Flexibly Monitoring Group Sequential Survival Trials When Testing is Based Upon a Weighted Log-Rank Statistic

Abstract

We consider the repeated group sequential testing of a survival endpoint with a time-varying treatment effect using a weighted log-rank statistic. The emphasis of this article is on the monitoring of this statistic where information growth is nonlinear. We propose using a constrained boundaries approach to maintain the planned operating characteristics of a group sequential design. A simulation study is presented to demonstrate the operating characteristics of the method together with a case study to illustrate the procedure. We show that when monitoring a weighted log-rank statistic, the entry and survival distribution needs to be estimated at interim analyses.     

Optimal Sequential Tests for Two Simple Hypotheses

Abstract

A general problem of testing two simple hypotheses about the distribution of a discrete-time stochastic process is considered. The main goal is to minimize an average sample number over all sequential tests whose error probabilities do not exceed some prescribed levels. As a criterion of minimization, the average sample number under a third hypothesis is used (modified Kiefer–Weiss problem). For a class of sequential testing problems, the structure of optimal sequential tests is characterized. An application to the Kiefer–Weiss problem for discrete-time stochastic processes is proposed. As another application, the structure of Bayes sequential tests for two composite hypotheses, with a fixed cost per observation, is given. The results are also applied for finding optimal sequential tests for discrete-time Markov processes. In a particular case of testing two simple hypotheses about a location parameter of an autoregressive process of order 1, it is shown that the sequential probability ratio test has the Wald–Wolfowitz optimality property.     

Real time change-point detection in a nonlinear quantile model

Most studies in real-time change-point detection either focus on the linear model or use the cumulative sum (CUSUM) method under classical assumptions on model errors. This article considers the sequential change-point detection in a nonlinear quantile model. A test statistic based on the CUSUM of the quantile process subgradient is proposed and studied. Under the null hypothesis that the model does not change, the asymptotic distribution of the test statistic is determined. Under the alternative hypothesis that at some unknown observation there is a change in the model, the proposed test statistic converges in probability to ∞. These results allow building the critical regions on open-end and on closed-end procedures. Simulation results, using a Monte Carlo technique, investigate the performance of the test statistic, especially for heavy-tailed error distributions. We also compare it with the classical CUSUM test statistic. An example on real data is also given.     

A test for second‐order stationarity of a time series based on the discrete Fourier transform

We consider a zero mean discrete time series, and define its discrete Fourier transform (DFT) at the canonical frequencies. It can be shown that the DFT is asymptotically uncorrelated at the canonical frequencies if and only if the time series is second‐order stationary. Exploiting this important property, we construct a Portmanteau type test statistic for testing stationarity of the time series. It is shown that under the null of stationarity, the test statistic has approximately a chi‐square distribution. To examine the power of the test statistic, the asymptotic distribution under the locally stationary alternative is established. It is shown to be a generalized non‐central chi‐square, where the non‐centrality parameter measures the deviation from stationarity. The test is illustrated with simulations, where is it shown to have good power.     

Monte Carlo方法在计算多级颗粒反应系统中的应用

对于具有级间返混的多级固体颗粒—流体(气-固、液-固、气-液-固)反应器系统,当各级反应条件(温度、或压力)不同时,其颗粒平均反应产率的计算是一个复杂而困难的问题.本文采用Monte Carlo方法对颗粒的停留时间及其反应转化率进行随机跟踪模拟.给出了计算程序.此法概念简单、程序简短、适用性广,对反应速率方程和各级内流动模型均无任何限制.尚可用于模拟计算更复杂的网络流动系统,及颗粒的其它物理或化学过程.此方法对一些特殊情况的模拟结果与确定型解进行了比较验证,符合很好.     

Asymptotically Pointwise Optimal Change Detection in Multiple Channels

Abstract

A Bayesian multichannel change-point detection problem is studied in the following general setting. A multidimensional stochastic process is observed; some or all of its components may experience changes in distribution, simultaneously or not. The loss function penalizes for false alarms and detection delays, and the penalty increases with each missed change-point. For wide classes of stochastic processes, with or without nuisance parameters and practically any joint prior distribution of change-points, asymptotically pointwise optimal (APO) rules are obtained, translating the classical concept of Bickel and Yahav to the sequential change-point detection. These APO rules are attractive because of their simple analytic form and straightforward computation. An application to a multidimensional autoregressive time series is shown.     

Goodness-of-fit tests for autoregressive processes

Goodness-of-fit tests for autoregressive processes can be based on the difference betwe en the empirical standardized spectral distribution of an observed time series and the standardized spectral distribution of the autoregressive process with parameters estimated from the series. The asymptotic covariance function of this difference, considered as a stochastic process on [0, π], is found. Methods to compute the asymptotic distribution of the Cramer--von Mises statistic are given.     

Stochastic modeling of a fluid batch mixed system with variable dispersion coefficient

This contribution presents the numerical solution of the one-dimensional stochastic differential equation with a diffusion coefficient which is a function of the spatial coordinate, as a model of a fluid batch system stirred by a rotating mechanical impeller. It is demonstrated that neither the Ito nor the Stratonovich methods of solution, which are widely used to solve equations of this kind, lead to a uniform distribution of concentration of the blended species component at steady-state. A method of solution is suggested, with the aid of the so-called transport integral, which converges to the uniform steady-state distribution, consistent with the physical conception of the course of the mixing process.     

Testing for the Presence of a Random Walk in Series with Structural Breaks

We consider tests for the presence of a random walk component in a stationary or trend stationary time series and extend them to series that contain structural breaks. The locally best invariant (LBI) test is derived and the asymptotic distribution is obtained. Then a modified test statistic is proposed. The advantage of this statistic is that its asymptotic distribution is not dependent on the location of the break point and its form is that of the generalized Cramer–von Mises distribution, with degrees of freedom depending on the number of break points. The performance of this modified test is shown, via some simulation experiments, to be comparable with that of the LBI test. An unconditional test, based on the assumption that there is a single break at an unknown point, is also examined. The use of the tests is illustrated with data on the flow of the Nile and US gross national product.     

The Marginal Density of a TMA(1) Process

This note reconsiders the marginal density of a threshold moving average process and proposes a simple yet effective numerical algorithm to implement that by solving an associated integral equation. This algorithm can also be applied to calculate stationary probability density or distribution functions of a few other types of nonlinear stationary stochastic processes numerically.     

Generalized Sequential Probability Ratio Test for Separate Families of Hypotheses

Abstract

In this article, we consider the problem of testing two separate families of hypotheses via a generalization of the sequential probability ratio test. In particular, the generalized likelihood ratio statistic is considered and the stopping rule is the first boundary crossing of the generalized likelihood ratio statistic. We show that this sequential test is asymptotically optimal in the sense that it achieves asymptotically the shortest expected sample size as the maximal type I and type II error probabilities tend to zero.     

Testing for EGARCH Against Stochastic Volatility Models

Abstract. It is shown that the EGARCH model is the degenerate case of Danielsson's [Journal of Econometrics (1994) Vol. 61, pp. 375–400] stochastic volatility model where the disturbance of the transition equation of conditional volatility has zero variance. The Lagrange multiplier test statistic is obtained for the EGARCH model against the stochastic volatility model by expressing the degenerate density under the null hypothesis by the Dirac delta function. The finite sample performance of the test is studied in a small Monte Carlo experiment.     

Outlier detection in ARMA models

Abstract. We consider an autoregressive moving‐average (ARMA) time series where the observations are perturbed by two kinds of outliers: an additive outlier (AO) or an innovation outlier (IO). Abraham and Yatawara [Journal of Time Series Analysis (1988) Vol. 9, pp. 109–19] investigate a sequential test which successively detects and identifies the outlier type. In this article, we propose an extension of this test, called ‘modified sequential test’, which performs the two procedures simultaneously and coherently. The asymptotic distribution of the test statistic is calculated under the null hypothesis that no outlier is present. Comparison of the two test procedures using simulation experiments shows that the proposed test gives a better power especially in the case of an IO.     

Mixed Portmanteau Tests for Time‐Series Models

Abstract. This paper obtains the joint limiting distribution of residuals and squared residuals of a general time‐series model. Based on this, we propose a mixed portmanteau statistic for testing the adequacy of fitted time‐series models. In some cases, it is shown that this statistic can be simply approximated by the sum of well‐known portmanteau statistics. The finite‐sample performance of the new test is compared with those of well‐known tests through simulations.     

Seasonal Moving-average Unit Root Tests in the Presence of a Linear Trend

To distinguish stochastic from deterministic seasonality, test procedures are developed for a unit root in the integrated seasonal moving-average (SMA) model when an underlying deterministic trend is present. Locally best invariant unbiase (LBIU) and point optimal invariant tests are considered. Their asymptotic distributions are developed and are found to differ from those for the no-linear-trend case. The limiting distribution of the LBIU statistic is expressed as a functional of Brownian motions. The procedures are extended to more general seasonal autoregressive moving-average (ARMA) models, and to the inclusion of exogenous regressors. Finite-sample distributions are also derived for the SMA(1) model. Simulations suggest that these distributions provide accurate approximations for more general ARMA models. A numerical example is included to illustrate the tests.     

Wear convergence of stochastic approximation processes with random indices

Conditions are given for weak convergence through random indices of a general stochastic approximation process which includes the Robbins-Monro and Kiefer-Wolfowitz processes. For a particular index, a sequential fixed-width bounded length confidence interval for the parameter being estimated is established. As an example, an optimal recursive estimator and confidence interval for the mode of a distribution function is constructed.     

气液固三相流化床中局部相含率的随机分析

本文利用Markov过程原理获得了描述三相流化床中局部相含率的随机模型,并进行了实验验证,发现拟合很好.     

Intermediate storage in batch/continuous processing systems under stochastic operation conditions

A mathematical model and model-based method is presented to design the intermediate storages aiming to buffer the operational differences between the batch and continuous subsystems in processing systems. The occurrence times of the inputs are assumed to be described by a Poisson process, while the amounts of the material transferred by the batch units allowed changing according to general probability distributions. Based on the stochastic differential equation model of operation, integral equations for determining the overflow and underflow probabilities of a finite storage are formulated for both infinite and finite operation horizons that provide the basis for the rational design of such intermediate storages. Analytical solutions to the integral equations for infinite horizons are derived in the cases of constant and exponentially distributed inputs. For the batch sizes described by general distribution functions, solutions to the integral equations are obtained in the form of approximating functions generated by stochastic simulation. A number of numerical experiments with exponential, normal and lognormal distributions of the batch sizes are presented and analyzed. The effects of process parameters on the design are also investigated.     

A small-sample overlapping variance-ratio test

Abstract.  The null distribution of the overlapping variance-ratio (OVR) test of the random-walk hypothesis is known to be downward biased and skewed to the right in small samples. As shown by Lo and MacKinlay (1989) , the test under-rejects the null on the left tail seriously when the sample size is small. This property adversely affects the applicability of the OVR test to macroeconomic time series, which usually have rather small samples. In this paper, we propose a modified overlapping variance-ratio statistic and derive its exact mean under the normality assumption. We propose to approximate the small-sample distribution of the modified statistic using a beta distribution that matches the (exact) mean and the (asymptotic) variance. A Monte Carlo experiment shows that the beta approximation performs well in small samples.     

ON THE ASYMPTOTIC DISTRIBUTION OF BARTLETT'S Up-STATISTIC

Abstract. In this paper the asymptotic behaviour of Bartlett's U _p -statistic for a goodness-of-fit test for stationary processes, is considered. The asymptotic distribution of the test process is given under the assumption that a central limit theorem for the empirical spectral distribution function holds. It is shown that the U_p -statistic tends to the supremum of a tied down Brownian motion. By a counterexample we refute the conjecture that this distribution is in general of the Kolmogorov-Smirnov type. The validity of the central limit theorem for the spectral distribution function is then discussed. Finally a goodness-of-fit test for ARMA-processes based on the estimated innovation sequence is given, and it is shown that this test statistic is asymptotically Kolmogorov-Smirnov distributed.