M–Estimtors and l–estimators of location: uniform integrability and asymptotic risk–efficient sequential versions

In the context of asymptotically minimum risk (sequential) point estimation of location of a symmetric distribution, M-and L-estimators are considered, and various properties of their sequential versions are studied. Asymptotic distributions of he allied stopping times are also derived. In this study, uniform integrability and moment convergence of (non-sequential) M- and L-estimators are established. These results have interest of therir own and provide the main tools for the proof of the other results presented. For the sequential estimators, their asymptotic risk efficiencies are shown to coincide with the asymptotic efficiencies of the respective non-sequential estimators; this enables one to construct the asymptotically minimax sequential M- and L-estimators in the model of contaminancy. Parallel results also hold for the rank estimators of location.     

Approximations to expected stopping times with applications to sequential estimation

In this paper we consider sharp asymptotic aproximations to renewal functions associated with general stochastic sequences. We establish a variant of Smith's renewal theorem with a bounded remainder term imposing moment conditions on the underlying stochastic sequence. This result is then applied to an asymptotic analysis of stopping times arising in sequential estimation problems. We show that the renewal theorem can be used for deriving approximations to the expected values of first passage times for random walks with dependent increments     

On sequential R estimation of location in the general behrens fisher model

For a general Behrens-Fisher model (relating to a shift in location between two symmetric distributions with possibly different shapes), asymptotic theory of sequential R-estimation is studied. Sequential versions of the usual two sample R-estimators and some alternative ones (based on the difference of one sample R-estimators) are considered and asymptotic risk efficiency results for these estimators are presented. In this context, under the general Behrens-Fisher model, for R-estimators, uniform integrability and moment convergence results are established.     

Asymptotic optimality of the wald sequential test

We give a generalization to the case of m hypotheses of a theorem of Lai and derive an asymptotic optimality property of the Wald sequential test for general, possibly dependent, random variables with respect to the r^th moment of observation time, under some r-quick convergence condition. We also extend the definition of this convergence and give an application to sequential analysis.     

Assessing Time‐Reversibility Under Minimal Assumptions

Abstract. This article considers a simple procedure for assessing whether a weakly dependent univariate stochastic process is time‐reversible. Our approach is based on a simple index of the deviation from zero of the median of the one‐dimensional marginal law of differenced data. An attractive feature of the method is that it requires no moment assumptions. Instead of relying on Gaussian asymptotic approximations, we consider using subsampling and resampling methods to construct confidence intervals for the time‐reversibility parameter, and show that such inference procedures are asymptotically valid under a mild mixing condition. The small‐sample properties of the proposed procedures are examined by means of Monte Carlo experiments and an application to real‐world data is also presented.     

Bootstrapping Sequential Change-Point Tests

Abstract

In this article we propose some bootstrapping methods to obtain critical values for sequential change-point tests. We consider a change in the mean with i.i.d. errors. Theoretical results show the asymptotic validity of the proposed bootstrap procedures. A simulation study compares the bootstrap and the asymptotic tests and shows that the studentized bootstrap test behave generally better than asymptotic tests if measured by α- resp. β-errors and its run length.     

Recursive estimation of quantitles using recursive kernel density estimators

Recursive estimation of quantiles may be obained via adaptive stochastic approximation

approximation theorms can be used to obtained the asympotic properties when the obervation are independent. for dependent sequences matingale theory cannot be applied straight forwardly as the tool for asympototic analysis.In this paper we consider both the case when the observation are i.i.d. and when they form a stationary and strongly regular process.the main result is sufficient condition for almost sure convergence in the strongly regular case.     

Sequential estimation of the autoregressive parameter in a first order autoregressive process

This paper considers the problem of sequential point estimation of the autoregressive parameter in a first order autoregressive model. The sequential estimator proposed here is based on the least squares estimator and is shown to be asymtotically risk efficient as the cost of estimation error tends to infinity, under certain regularity conditions. Furthermore, nonlinear renewal theory is used to obtain a second order approximation to the expected stopping time. The asymptotic normality and uniform integrability of the standardized stopping time are also established.     

On Truncatd sequential estimation of the drifting parametermean in the first order autoregressive models

This paper considers the problem of sequential point estimation of the drifting parameter mean in the first order autoregression process. The truncated sequential procedure proposed here is based on the least squares estimator and is shown to ensure the preassigned mean square accuracy of the estimates. The uniform in parameter asymptotic normality of the sequential estimator is established.     

Inference on Multivariate Heteroscedastic Time Varying Random Coefficient Models

In this article, we introduce the general setting of a multivariate time series autoregressive model with stochastic time‐varying coefficients and time‐varying conditional variance of the error process. This allows modelling VAR dynamics for non‐stationary time series and estimation of time‐varying parameter processes by the well‐known rolling regression estimation techniques. We establish consistency, convergence rates, and asymptotic normality for kernel estimators of the paths of coefficient processes and provide pointwise valid standard errors. The method is applied to a popular seven‐variable dataset to analyse evidence of time variation in empirical objects of interest for the DSGE (dynamic stochastic general equilibrium) literature.     

On Uniform Asymptotic Normality of Sequential Estimators for the Parameters in a Stable AR(1)

Abstract

A sequential procedure for estimating two autoregressive parameters is constructed. The uniform asymptotic normality of estimators is established.     

Random Walks with Drift – A Sequential Approach

Abstract. In this paper, sequential monitoring schemes to detect nonparametric drifts are studied for the random walk case. The procedure is based on a kernel smoother. As a by‐product we obtain the asymptotics of the Nadaraya–Watson estimator and its associated sequential partial sum process under non‐standard sampling. The asymptotic behaviour differs substantially from the stationary situation, if there is a unit root (random walk component). To obtain meaningful asymptotic results, we consider local nonparametric alternatives for the drift component. It turns out that the rate of convergence at which the drift vanishes determines whether the asymptotic properties of the monitoring procedure are determined by a deterministic or random function. Furthermore, we provide a theoretical result about the optimal kernel for a given alternative.     

Empirical Characteristic Functions-Based Estimation and Distance Correlation for Locally Stationary Processes

In this article, we propose a kernel-type estimator for the local characteristic function of locally stationary processes. Under weak moment conditions, we prove joint asymptotic normality for local empirical characteristic functions. For time-varying linear processes, we establish a central limit theorem under the assumption of finite absolute first moments of the process. Additionally, we prove weak convergence of the local empirical characteristic process. We apply our asymptotic results to parameter estimation. Furthermore, by extending the notion of distance correlation to locally stationary processes, we are able to provide asymptotic theory for local empirical distance correlations. Finally, we provide a simulation study on minimum distance estimation for α-stable distributions and illustrate the pairwise dependence structure over time of log returns of German stock prices via local empirical distance correlations.     

On the stochastic simulation of particulate systems

The stochastic chemical kinetics approach provides one method of formulating the stochastic crystallization population balance equation (PBE). In this formulation, crystal nucleation and growth are modelled as sequential additions of solubilized ions or molecules (units) to either other units or an assembly of any number of units. Monte Carlo methods provide one means of solving this problem. In this paper, we assess the limitations of such methods by both (1) simulating models for isothermal and nonisothermal size-independent nucleation, growth and agglomeration; and (2) performing parameter estimation using these models. We also derive the macroscopic (deterministic) PBE from the stochastic formulation, and compare the numerical solutions of the stochastic and deterministic PBEs. The results demonstrate that even as we approach the thermodynamic limit, in which the deterministic model becomes valid, stochastic simulation provides a general, flexible solution technique for examining many possible mechanisms. Thus the stochastic simulation permits the user to focus more on modelling issues as opposed to solution techniques.     

Time sequential testing with fixed study time

We investigate sampling plans in time sequential testing af snrvival distributions, when the study time is fixed. We obtain the asymptotic value of optimal sampling plans as the error probabilities of the test approach zero.     

Discussion on “Sequential Estimation for Time Series Models” by T. N. Sriram and Ross Iaci

Abstract

In this discussion, we focus on a sequential procedure for point and confidence region estimating parameters in autoregressive processes AR(1). The uniform asymptotic normality of estimators is considered.     

Simple Regressions with Linear Time Trends

Simple regressions of two trend stationary time series are considered. As the linear trend dominates the stochastic components the rates of convergence and the limiting distributions of ordinary least squares statistics are exactly the same as in the case of cointegrated regressions with drifts. In particular, asymptotic standard normal t statistics are readily available. Hence, asymptotic inference requires no distinction between simple regressions of trend stationary series and of cointegrated variables with drifts.     

Strictly stationary solutions of ARMA equations with fractional noise

We obtain necessary and sufficient conditions for the existence of strictly stationary solutions of ARMA equations with fractional noise. Here, the underlying noise sequence of the fractional noise is assumed to be i.i.d. but no a priori moment assumptions are made. We also characterize for which i.i.d. driving noise sequences the series defining fractional noise converges almost surely. In the proofs, we use growth estimates for the moments of random walks developed by Manstavi?ius (1982) and techniques related to those of Brockwell and Lindner (2010) for the existence of strictly stationary ARMA processes with i.i.d. noise.     

Moment tests for window length selection in singular spectrum analysis of short– and long–memory processes

In this paper, we propose a new methodology for selecting the window length in Singular Spectral Analysis in which the window length is determined from the data prior to the commencement of modelling. The selection procedure is based on statistical tests designed to test the convergence of the autocovariance function. A classical time series portmanteau type statistic and two test statistics derived using a conditional moment principle are considered. The first two are applicable to short–memory processes, and the third is applicable to both short– and long–memory processes. We derive the asymptotic null and alternative distributions of the statistics under fairly general regularity conditions. Consistency of the tests implies that the selection criteria will identify true convergence with a finite window length with probability arbitrarily close to one as the sample size increases. Results obtained using Monte Carlo simulation point to the relevance of the asymptotic theory and show that the conditional moment tests will choose a window length consistent with the Whitney embedding theorem. Application to observations on the Southern Oscillation Index shows how observed experimental behaviour can be reflected in features seen with real world data sets.     

BOOTSTRAPPING STATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. In this paper we develop an asymptotic theory for application of the bootstrap to stationary stochastic processes of autoregressive moving-average (ARMA) type, with known order ( p, q ). We give a proof of the asymptotic validity of the bootstrap proposal applied to M estimators for the unknown parameter vector of the process. For this purpose we derive an asymptotic expansion for M estimators in ARMA models and construct an estimate for the unknown distribution function of the residuals which in principle are not observable. A small simulation study is also included.