Transient analysis of one-sided Lévy-driven queues

In this article, we analyze the transient behavior of the workload process in a Lévy-driven queue. We are interested in the value of the workload process at a random epoch; this epoch is distributed as the sum of independent exponential random variables. We consider both cases of spectrally one-sided Lévy input processes, for which we succeed in deriving explicit results. As an application, we approximate the mean and the Laplace transform of the workload process after a deterministic time.     

Stochastic comparison of processes generated by random interruptions of monotone functions and related results

Many stochastic processes considered in applied probability models, and, in particular, in reliability theory, are processes of the following form: Shocks occur according to some point process, and each shock causes the process to have a random jump. Between shocks the process increases or decreases in some deterministic fashion. In this paper we study processes for which the rate of increase or decrease between shocks depends only on the height of the process. For such processes we find conditions under which the processes can be stochastically compared. We also study hybrid processes in which periods of increase and periods of decrease alternate. A further result yields a stochastic comparison of processes that start with a random jump, rather than processes in which there is at the beginning some random delay time before the first jump.Supported by NSF Grant DMS 9303891.     

An Asymptotic Linear Representation for the Breslow Estimator

We provide an asymptotic linear representation for the Breslow estimator of the baseline cumulative hazard function in the Cox model. Our 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.     

Random sampling of long-memory stationary processes

This paper investigates the second order properties of a stationary process after random sampling. While a short memory process gives always rise to a short memory one, we prove that long-memory can disappear when the sampling law has heavy enough tails. We prove that under rather general conditions the existence of the spectral density is preserved by random sampling. We also investigate the effects of deterministic sampling on seasonal long-memory.     

On random grouping in goodness of fit tests of discrete distributions

We consider the problem of random grouping of data from discrete distributions to form χ² goodness of fit tests. In general the random partitions need not converge to the partition one gets using the true distribution function and the same partitioning scheme. We present a method of guaranteeing the convergence of the random partition and yielding the usual χ² asymptotics.     

Empirical-distribution-function goodness-of-fit tests for discrete models

We present a simple framework for studying empirical-distribution-function goodness-of-fit tests for discrete models. A key tool is a weak-convergence result for an estimated discrete empirical process, regarded as a random element in some suitable sequence space. Special emphasis is given to the problem of testing for a Poisson model and for the geometric distribution. Simulations show that parametric bootstrap versions of the tests maintain a nominal level of significance very closely even for small samples where reliance upon asymptotic critical values is doubtful.     

Poisson bounds for moments of shot noise processes

Starting from general moment formulas for shot noise processes (v(t)), for which a simple proof is given by means of the CAMPBELL theorem concerning marked point pro¬cesses, and using certain monotonicity property, bounds are obtained for E[v(t)]² resp. for the variance Var v(t) under the additional assumption that the intershot times are inde¬pendent random variables with distribution having the property 1SFBXJE (NWTJE)     

Asymptotic efficiency of model selection criteria: the nonzero mean gaussian ar(∞) case

Motivated by Shibata’s (1980) asymptotic efficiency results this paper dis-cusses the asymptotic efficiency of the order selected by a selection procedure for an infinite order autoregressive process with nonzero mean and unob servable errors that constitute a sequence of independent Gaussian random variables with mean zero and variance σ² The asymptotic efficiency is established for AIC–type selection criteria such as AIC’, FPE, and S_n(k). In addition, some asymptotic results about the estimators of the parameters of the process and the error–sequence are presented.     

Point Processes with a Generalized Order Statistic Property

Mixed Poisson processes are characterized by a well-known order statistic property: their occurrence times are distributed like ordinary uniform order statistics, given the state of the process at a certain time. We study a generalization of this property using a generalized model of ordered random variables, including a sequence of real parameters.     

Uniform-in-bandwidth kernel estimation for censored data

We present a sharp uniform-in-bandwidth functional limit law for the increments of the Kaplan–Meier empirical process based upon right-censored random data. We apply this result to obtain limit laws for nonparametric kernel estimators of local functionals of lifetime densities, which are uniform with respect to the choices of bandwidth and kernel. These are established in the framework of convergence in probability, and we allow the bandwidth to vary within the complete range for which the estimators are consistent. We provide explicit values for the asymptotic limiting constant for the sup-norm of the estimation random error.     

On Tests of Uniformity for Randomly Distributed Arcs on a Circle

Let n points be independently distributed on a circle. Moving in a counter-clockwise direction, place arcs of length a on the circle with the i^th are starting at the i^th point. We describe three simple tests of the hypothesis of uniformity based on vacancy, on number of spacings and on the length of the maximal spacing. The tests do not require knowledge of the random points. The asymptotic power of these tests is investigated, and it is shown that vacancy-based tests perform best of the three.     

A Strong Invariance Theorem of the Tail Empirical Copula Processes

We study the behavior of bivariate empirical copula process 𝔾 _n (·, ·) on pavements [0, k _n /n]² of [0, 1]², where k _n is a sequence of positive constants fulfilling some conditions. We provide a upper bound for the strong approximation of 𝔾 _n (·, ·) by a Gaussian process when k _n /n↘γ as n → ∞, where 0 ≤ γ ≤1.     

Asymptotic expansions in non-linear renewal theory

We obtain first order asymptotic expansions for the distribution of the excess of a standard normal random walk over a curved boundary and the error probabilities of some repeated significance tests. The key step in the analysis is an asymptotic expansion for the conditional probability that the random walk has not crossed the boundary before the N step, given that it is near the boundary after the n^th step.     

La méthode de vraisemblance pour les processus linéaires spatiaux définis sur un treillis triangulaire

Spatial linear processes {X_s, s ? T} where T is a triangular lattice in R² are considered. Special attention is given to the class of spatial moving-average processes. Precisely, for each site s T, the variable X_s is defined as a linear combination of real-valued random shocks located at the vertices of regular concentric hexagons centered at s. For Gaussian random shocks, the process is also Gaussian, and estimates of its parameters are obtained by maximizing the exact likelihood. For non-Gaussian random shocks, the exact likelihood is difficult to obtain; however, the Gaussian likelihood is still used giving the pseudo-Gaussian likelihood estimates. The behaviour of these estimates is analyzed through the study of asymptotic properties and some simulation experiments based on an isotropic model defined with one coefficient.     

The X2-goodness-of-fit tests with moment type estimators

In the x²-goodness-of-fit test the underlying null hypothesis usually involves unknown parameters. In this article we study the asymptotic distribution of the Pearson statistic when the unknown parameters are estimated by a moment type estimator based on the ungrouped data. As is expected the usual Pearson statistic is no longer asymptotically x²-distributed in this situation. We propose a statistic [Qcirc] which under certain regularity conditions is asymptotically x²-distributed. We also propose a statistic Q? for the goodness-of-fit test when the class boundaries are random. The asymptotic powers of [Qcirc] and [Qcirc]? tests are discussed.     

Nonstationary Data Analysis by Time Deformation

ABSTRACT

In this article we discuss methodology for analyzing nonstationary time series whose periodic nature changes approximately linearly with time. We make use of the M-stationary process to describe such data sets, and in particular we use the discrete Euler(p) model to obtain forecasts and estimate the spectral characteristics. We discuss the use of the M-spectrum for displaying linear time-varying periodic content in a time series realization in much the same way that the spectrum shows periodic content within a realization of a stationary series. We also introduce the instantaneous frequency and spectrum of an M-stationary process for purposes of describing how frequency changes with time. To illustrate our techniques we use one simulated data set and two bat echolocation signals that show time varying frequency behavior. Our results indicate that for data whose periodic content is changing approximately linearly in time, the Euler model serves as a very good model for spectral analysis, filtering, and forecasting. Additionally, the instantaneous spectrum is shown to provide better representation of the time-varying frequency content in the data than window-based techniques such as the Gabor and wavelet transforms. Finally, it is noted that the results of this article can be extended to processes whose frequencies change like at^α, a > 0, ?∞ < α < ? ∞.     

Rényi statistics for testing composite hypotheses in general exponential models

We introduce a family of Rényi statistics of orders r?∈?R for testing composite hypotheses in general exponential models, as alternatives to the previously considered generalized likelihood ratio (GLR) statistic and generalized Wald statistic. If appropriately normalized exponential models converge in a specific sense when the sample size (observation window) tends to infinity, and if the hypothesis is regular, then these statistics are shown to be χ²-distributed under the hypothesis. The corresponding Rényi tests are shown to be consistent. The exact sizes and powers of asymptotically α-size Rényi, GLR and generalized Wald tests are evaluated for a concrete hypothesis about a bivariate Lévy process and moderate observation windows. In this concrete situation the exact sizes of the Rényi test of the order r?=?2 practically coincide with those of the GLR and generalized Wald tests but the exact powers of the Rényi test are on average somewhat better.     

The asymptotic distributions of maximum likelihood ratio test and maximally selected χ2-test in binomial observations

We considered binomial distributed random variables whose parameters are unknown and some of those parameters need to be estimated. We studied the maximum likelihood ratio test and the maximally selected χ²-test to detect if there is a change in the distributions among the random variables. Their limit distributions under the null hypothesis and their asymptotic distributions under the alternative hypothesis were obtained when the number of the observations is fixed. We discussed the properties of the limit distribution and found an efficient way to calculate the probability of multivariate normal random variables. Finally, those results for both tests have been applied to examples of Lindisfarne's data, the Talipes Data. Our conclusions are consistent with other researchers' findings.     

Convergence analysis of parametric identification methods for jump processes

The asymptotic properties of recursive estimate for the parameter of the signal process of a jump process are investigated. When increasing the observation period, conditions are proposed that guarantee the consistency and asymptotic normality of the estimate. An estimate of the unknown parameter for the random intensity of an extended gamma process model is discussed by way of example.     

Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum

This paper defines and studies a new class of non-stationary random processes constructed from discrete non-decimated wavelets which generalizes the Cramér (Fourier) representation of stationary time series. We define an evolutionary wavelet spectrum (EWS) which quantifies how process power varies locally over time and scale. We show how the EWS may be rigorously estimated by a smoothed wavelet periodogram and how both these quantities may be inverted to provide an estimable time-localized autocovariance. We illustrate our theory with a pedagogical example based on discrete non-decimated Haar wavelets and also a real medical time series example.