Some remarks on minimax sequential tests for stationary ornstein–uhilenbeck processes

In this paper sequential tests of one sided hypotheses H₁ : μ ≤ μ₀ and H₂ : μ > μ₀ for the velocity parameter of a stationary Ornstein-Uhlenbeck process(OUP) are discussed. When the variance parameter of the OUP is known the results obtained here complement a recent work of Roters (1991) by proving the existence of minimax sequential tests under concave and symmetric (about μ₀) loss l(μ)with l(μ₀) > 0 and linear costs of observation. For unkown variance parameter minimax sequential tests are derived for all loss functions discussed here and in the above mentioned paper.     

Optimality Of sequential prorability ratio tests for a class of continuous parameter stochastic processes

For a wide class of stochastic processes including processes belonging to the curved exponential families, it is proved that Wald SPRT is optimal in the sense of minimizing the expectations of increasing processes associated with the stochastic process. The increasing processes are given by the Doob-Meyer decompositions of the log-probability ratio under the two hypotheses.     

Asymptotically optimal sequential tests for nonhomogeneous processes

It is shown that under certain conditions the matrix sequential probability ratio test (SPRT) and the combinations of "rejecting" SPRTs minimize all moments of the stopping time distribution in the problem of sequential testing of several simple hypotheses for nonhomogeneous processes when probabilities of errors tend to zero. We consider the general case of observation process with discrete or continuous time parameter and asymmetric (relative to probabilities of errors) classes of tests.     

Bias reduction via resampling for estimation following sequential tests

It is well known that maximum likelihood (ML) estimation results in biased estimates when estimating parameters following a sequential test. Existing bias correction methods rely on explicit calculations of the bias that are often difficult to derive. We suggest a simple alternative to the existing methods. The new approach relies on approximating the bias of the estimate using a bootstrap method. It requires bootstrapping the sequential testing process by resampling observations from a distribution based on the ML estimate. Each bootstrap process will give a new ML estimate, and the corresponding bootstrap mean can be used to calibrate the estimate. An advantage of the new method over the existing methods is that the same procedure can be used under different stopping rules and different study designs. Simulation results suggest that this method performs competitively with existing methods.     

Asymptotics for stationary very nearly unit root processes

Abstract. This article considers a mean zero stationary first‐order autoregressive (AR) model. It is shown that the least squares estimator and t statistic have Cauchy and standard normal asymptotic distributions, respectively, when the AR parameter ρ_n is very near to one in the sense that 1 ? ρ_n = o(n^?1).     

Repeated significance tests for stationary arm processes

We consider testing problems for the parameters of autoregressive moving average (ARMA)-processes. For a broad class of distributions of the white noise process, which consists of i.i.d.random variables,we prove locally asymptotic Wiener (LAW) structure in extension of locally asymptotic normality (LAN) for our model. On this basis invariance principles can be established in a very convenient way. We are able to give asymptotic results for the proposed repeated significance test(RST) procedure under hypothesis as well as under contiguous alternatives.Finally we introduce RST-procedures with estimated scores     

Second-order properties of locally stationary processes

Abstract.  In this article, we investigate an optimal property of the maximum likelihood estimator of Gaussian locally stationary processes by the second-order approximation. In the case where the model is correctly specified, it is shown that appropriate modifications of the maximum likelihood estimator for Gaussian locally stationary processes is second-order asymptotically efficient. We also discuss second-order robustness properties.     

Transformation to approximate independence for locally stationary Gaussian processes

We provide new approximations for the likelihood of a time series under the locally stationary Gaussian process model. The likelihood approximations are valid even in cases when the evolutionary spectrum is not smooth in the rescaled time domain. We describe a broad class of models for the evolutionary spectrum for which the approximations can be computed particularly efficiently. In developing the approximations, we extend to the locally stationary case the idea that the discrete Fourier transform is a decorrelating transformation for stationary time series. The approximations are applied to fit non‐stationary time‐series models to high‐frequency temperature data. For these data, we fit evolutionary spectra that are piecewise constant in time and use a genetic algorithm to search for the best partition of the time interval.     

Bootstrapping unit root tests for integrated processes

Abstract. In this paper, we consider two bootstrap algorithms for testing unit roots under the condition that the observed process is unit root integrated. The first method consists of generating the resampled data after fitting an autoregressive model to the first differences of the observations. The second method consists of applying the stationary bootstrap to the first differences. Both procedures are shown to give methods that approach the correct asymptotic distribution under the null hypothesis of a unit root. We also present a Monte-Carlo study comparing the two methods for some ARIMA models.     

Optimal two-sided group sequential tests

We derive two-sided group sequential tests for normal responses with known variance which minimise expected sample size; minimisation is at a single value of the normal mean or integrated with respect to a normal density. In deriving these tests we use a method analogous to that used by Eales and Jennison (1992) to compute optimal one-sided tests. We present a comparison of our optimal tests with existing tests.     

Locally stationary harmonizable complex improper stochastic processes

This article concerns continuous‐time second‐order complex‐valued improper stochastic processes that are harmonizable and locally stationary in Silverman's sense. We study necessary and sufficient conditions for the property of local stationarity in the time and frequency domains. A sufficient condition by Silverman is generalized and extended to the improper case. We obtain a result on the absolute continuity of the complementary spectral measure with respect to the spectral measure, which is related to a spectral characterization of improper wide‐sense stationary processes.     

Testing non‐parametric hypotheses for stationary processes by estimating minimal distances

In this article, new tests for non‐parametric hypotheses in stationary processes are proposed. Our approach is based on an estimate of the L²‐distance between the spectral density matrix and its best approximation under the null hypothesis. We explain the main idea in the problem of testing for a constant spectral density matrix and in the problem of comparing the spectral densities of several correlated stationary time series. The method is based on direct estimation of integrals of the spectral density matrix and does not require the specification of smoothing parameters. We show that the limit distribution of the proposed test statistic is normal and investigate the finite sample properties of the resulting tests by means of a small simulation study.     

A Modification of schwarz's sequential likelihood ratio tests in multivariate sequential analysis

By developing and applying certain results on boundary crossing probabilities of generalized likelihood ratio statistics in multiparameter exponential families, we show that Lai's (1988a) modification of Schwarz's (1962) sequential likelihood ratio tests of one-sided hypotheses in the one-parameter case can be extended to general hypotheses in the multiparameter case. Such tests are shown to be asymptotically optimal, from both frequentist and Bayesian points of view, and numerical results are also given to demonstrate their performance.     

Necessary and sufficient conditions for bounded stopping time of sequential bayes tests in one parameter exponential families1

The problem of sequently testing one-sided hypotheses about the parameter in a one-parameter exponential family, continuous with respect to Lebesgue measure, is consiered in a Bayesian framework. The paper gives a simple necessary and sufficient condition for the Bayes sampling rule to be bounded. The risk fucntion is taken to be a constant times the number of observations plus a weighted probability of error. The sufficient condition for boundness is generalized to other risk functions as wells.     

A class of sequential grouped signed rank tests

A class of nonparametric sequential tests for testing symmetry about a specified value is developed for the case where observations are taken in groups. Signed rank statistics are computed indepently for each group and the test is based on the sum of these statistics used with linear boundaries. The procedure is formulated as a Markov chain so that the exact small sample properties of the test can be evaluated. The asymtotic OC and ASN functions are developed for local alternatives. For the case where Wilcoxon scores are used tables are given which make it easy to design a test to achieve specified error probabilities.     

Reducing size distortions of parametric stationarity tests

The use of asymptotic critical values in stationarity tests against the alternative of a unit root process is known to lead to over-rejections in finite samples when the considered process is stationary but highly persistent. We claim that, in recent parametric tests, this is caused by estimation errors which result when the autoregressive parameters used to describe the short-run dynamics of the process are replaced by estimators. We suggest a modification that corrects for these errors. Simulation results show that the modified test works reasonably well when the persistence is moderate and there is no time trend in the model but it is less effective when the model contains a time trend. An empirical illustration with inflation rate data is provided.     

On the point of truncation of bayes sequential tests for a discrete univariate distribution of the exponential type

A theoretical result due to S. N. Ray (1965) and Mann and Bratcher (1979) is refined and applied to several discrete one dimensional univariate distributions of the exponential type. In these cases a means of obtaining upper bounds to the exact stage of truncation of a Bayes sequential test is given.     

Sequential estimation for the parameters of a stationary auto regressive model

This paper considers the problem of sequential point estimation and fixed accuracy confidence set procedures of autoregressive parameters in a ρ-th order stationary autoregressive model. The sequential estimator proposed here is based on the least squares estimator and is shown to be risk efficient as the cost of estimation error tends to infinity. Furthermore, the proposed procedure for fixed-width confidence set is shown to be both asymptotically consistent and asymptotically efficient as the width approaches zero.     

Some comments on specification tests in nonparametric absolutely regular processes

In this note, several aspects of a recently proposed specification test in nonparametric models driven by an absolutely regular process are discussed. In particular, we give a more detailed asymptotic analysis of tests based on kernel methods under fixed alternatives using a central limit theorem for U-statistics with n-dependent nondegenerate kernel. As a by-product, it is demonstrated that several results regarding the asymptotic distribution or goodness-of-fit tests are incorrectly stated in the literature. Our result also indicates that results on the asymptotic equivalence of nonparametric autoregression and nonparametric regression cannot be used for the asymptotic analysis of goodness-of-fit tests under fixed alternatives.