This article develops a statistical test for the null hypothesis of strict stationarity of a discrete time stochastic process. When the null hypothesis is true, the second order cumulant spectrum is zero at all the discrete Fourier frequency pairs present in the principal domain of the cumulant spectrum. The test uses a frame (window) averaged sample estimate of the second order cumulant spectrum to build a test statistic that has an asymptotic complex standard normal distribution. We derive the test statistic, study the size and power properties of the test, and demonstrate its implementation with intraday stock market return data. The test has conservative size properties and good power to detect varying variance and unit root in the presence of varying variance.
Estimation and inference about the change in the mean of a stationary random process was amply examined in literature. Many articles studies the estimation and inference about the change in the mean of a random process. @cite_17 proposed a likelihood ratio based procedure to detect a change in the mean while assuming constant variance. In a different approach, @cite_11 developed CUSUM and EWMA charts that detect changes in probability distributions via their scale parameter. Also, tests of a unit root with or without drift against an alternative hypothesis of trend stationarity, can be found, for example, in @cite_8 and @cite_2 .
@cite_8 @cite_11 @cite_2 @cite_17
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Detection of change-points in normal means is a well-studied problem. The parallel problem of detecting changes in variance has had less attention. The form of the generalized likelihood ratio test statistic has long been known, but its null distribution resisted exact analysis. In this paper, we formulate the change-point problem for a sequence of chi-square random variables. We describe a procedure that is exact for the distribution of the likelihood ratio statistic for all even degrees of freedom, and gives upper and lower bounds for odd (and also for non-integer) degrees of freedom. Both the liberal and conservative bounds for X 2 1 degrees of freedom are shown through simulation to be reasonably tight. The important problem of testing for change in the normal variance of individual observations corresponds to the X 2 1 case. The non-null case is also covered, and confidence intervals for the true change point are derived. The mehhodology is illustrated with an application to quality control in a deep level gold mine. Other applications include ambulatory monitoring of medical data and econometrics. In the past few years there has been increased interest in using data-mining techniques to extract interesting patterns from time series data generated by sensors monitoring temporally varying phenomenon. Most work has assumed that raw data is somehow processed to generate a sequence of events, which is then mined for interesting episodes. In some cases the rule for determining when a sensor reading should generate an event is well known. However, if the phenomenon is ill-understood, stating such a rule is difficult. Detection of events in such an environment is the focus of this paper. Consider a dynamic phenomenon whose behavior changes enough over time to be considered a qualitatively significant change. The problem we investigate is of identifying the time points at which the behavior change occurs. In the statistics literature this has been called the change-point detection problem. The standard approach has been to (a) upriori determine the number of change-points that are to be discovered, and (b) decide the function that will be used for curve fitting in the interval between successive change-points. In this paper we generalize along both these dimensions. We propose an iterative algorithm that fits a model to a time segment, and uses a likelihood criterion to determine if the segment should be partitioned further, i.e. if it contains a new changepoint. In this paper we present algorithms for both the batch and incremental versions of the problem, and evaluate their behavior with synthetic and real data. Finally, we present initial results comparing the change-points detected by the batch algorithm with those detected by people using visual inspection. A great deal of effort has been devoted to reducing the risk of spurious scientific discoveries, from the use of sophisticated validation techniques, to deep statistical methods for controlling the false discovery rate in multiple hypothesis testing. However, there is a fundamental disconnect between the theoretical results and the practice of data analysis: the theory of statistical inference assumes a fixed collection of hypotheses to be tested, or learning algorithms to be applied, selected non-adaptively before the data are gathered, whereas in practice data is shared and reused with hypotheses and new analyses being generated on the basis of data exploration and the outcomes of previous analyses. In this work we initiate a principled study of how to guarantee the validity of statistical inference in adaptive data analysis. As an instance of this problem, we propose and investigate the question of estimating the expectations of @math adaptively chosen functions on an unknown distribution given @math random samples. We show that, surprisingly, there is a way to estimate an exponential in @math number of expectations accurately even if the functions are chosen adaptively. This gives an exponential improvement over standard empirical estimators that are limited to a linear number of estimates. Our result follows from a general technique that counter-intuitively involves actively perturbing and coordinating the estimates, using techniques developed for privacy preservation. We give additional applications of this technique to our question. Abstract : A tracking problem is considered. Observations are taken on the successive positions of an object traveling on a path, and it is desired to estimate its current position. The objective is to arrive at a simple formula which implicitly accounts for possible changes in direction and discounts observations taken before the latest change. To develop a reasonable procedure, a simpler problem is studied. Successive observations are taken on n independently and normally distributed random variables X sub 1, X sub 2, ..., X sub n with means mu sub 1, mu sub 2, ..., mu sub n and variance 1. Each mean mu sub i is equal to the preceding mean mu sub (i-1) except when an occasional change takes place. The object is to estimate the current mean mu sub n. This problem is studied from a Bayesian point of view. An 'ad hoc' estimator is described, which applies a combination of the A.M.O.C. Bayes estimator and a sequence of tests designed to locate the last time point of change. The various estimators are then compared by a Monte Carlo study of samples of size 9. This Bayesian approach seems to be more appropriate for the related problem of testing whether a change in mean has occurred. This test procedure is simpler than that used by Page. The power functions of the two procedures are compared. (Author)