Change-point mle in the rate of exponential sequences with application to Indonesian seismological data

In this paper, we derive explicit computable expressions for the asymptotic distribution of the maximum likelihood estimate of an unknown change-point in a sequence of independently and exponentially distributed random variables. First we state and prove a theorem that shows asymptotic equivalence of the change-point mle for the cases of both known and unknown parameters, respectively. Thereafter, the computational form of the asymptotic distribution of the change-point mle is derived for the case of known parameter situation only. Simulations show that the distribution for the known case applies very well to the case where the parameters are estimated. Further, it is seen from simulations that the derived unconditional mle shows better performance compared to the conditional solution of Cobb. Application of change detection methodology and the derived estimation methodology show strong support in favor the dynamic triggering hypothesis for seismic faults in Sumatra, Indonesia region.     

Convergence rates of change-point estimators and tail probabilities of the first-passage-time process

In the classical setting of the change-point problem, the maximum-likelihood estimator and the traditional confidence region for the change-point parameter are considered. It is shown that the probability of the correct decision, the coverage probability and the expected size of the confidence set converge exponentially fast as the sample size increases to infinity. For this purpose, the tail probabilities of the first passage times are studied. General inequalities are established, and exact asymptotics are obtained for the case of Bernoulli distributions. A closed asymptotic form for the expected size of the confidence set is derived for this case via the conditional distribution of the first passage times.     

Testing for changes in autocovariances of nonparametric time series models

In the literature on change-point analysis, much attention has been paid to detecting changes in certain marginal characteristics, such as mean, variance, and marginal distribution. For time series data with nonparametric time trend, we study the change-point problem for the autocovariance structure of the unobservable error process. To derive the asymptotic distribution of the cumulative sum test statistic, we develop substantial theory for uniform convergence of weighted partial sums and weighted quadratic forms. Our asymptotic results improve upon existing works in several important aspects. The performance of the test statistic is examined through simulations and an application to interest rates data.     

Inference for post-change mean by a CUSUM procedure

In this paper, the inference problem for the post-change mean is considered after a change is detected by a CUSUM process in a sequence of independent normal variables. The change-point is estimated as the maximum likelihood estimate at the reference value and the post-change mean is estimated as the sample mean after the change-point estimate. By assuming the change-point is large and the monitoring limit approaches infinity, the first-order bias of the post-change mean estimate and a corrected asymptotic normal pivot are derived conditioning on that a change is detected. Local approximations for small reference value and post-change mean are obtained for numerical evaluation.     

The LAD estimation of the change-point linear model with randomly censored data

Abstract

In this paper, a change-point linear model with randomly censored data is investigated. We propose the least absolute deviation estimation procedure for regression and change-point parameters simultaneously. The asymptotic properties of the change-point and regression parameter estimators are obtained. We show that the resulting regression parameter estimator is asymptotically normal, and the change-point estimator converges weakly to the minimizer of a given random process. The extensive simulation studies and the analysis of an acute myocardial infarction data set are conducted to illustrate the finite sample performance of the proposed method.     

THE EFFECTS OF CORRELATION AMONG OBSERVATIONS ON THE ACCURACY OF APPROXIMATING THE DISTRIBUTION OF SAMPLE MEAN BY ITS ASYMPTOTIC DISTRIBUTION1

In this paper, we investigate the effects of correlation among observations on the accuracy of approximating the distribution of sample mean by its asymptotic distribution. The accuracy is investigated by the Berry-Esseen bound (BEB), which gives an upper bound on the error of approximation of the distribution function of the sample mean from its asymptotic distribution for independent observations. For a given sample size (n₀) the BEB is obtained when the observations are independent. Let this be BEB. We then find the sample size (n_*) required to have BEB below BEB₀, when the observations are dependent. Comparison of n_* with n₀ reveals the effects of correlation among observations on the accuracy of the asymptotic distribution as an approximation. It is shown that the effects of correlation among observations are not appreciable if the correlation is moderate to small but it can be severe for extreme correlations.     

Likelihood-based estimation for Gaussian MRFs

Markov random fields (MRFs) express spatial dependence through conditional distributions, although their stochastic behavior is defined by their joint distribution. These joint distributions are typically difficult to obtain in closed form, the problem being a normalizing constant that is a function of unknown parameters. The Gaussian MRF (or conditional autoregressive model) is one case where the normalizing constant is available in closed form; however, when sample sizes are moderate to large (thousands to tens of thousands), and beyond, its computation can be problematic. Because the conditional autoregressive (CAR) model is often used for spatial-data modeling, we develop likelihood-inference methodology for this model in situations where the sample size is too large for its normalizing constant to be computed directly. In particular, we use simulation methodology to obtain maximum likelihood estimators of mean, variance, and spatial-depencence parameters (including their asymptotic variances and covariances) of CAR models.     

Truncated sequential change-point detection based on renewal counting processes II

The standard approach in change-point theory is to base the statistical analysis on a sample of fixed size. Alternatively, one observes some random phenomenon sequentially and takes action as soon as one observes some statistically significant deviation from the “normal” behaviour. The present paper is a continuation of Gut and Steinebach [2002. Truncated sequential change-point detection based on renewal counting processes. Scand. J. Statist. 29, 693–719] the main point being that here we look in more detail into the behaviour of the relevant stopping times, in particular the time it takes from the actual change-point until the change is detected, more precisely, we prove asymptotics for stopping times under alternatives.     

Minimax optimality of the Shiryayev–Roberts change-point detection rule

The result of Pollak [1985. Optimal detection of a change in distribution. Ann. Statist. 13, 206–227] proving the asymptotic optimality in sequential change-point detection of a suitable Shirayayev–Roberts stopping rule up to terms that vanish in the limit is generalized from the case of two completely specified distributions to that of a composite alternative hypothesis in a multidimensional exponential family. An explicit asymptotic lower bound on the expected Kullback–Leibler information required to detect a change-point is derived and is shown to be attained by a Shirayayev–Roberts stopping rule.     

Estimation Mean Change-Point in ARCH Models with Heavy-Tailed Innovations

The quest of the mean change-point in ARCH models with innovations in the domain of attraction of a κ-stable law appears to still be ongoing. We derive the asymptotic distribution of the residuals CUSUM of squares test (RCUSQ) statistic and find it depends on the stable index κ which is often typically unknown and difficult to estimate. Therefore, the subsampling method is proposed to detect changes without estimating κ. The tests are easy to use and are found to perform well in a Monte Carlo experiment.     

Nonparametric asymptotic confidence intervals for extreme quantiles

In this paper, we propose new asymptotic confidence intervals for extreme quantiles, that is, for quantiles located outside the range of the available data. We restrict ourselves to the situation where the underlying distribution is heavy-tailed. While asymptotic confidence intervals are mostly constructed around a pivotal quantity, we consider here an alternative approach based on the distribution of order statistics sampled from a uniform distribution. The convergence of the coverage probability to the nominal one is established under a classical second-order condition. The finite sample behavior is also examined and our methodology is applied to a real dataset.     

When is n large enough? Looking for the right sample size to estimate proportions

The central limit theorem indicates that when the sample size goes to infinite, the sampling distribution of means tends to follow a normal distribution; it is the basis for the most usual confidence interval and sample size formulas. This study analyzes what sample size is large enough to assume that the distribution of the estimator of a proportion follows a Normal distribution. Also, we propose the use of a correction factor in sample size formulas to ensure a confidence level even when the central limit theorem does not apply for these distributions.     

Change-Point Tests for the Error Distribution in Non-parametric Regression

Abstract.  Several testing procedures are proposed that can detect change-points in the error distribution of non-parametric regression models. Different settings are considered where the change-point either occurs at some time point or at some value of the covariate. Fixed as well as random covariates are considered. Weak convergence of the suggested difference of sequential empirical processes based on non-parametrically estimated residuals to a Gaussian process is proved under the null hypothesis of no change-point. In the case of testing for a change in the error distribution that occurs with increasing time in a model with random covariates the test statistic is asymptotically distribution free and the asymptotic quantiles can be used for the test. This special test statistic can also detect a change in the regression function. In all other cases the asymptotic distribution depends on unknown features of the data-generating process and a bootstrap procedure is proposed in these cases. The small sample performances of the proposed tests are investigated by means of a simulation study and the tests are applied to a data example.     

Sample size determination for testing equality in Poisson frequency data under an AB/BA crossover trial

Assuming that the frequency of occurrence follows the Poisson distribution, we develop sample size calculation procedures for testing equality based on an exact test procedure and an asymptotic test procedure under an AB/BA crossover design. We employ Monte Carlo simulation to demonstrate the use of these sample size formulae and evaluate the accuracy of sample size calculation formula derived from the asymptotic test procedure with respect to power in a variety of situations. We note that when both the relative treatment effect of interest and the underlying intraclass correlation between frequencies within patients are large, the sample size calculation based on the asymptotic test procedure can lose accuracy. In this case, the sample size calculation procedure based on the exact test is recommended. On the other hand, if the relative treatment effect of interest is small, the minimum required number of patients per group will be large, and the asymptotic test procedure will be valid for use. In this case, we may consider use of the sample size calculation formula derived from the asymptotic test procedure to reduce the number of patients needed for the exact test procedure. We include an example regarding a double‐blind randomized crossover trial comparing salmeterol with a placebo in exacerbations of asthma to illustrate the practical use of these sample size formulae. Copyright © 2013 John Wiley & Sons, Ltd.     

Detection of a change-point in student-t linear regression models

In this work the Schwarz Inforamtion Criterion (SIC) is used in order to locate a change-point in linear regression models with independent errors distributed according to the Student-t distribution. The methodology is applied to data sets from the financial area.     

Linear statistics in change-point estimation and their asymptotic behaviour

ABSTRACT The limiting behaviour of Bayes procedures in the asymptotic setting of the change-point estimation problem is studied. It is shown that the distribution of the difference between the Bayes estimator and the parameter converges to the distribution of a fairly complicated random variable. A class of linear statistics is introduced, and the form of the Bayes estimator within this class is deduced. The asymptotic properties of this linear estimator are investigated in two different settings for the prior distribution.     

Goodness-of-fit tests for binomial AR(1) processes

The binomial AR(1) model describes a nonlinear process with a first-order autoregressive (AR(1)) structure and a binomial marginal distribution. To develop goodness-of-fit tests for the binomial AR(1) model, we investigate the observed marginal distribution of the binomial AR(1) process, and we tackle its autocorrelation structure. Motivated by the family of power-divergence statistics for handling discrete multivariate data, we derive the asymptotic distribution of certain categorized power-divergence statistics for the case of a binomial AR(1) process. Then we consider Bartlett's formula, which is widely used in time series analysis to provide estimates of the asymptotic covariance between sample autocorrelations, but which is not applicable when the underlying process is nonlinear. Hence, we derive a novel Bartlett-type formula for the asymptotic distribution of the sample autocorrelations of a binomial AR(1) process, which is then applied to develop tests concerning the autocorrelation structure. Simulation studies are carried out to evaluate the size and power of the proposed tests under diverse alternative process models. Several real examples are used to illustrate our methods and findings.     

Triple sampling procedure for estimating the scale parameters of pareto distributions

This paper sheds light on the large sample performance of the three stage sam- pling procedure, as it pertains to estimating the scale parameter(s) of the Pareto distribution(s). This group sampling procedure merges the efficiency of the purely sequential procedure of Anscombe (1953) and Chow and Robbins (1965) with substan-tial savings in the number of sampling operations, as noted by Hall (1981). Both its simplicity and its economical features provide visible advantages over the one-by-one sampling as an alternative. In this paper we develop some asymptotic properties for the final stage sample size of the triple stage sampling originated by Hall (1981). These results are used to study both the point and the interval estimation problems for the scale parameters of the Pareto distributions. Since our results are asymptotic in nature, a simulation study is given to discuss the moderate sample size peformance of the proposed procedures.     

Empirical Likelihood Ratio Test for a Change-Point in Linear Regression Model

A nonparametric method based on the empirical likelihood is proposed to detect the change-point in the coefficient of linear regression models. The empirical likelihood ratio test statistic is proved to have the same asymptotic null distribution as that with classical parametric likelihood. Under some mild conditions, the maximum empirical likelihood change-point estimator is also shown to be consistent. The simulation results show the sensitivity and robustness of the proposed approach. The method is applied to some real datasets to illustrate the effectiveness.     

Bayesian non-parametric inference for species variety with a two-parameter Poisson–Dirichlet process prior

Summary.  A Bayesian non-parametric methodology has been recently proposed to deal with the issue of prediction within species sampling problems. Such problems concern the evaluation, conditional on a sample of size n , of the species variety featured by an additional sample of size m . Genomic applications pose the additional challenge of having to deal with large values of both n and m . In such a case the computation of the Bayesian non-parametric estimators is cumbersome and prevents their implementation. We focus on the two-parameter Poisson–Dirichlet model and provide completely explicit expressions for the corresponding estimators, which can be easily evaluated for any sizes of n and m . We also study the asymptotic behaviour of the number of new species conditionally on the observed sample: such an asymptotic result, combined with a suitable simulation scheme, allows us to derive asymptotic highest posterior density intervals for the estimates of interest. Finally, we illustrate the implementation of the proposed methodology by the analysis of five expressed sequence tags data sets.