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An expansion is derived for the density of the first time a Brownian path crosses a perturbed linear boundary α+εf(t). When the perturbation f(t) is a finite mixture of negative exponentials of either sign the expansion is shown to converge for all values of the perturbation parameter ε. Numerical examples suggest that the technique works well for a wider choice of f(t), including cases where f(t) is periodic.
Stein's method for compound Poisson approximation was introduced by Barbour, Chen and Loh. One difficulty in applying the method is that the bounds on the solutions of the Stein equation are by no means as good as for Poisson approximation. We show that, for the Kolmogorov metric and under a condition on the parameters of the approximating compound Poisson distribution, bounds comparable with those obtained for the Poisson distribution can be recovered.
Signed Poisson approximation is a signed measure, has the structure of the Poisson distribution and can be regarded as a special sort of asymptotic expansion when expansion is in the exponent. For certain lattice distributions signed Poisson approximation combines advantages of both the normal and Poisson approximations. For the generalized binomial distribution estimates with respect to the total variation and Wasserstein distances are obtained. The results are exemplified by Bernoulli decomposable variables.
Levy and Taqqu (2000) considered a renewal reward process with both inter-renewal times and rewards that have heavy tails with exponents α and β, respectively. When 1<α<β< 2 and the renewal reward process is suitably normalized, the authors found that it converges to a symmetric β-stable process Zβ(t), t∈[0,1] which possesses stationary increments and is self-similar. They identified the limit process through its finite-dimensional characteristic functions. We provide an integral representation for the process and show that it does not belong to the family of linear fractional stable motions.
We study a concentration property of product probability measures with respect to the supremum distance. This property is shown to be equivalent to the conditions given by de Haan and Ridder for the stochastic boundedness of centred extreme samples.
We derive the chaotic expansion of the product of nth- and first-order multiple stochastic integrals with respect to certain normal martingales. This is done by application of the classical and quantum product formulae for multiple stochastic integrals. Our approach extends existing results on chaotic calculus for normal martingales and exhibits properties, relative to multiple stochastic integrals, polynomials and Wick products, that characterize the Wiener and Poisson processes.
Orthogeodesic models admit marginal local cuts and therefore separate inference on subparameters is asymptotically justified. Doubly-flat orthogeodesic models admit local cuts marginally and conditionally. Two important empirical models for panel data are used to illustrate this property and demonstrate its usefulness.
Let , , be a stochastic process. Suppose that the process may not be continuously observed, yet an inference which is related to its probabilistic parameters, or to its sample path, is required. The main purpose of this paper is to study sampling plans. A sampling plan is a method for deciding about time instants at which the process is observed. We study the effect of various sampling plans and sampling rates on the expected time to an alarm in change-point problems (of the mean). Our main effort is studying the asymptotic variance of the sum of the sampled observations until time t. This variance determines asymptotically the expected time to an alarm. As a by-product, we obtain the asymptotic variances of natural estimators for and for . Obviously, as the sampling rate is increased, a better estimation is possible. Our study enables us to decide on the `right' sampling rate. This is analogous to the problem of deciding on the `right' sample size in the case of independently and identically distributed observations.
There is a firm belief in the literature on statistical applications of wavelets that adaptive procedures developed for Fourier series, labelled by that literature as `linear', are inadmissible because they are created for estimation of smooth functions and cannot attain optimal rates of mean integrated squared error convergence whenever an underlying function is spatially inhomogeneous, for instance, when it contains spikes/jumps and smooth parts. I use the recent remarkable results by Hall, Kerkyacharian and Picard on block-thresholded wavelet estimation to present a counterexample to that belief.
We establish the consistency and asymptotic normality of a certain minimum contrast estimator, introduced by Taniguchi (1979), for Gaussian long-range dependent processes. The estimator is based on regression over the log-periodogram in a parametric setting.
Suppose that U is a U-statistic of degree 2 based on N random observations drawn without replacement from a finite population. For the distribution of a standardized version of U we construct an Edgeworth expansion with remainder O(N-1) provided that the linear part of the statistic satisfies a Cramér type condition.