the Lévy process in terms of its Laplace exponent, we obtain for the tail probability of the stochastic present value of aggregate claims a precise asymptotic formula, which holds uniformly for all time horizons. We further prove that the corresponding ruin probability also satisfies the same asymptotic formula.     

Waiting Time Asymptotics in the Single Server Queue with Service in Random Order

We consider the single server queue with service in random order. For a large class of heavy-tailed service time distributions, we determine the asymptotic behavior of the waiting time distribution. For the special case of Poisson arrivals and regularly varying service time distribution with index ?ν, it is shown that the waiting time distribution is also regularly varying, with index 1?ν, and the pre-factor is determined explicitly. Another contribution of the paper is the heavy-traffic analysis of the waiting time distribution in the M/G/1 case. We consider not only the case of finite service time variance, but also the case of regularly varying service time distribution with infinite variance.     

Exact inference for a simple step-stress model from the exponential distribution under time constraint

In reliability and life-testing experiments, the researcher is often interested in the effects of extreme or varying stress factors such as temperature, voltage and load on the lifetimes of experimental units. Step-stress test, which is a special class of accelerated life-tests, allows the experimenter to increase the stress levels at fixed times during the experiment in order to obtain information on the parameters of the life distributions more quickly than under normal operating conditions. In this paper, we consider the simple step-stress model from the exponential distribution when there is time constraint on the duration of the experiment. We derive the maximum likelihood estimators (MLEs) of the parameters assuming a cumulative exposure model with lifetimes being exponentially distributed. The exact distributions of the MLEs of parameters are obtained through the use of conditional moment generating functions. We also derive confidence intervals for the parameters using these exact distributions, asymptotic distributions of the MLEs and the parametric bootstrap methods, and assess their performance through a Monte Carlo simulation study. Finally, we present two examples to illustrate all the methods of inference discussed here.     

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??In this paper, precise large deviations of nonnegative, non-identical distributions and negatively associated random variables are investigated. Under certain conditions, the lower bound of the precise large deviations for the non-random sum is solved and the uniformly asymptotic results for the corresponding random sum are obtained. At the same time, we deeply discussed the compound renewal risk model, in which we found that the compound renewal risk model can be equivalent to renewal risk model under certain conditions. The relative research results of precise large deviations are applied to the more practical compound renewal risk model, and the theoretical and practical values are verified. In addition, this paper also shows that the impact of this dependency relationship between random variables to precise large deviations of the final result is not significant.     

Innovations algorithm asymptotics for periodically stationary time series with heavy tails

The innovations algorithm can be used to obtain parameter estimates for periodically stationary time series models. In this paper we compute the asymptotic distribution for these estimates in the case where the underlying noise sequence has infinite fourth moment but finite second moment. In this case, the sample covariances on which the innovations algorithm are based are known to be asymptotically stable. The asymptotic results developed here are useful to determine which model parameters are significant. In the process, we also compute the asymptotic distributions of least squares estimates of parameters in an autoregressive model.     

Asymptotic Properties of a Class of Mixture Models for Failure Data: The Interior and Boundary Cases

We analyse an exponential family of distributions which generalises the exponential distribution for censored failure time data, analogous to the way in which the class of generalised linear models generalises the normal distribution. The parameter of the distribution depends on a linear combination of covariates via a possibly nonlinear link function, and we allow another level of heterogeneity: the data may contain "immune" individuals who are not subject to failure. Thus the data is modelled by a mixture of a distribution from the exponential family and a "mass at infinity" representing individuals who never fail. Our results include large sample distributions for parameter estimators and for hypothesis test statistics obtained by maximising the likelihood of a sample. The asymptotic distribution of the likelihood ratio test statistic for the hypothesis that there are no immunes present in the population is shown to be "non-standard"; it is a 50-50 mixture of a chi-squared distribution on 1 degree of freedom and a point mass at 0. Our analysis clearly shows how "negligibility" of individual covariate values and "sufficient followup" conditions are required for the asymptotic properties.     

Density of Space-Time Distribution of Brownian First Hitting of a Disc and a Ball

We compute the joint distribution of the site and the time at which a d-dimensional standard Brownian motion ((B˙t)) hits the surface of the ball ((U(a) ={—x—<a})) for the first time. The asymptotic form of its density is obtained when either the hitting time or the starting site ((B˙0)) becomes large. Our results entail that if Brownian motion is started at ((x)) and conditioned to hit ((U(a))), at time t, the distribution of the hitting site approaches the uniform distribution or the point mass at ((ax/—x—)) according as ((—x—/t)) tends to zero or infinity; in each case we provide a precise asymptotic estimate of the density. In the case when ((—x—/t)) tends to a positive constant we show the convergence of the density and derive an analytic expression of the limit density.     

Mode estimation for discrete distributions

The problem of estimating the mode of a discrete distribution is considered. New characterizations of discrete unimodal and multi-modal distributions are obtained. The proposed mode estimator is essentially the sample mode, modulo appropriate modifications when the sample mode is not well defined. In the case of i.i.d. observations coming from a unimodal discrete distribution, our proposed mode estimator is shown to possess a number of strong asymptotic properties. Many of these results extend to the case of multi-modal discrete distributions as well. Our method also applies — and we have similar asymptotic results — to the problem of mode estimation based on finitely many observations on a Markov chain whose equilibrium distribution is the underlying unimodal distribution. For unimodal discrete distributions, we also propose a consistent large sample test of mode based on the proposed statistic. Applications of mode estimation problem in Monte-Carlo optimization problem using the Hastings Metropolis chain and in prediction problem using binary response variable, specially in the context of dose-response experiments, are also illustrated.     

Modified Unit Root Tests with Nuisance Parameter Free Asymptotic Distributions

It is well-known that the asymptotic distributions of the Dickey-Fuller (DF) tests for a unit root with linear process errors are not free of nuisance parameters. In this paper, we introduce a consistent estimator for the nuisance parameters and then use it to modify the DF tests, denoted as R-tests. Under fairly mild moment and summability conditions on the errors, we show that the asymptotic distributions of the R-tests are of the same as the Dickey-Fuller distributions. In Monte Carlo experiments, the R-tests are shown to have improved size properties.     

Near-exact distributions for the likelihood ratio test statistic to test equality of several variance-covariance matrices in elliptically contoured distributions

The exact distribution of the likelihood ratio test statistic to test the equality of several variance-covariance matrices has a non-manageable form. On the other hand, the existing asymptotic approximations do not exhibit the necessary precision for many applications. For these reasons, the development of near-exact approximations to the distribution of this statistic, arising from a different method of approximating distributions, emerges as a desirable goal. These distributions, while being manageable are much closer to the exact distribution than the usual asymptotic distributions and opposite to these, are also asymptotic for increasing number of variables and matrices involved. Computational modules to implement the near-exact distributions are made available on a web-site.     

Estimates of oscillatory integrals with stationary phase and singular amplitude: Applications to propagation features for dispersive equations

In this paper, we study time‐asymptotic propagation phenomena for a class of dispersive equations on the line by exploiting precise estimates of oscillatory integrals. We propose first an extension of the van der Corput Lemma to the case of phases which may have a stationary point of real order and amplitudes allowed to have an integrable singular point. The resulting estimates provide optimal decay rates which show explicitly the influence of these two particular points. Then we apply these abstract results to solution formulas of a class of dispersive equations on the line defined by Fourier multipliers. Under the hypothesis that the Fourier transform of the initial data has a compact support or an integrable singular point, we derive uniform estimates of the solutions in space‐time cones, describing their motions when the time tends to infinity. The method permits also to show that symbols having a restricted growth at infinity may influence the dispersion of the solutions: we prove the existence of a cone, depending only on the symbol, in which the solution is time‐asymptotically localized. This corresponds to an asymptotic version of the notion of causality for initial data without compact support.     

Exact and asymptotic distributions of LULU smoothers

This paper considers a class of non-linear smoothers, called LULU smoothers, introduced by Rohwer in the late eighties in the mathematics literature, and since then investigated fairly extensively by a number of authors for its mathematical properties. They have been successfully applied in various engineering and scientific problems. However, to date their distribution theory has not received any attention in the literature. In this paper we derive their exact as well as asymptotic distributions and show their relationship to the upper order statistics.     

Convexity properties,moment inequalities and asymptotic exponential approximations for delay distributions in G1/G/1 systems

Pollaczek distributions pervade the class of delay distibutions in G1/G/1 systems with concave service time distributions. When the service time distribution has finite support and the delay distribution is absolutely continuous on (0, ∞), one can find a distribution with a pure exponential tail that satisfies the corresponding Wiener-Hopf integral equation except for values of the argument that belong to the support in question or to a translate thereof. Again for an exponentially decaying delay distribution, one can formulate sufficient moment inequalities which ensure the existence of asymptotic upper and lower bounds derived from M/D/1 and M/M/1 delay distributions which agree with the former in terms of the first two moments.     

Asymptotic properties of sample quantiles of discrete distributions

The asymptotic distribution of sample quantiles in the classical definition is well-known to be normal for absolutely continuous distributions. However, this is no longer true for discrete distributions or samples with ties. We show that the definition of sample quantiles based on mid-distribution functions resolves this issue and provides a unified framework for asymptotic properties of sample quantiles from absolutely continuous and from discrete distributions. We demonstrate that the same asymptotic normal distribution result as for the classical sample quantiles holds at differentiable points, whereas a more general form arises for distributions whose cumulative distribution function has only one-sided differentiability. For discrete distributions with finite support, the same type of asymptotics holds and the sample quantiles based on mid-distribution functions either follow a normal or a two-piece normal distribution. We also calculate the exact distribution of these sample quantiles for the binomial and Poisson distributions. We illustrate the asymptotic results with simulations.     

The asymptotic of the necessary sample size in testing the hypotheses on the shape parameter of a distribution close to the normal one

We study the problem of testing the hypothesis on the “approximate normality” formulated in terms of large values of the shape parameter of an asymptotically normal underlying distribution. Considering the examples of gamma-and generalized Birnbaum—Saunders distributions, we propose one way to obtain the asymptotic of the necessary sample size for testing the mentioned hypothesis. Our approach differs from those based on contiguous alternatives or on the use of the large deviations theory for distributions of sums of independent random variables. Our method yields remarkably precise approximate formulas, what is illustrated by numerical data.     

Detection of Singularities by Discrete Multiscale Directional Representations

One of the most remarkable properties of the continuous curvelet and shearlet transforms is their sensitivity to the directional regularity of functions and distributions. As a consequence of this property, these transforms can be used to characterize the geometry of edge singularities of functions and distributions by their asymptotic decay at fine scales. This ability is a major extension of the conventional continuous wavelet transform which can only describe pointwise regularity properties. However, while in the case of wavelets it is relatively easy to relate the asymptotic properties of the continuous transform to properties of discrete wavelet coefficients, this problem is surprisingly challenging in the case of discrete curvelets and shearlets where one wants to handle also the geometry of the singularity. No result for the discrete case was known so far. In this paper, we derive non-asymptotic estimates showing that discrete shearlet coefficients can detect, in a precise sense, the location and orientation of curvilinear edges. We discuss connections and implications of this result to sparse approximations and other applications.