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A new goodness-of-fit (GoF) test is proposed and investigated for the Gaussianity of the observed functional data. The test statistic is the Cramér-von Mises distance between the observed empirical characteristic functional (CF) and the theoretical CF corresponding to the null hypothesis stating that the functional observations (process paths) were generated from a specific parametric family of Gaussian processes, possibly with unknown parameters. The asymptotic null distribution of the proposed test statistic is derived also in the presence of these nuisance parameters, the consistency of the classical parametric bootstrap is established, and some particular choices of the necessary tuning parameters are discussed. The empirical level and power are investigated in a simulation study involving GoF tests of an Ornstein–Uhlenbeck process, Vašíček model, or a (fractional) Brownian motion, both with and without nuisance parameters, with suitable Gaussian and non-Gaussian alternatives.

Motivated by a neuroscience question about synchrony detection in spike train analysis, we deal with the independence testing problem for point processes. We introduce nonparametric test statistics, which are rescaled general U-statistics, whose corresponding critical values are constructed from bootstrap and randomization/permutation approaches, making as few assumptions as possible on the underlying distribution of the point processes. We derive general consistency results for the bootstrap and for the permutation w.r.t. Wasserstein's metric, which induces weak convergence as well as convergence of second-order moments. The obtained bootstrap or permutation independence tests are thus proved to be asymptotically of the prescribed size, and to be consistent against any reasonable alternative. A simulation study is performed to illustrate the derived theoretical results, and to compare the performance of our new tests with existing ones in the neuroscientific literature.

In this paper, we present a test for the maximal rank of the matrix-valued volatility process in the continuous Itô semimartingale framework. Our idea is based upon a random perturbation of the original high frequency observations of an Itô semimartingale, which opens the way for rank testing. We develop the complete limit theory for the test statistic and apply it to various null and alternative hypotheses. Finally, we demonstrate a homoscedasticity test for the rank process.

In this paper, we prove the asymptotic normality of the conditional distribution of the number of runs given the number of successes for a sequence of independent Bernoulli random variables. In our proof, the Frobenius-Harper technique is used to represent the number of runs as the sum of independent and not necessarily identically distributed Bernoulli random variables. Then, an asymptotic conditional test for independence is provided. Our simulation results exhibit that the test based on conditional distribution performs better that one based on unconditional distribution, over the entire range of success probability and first order correlation. In addition, the UMVUEs of the factorial moments and the probabilities of the number of runs are presented in this paper.

Customers arrive sequentially at times x 1 < x 2 < · · · < x n and stay for independent random times Z 1, …, Z n > 0. The Z-variables all have the same distribution Q. We are interested in situations where the data are incomplete in the sense that only the order statistics associated with the departure times x i + Z i are known, or that the only available information is the order in which the customers arrive and depart. In the former case we explore possibilities for the reconstruction of the correct matching of arrival and departure times. In the latter case we propose a test for exponentiality.

We formulate nonparametric and semiparametric hypothesis testing of multivariate stationary linear time series in a unified fashion and propose new test statistics based on estimators of the spectral density matrix. The limiting distributions of these test statistics under null hypotheses are always normal distributions, and they can be implemented easily for practical use. If null hypotheses are false, as the sample size goes to infinity, they diverge to infinity and consequently are consistent tests for any alternative. The approach can be applied to various null hypotheses such as the independence between the component series, the equality of the autocovariance functions or the autocorelation functions of the component series, the separability of the covariance matrix function and the time reversibility. Furthermore, a null hypothesis with a nonlinear constraint like the conditional independence between the two series can be tested in the same way.

We derive an approximation of a density estimator based on weakly dependent random vectors by a density estimator built from independent random vectors. We construct, on a sufficiently rich probability space, such a pairing of the random variables of both experiments that the set of observations {X1,... , Xn} from the time series model is nearly the same as the set of observations {Y1,... , Yn} from the i.i.d. model. With a high probability, all sets of the form ({X1,... , Xn}Δ {Y1,... , Yn})∩ ([a1, b1] × ⋯ × [ad, bd]) contain no more than O({[n1/2Π (bi- ai)] + 1} log(n)) elements, respectively. Although this does not imply very much for parametric problems, it has important implications in nonparametric statistics. It yields a strong approximation of a kernel estimator of the stationary density by a kernel density estimator in the i.i.d. model. Moreover, it is shown that such a strong approximation is also valid for the standard bootstrap and the smoothed bootstrap. Using these results we derive simultaneous confidence bands as well as supremum-type nonparametric tests based on reasoning for the i.i.d. model.

Consider a first-order autoregressive process Xt= β Xt - 1+ εt, where {εt} are independent and identically distributed random errors with mean 0 and variance 1. It is shown that when β = 1 the standard bootstrap least squares estimate of β is asymptotically invalid, even if the error distribution is assumed to be normal. The conditional limit distribution of the bootstrap estimate at β = 1 is shown to converge to a random distribution.

I explain so-called quantum nonlocality experiments and discuss how to optimize them. Statistical tools from missing data maximum likelihood are crucial. New results are given on CGLMP, CH and ladder inequalities. Open problems are also discussed.

Asymptotic validity of the bootstrap is established for the least squares estimate of the parameter of an explosive first-order autoregressive process. It is noted that nonnormal limit distributions are obtained for both the traditional and the bootstrap estimates. The theoretical bootstrap validity results are supported by appropriate simulation.