Estimating the Number of Change Points in Exponential Families Distributions

A simple method is proposed to detect the number of change points in a sequence of independent exponential family random variables. An estimator to maximize some criterion, say SC ( k ), which is to maximize the log- likelihood function with some penalty term, is used in detection. Under some mild assumptions, the consistency of the estimator for the true number of change points and the boundedness between the estimated change locations and the true change location are obtained. Some simulated results are given, and the Nile problem is investigated by this method.     

Large Deviations Limit Theorems for the Kernel Density Estimator

We establish pointwise and uniform large deviations limit theorems of Chernoff-type for the non-parametric kernel density estimator based on a sequence of independent and identically distributed random variables. The limits are well-identified and depend upon the underlying kernel and density function. We derive then some implications of our results in the study of asymptotic efficiency of the goodness-of-fit test based on the maximal deviation of the kernel density estimator as well as the inaccuracy rate of this estimate     

Hypotheses Testing: Poisson Versus Self-exciting

Abstract.  We consider the problem of hypotheses testing with the basic simple hypothesis: observed sequence of points corresponds to the stationary Poisson process with known intensity. The alternatives are stationary self-exciting point processes. We consider one-sided parametric and one-sided non-parametric composite alternatives and construct locally asymptotically uniformly most powerful tests. The results of numerical simulations of the tests are presented.     

Correctings tests for trend in proportions for skewness

We develop the score test for the hypothesis that a parameter of a Markov sequence is constant over time, against the alternatives that it varies over time, i.e., θ_t = θ + U_t; t = 1,2,…, where {U_t; t = 1,2,...} is a sequence of independently and identically distributed random variables with mean zero and variance σ^z _u and θ is a fixed constant. The asymptotic null distribution of the test statistic is proved to be normal. We illustrate our procedure by examples and a real life data analysis.     

Accurate tests for the equality of coefficients of variation

ABSTRACT

A frequently encountered statistical problem is to determine if the variability among k populations is heterogeneous. If the populations are measured using different scales, comparing variances may not be appropriate. In this case, comparing coefficient of variation (CV) can be used because CV is unitless. In this paper, a non-parametric test is introduced to test whether the CVs from k populations are different. With the assumption that the populations are independent normally distributed, the Miller test, Feltz and Miller test, saddlepoint-based test, log likelihood ratio test and the proposed simulated Bartlett-corrected log likelihood ratio test are derived. Simulation results show the extreme accuracy of the simulated Bartlett-corrected log likelihood ratio test if the model is correctly specified. If the model is mis-specified and the sample size is small, the proposed test still gives good results. However, with a mis-specified model and large sample size, the non-parametric test is recommended.     

Changepoint Analysis as a Method for Isotonic Inference

Concavity and sigmoidicity hypotheses are developed as a natural extension of the simple ordered hypothesis in normal means. Those hypotheses give reasonable shape constraints for obtaining a smooth response curve in the non-parametric inputoutput analysis. The slope change and inflection point models are introduced correspondingly as the corners of the polyhedral cones defined by those isotonic hypotheses. Then a maximal contrast type test is derived systematically as the likelihood ratio test for each of those changepoint hypotheses. The test is also justified for the original isotonic hypothesis by a complete class lemma. The component variables of the resulting test statistic have second or third order Markov property which, together with an appropriate non-linear transformation, leads to an exact and very efficient algorithm for the probability calculation. Some considerations on the power of the test are given showing this to be a very promising way of approaching to the isotonic inference.     

A parametric empirical Bayes approach to the analysis of capture-recapture experiments

Empirical Bayes methods and a bootstrap bias adjustment procedure are used to estimate the size of a closed population when the individual capture probabilities are independently and identically distributed with a Beta distribution. The method is examined in simulations and applied to several well-known datasets. The simulations show the estimator performs as well as several other proposed parametric and non-parametric estimators.     

Convergence in distribution of multiple change point estimators

In this paper we establish the asymptotic distribution for a class of multiple change point estimators in the following setup: a finite sequence of independent random variables consists of segments given by a known number of so-called change points such that the underlying distribution differs from segment to segment. In a nonparametric framework the proposed estimator is defined as the maximizing point of weighted multivariate U-statistic processes. We show that the proposed estimators converge in distribution to a maximizer of a sum of random walks with drift.     

Multiscale detection and location of multiple variance changes in the presence of long memory

Procedures for detecting change points in sequences of correlated observations (e.g., time series) can help elucidate their complicated structure. Current literature on the detection of multiple change points emphasizes the analysis of sequences of independent random variables. We address the problem of an unknown number of variance changes in the presence of long-range dependence (e.g., long memory processes). Our results are also applicable to time series whose spectrum slowly varies across octave bands. An iterated cumulative sum of squares procedure is introduced in order to look at the multiscale stationarity of a time series; that is, the variance structure of the wavelet coefficients on a scale by scale basis. The discrete wavelet transform enables us to analyze a given time series on a series of physical scales. The result is a partitioning of the wavelet coefficients into locally stationary regions. Simulations are performed to validate the ability of this procedure to detect and locate multiple variance changes. A ‘time’ series of vertical ocean shear measurements is also analyzed, where a variety of nonstationary features are identified.     

Unbiased Estimation of P(X > Y) for Exponential Populations Using Order Statistics with Application in Ranked Set Sampling

This paper addresses the problem of unbiased estimation of P[X > Y] = θ for two independent exponentially distributed random variables X and Y. We present (unique) unbiased estimator of θ based on a single pair of order statistics obtained from two independent random samples from the two populations. We also indicate how this estimator can be utilized to obtain unbiased estimators of θ when only a few selected order statistics are available from the two random samples as well as when the samples are selected by an alternative procedure known as ranked set sampling. It is proved that for ranked set samples of size two, the proposed estimator is uniformly better than the conventional non-parametric unbiased estimator and further, a modified ranked set sampling procedure provides an unbiased estimator even better than the proposed estimator.     

Comparison of three estimators of the number of species

We consider estimation of the number of cells in a multinomial distribution. This is one version of the species problem: there are many applications, such as the estimation of the number of unobserved species of animals; estimation of vocabulary size, etc. We describe the results of a simulation comparison of three principal frequent-ist' procedures for estimating the number of cells (or species). The first procedure postulates a functional form for the cell probabilities; the second procedure approxi mates the distribution of the probabilities by a parametric probability density function; and the third procedure is based on an estimate of the sample coverage, i.e. the sum of the probabilities of the observed cells. Among the procedures studied, we find that the third (non-parametric) method is globally preferable; the second (functional parametric) method cannot be recommended; and that, when based on the inverse Gaussian density, the first method is competitive in some cases with the third method. We also discuss Sichel's recent generalized inverse Gaussian-based procedure which, with some refine ment, promises to perform at least as well as the non-parametric method in all cases.     

Estimation of non-parametric regression for dasometric measures

The aim of this paper is to describe a simulation procedure to compare parametric regression against a non-parametric regression method, for different functions and sets of information. The proposed methodology improves lack of fit at the edges of the regression curves, and an acceptable result is obtained for the no-parametric estimation in all studied cases. Larger differences appear at the edges of the estimation. The results are applied to the study of dasometric variables, which do not fulfil the normality hypothesis needed for parametric estimation. The kernel regression shows the relationship between the studied variables, which would not be detected with more rigid parametric models.     

Sequential change detection in the presence of unknown parameters

It is commonly required to detect change points in sequences of random variables. In the most difficult setting of this problem, change detection must be performed sequentially with new observations being constantly received over time. Further, the parameters of both the pre- and post- change distributions may be unknown. In Hawkins and Zamba (Technometrics 47(2):164–173, 2005), the sequential generalised likelihood ratio test was introduced for detecting changes in this context, under the assumption that the observations follow a Gaussian distribution. However, we show that the asymptotic approximation used in their test statistic leads to it being conservative even when a large numbers of observations is available. We propose an improved procedure which is more efficient, in the sense of detecting changes faster, in all situations. We also show that similar issues arise in other parametric change detection contexts, which we illustrate by introducing a novel monitoring procedure for sequences of Exponentially distributed random variable, which is an important topic in time-to-failure modelling.     

Wavelet Analysis of Change Points in Nonparametric Hazard Rate Models Under Random Censorship

In this article, we consider detection and estimation of change points in nonparametric hazard rate models. Wavelet methods are utilized to develop a testing procedure for change points detection. The asymptotic properties of the test statistic are explored. When there exist change points in hazard function, we also propose estimators for the number, the locations, and the jump sizes of the change points. The asymptotic properties of these estimators are systematically derived. Some simulation examples are conducted to assess the finite sample performance of the proposed approach and to make comparisons with some existing methods. A real data analysis is provided to illustrate the new approach.     

Local Linear Kernel Regression with Long-Range Dependent Errors

This paper considers the use of a local linear kernel regression method to test whether the mean function of a sequence of long-range dependent processes has discontinuities or change-points. It proposes a non-parametric estimation procedure and then establishes an asymptotic theory for the estimation procedure. Examples, simulated and real, illustrate the estimation procedure.     

Assessing a changepoint in a sequence of repeated measurements with application to a low-energy emission therapy sleep study

We describe a test statistic for a changepoint in a sequence of repeated measurements. This statistic is non-parametric, and may be motivated from either frequentist or Bayesian considerations. We apply this procedure to data arising from a clinical sleep study of low-energy emission therapy.     

Generalized run tests for statistical process control

ABSTRACT

In a sequence of elements, a run is defined as a maximal subsequence of like elements. The number of runs or the length of the longest run has been widely used to test the randomness of an ordered sequence. Based on two different sampling methods and two types of test statistics used, run tests can be classified into one of four cases. Numerous researchers have derived the probability distributions in many different ways, treating each case separately. In the paper, we propose a unified approach which is based on recurrence arguments of two mutually exclusive sub-sequences. We also consider the sequence of nominal data that has more than two classes. Thus, the traditional run tests for a binary sequence are special cases of our generalized run tests. We finally show that the generalized run tests can be applied to many quality management areas, such as testing changes in process variation, developing non-parametric multivariate control charts, and comparing the shapes and locations of more than two process distributions.     

Dynamic functional data analysis with non-parametric state space models

In this article, we introduce a new method for modelling curves with dynamic structures, using a non-parametric approach formulated as a state space model. The non-parametric approach is based on the use of penalised splines, represented as a dynamic mixed model. This formulation can capture the dynamic evolution of curves using a limited number of latent factors, allowing an accurate fit with a small number of parameters. We also present a new method to determine the optimal smoothing parameter through an adaptive procedure, using a formulation analogous to a model of stochastic volatility (SV). The non-parametric state space model allows unifying different methods applied to data with a functional structure in finance. We present the advantages and limitations of this method through simulation studies and also by comparing its predictive performance with other parametric and non-parametric methods used in financial applications using data on the term structure of interest rates.     

Eine bemerkung zur stetigkeit zufälliger felder

In the paper a sequence of bounded regions containing n independent identically and uniformly on D_n distributed points is considered. It is assumed that the d–dimensional volume v(D_n) is asymptotically proportional to n. Under these conditions it is shown that the number of pairs of points within a distance r>0 of each other is asymptotically normally distributed. For proving this among other things a lemma of BOLTHAUSEN is used, whereas even strong estimates for U–statistics are insufficient. The obtained result is applied for testing the hypothesis of randomness     

Non-parametric Tests for the Comparison of Point Processes Based on Incomplete Data

We consider the comparison of point processes in a discrete observation situation in which each subject is observed only at discrete time points and no history information between observation times is available. A class of non-parametric test statistics for the comparison of point processes based on this kind of data is presented and their asymptotic distributions are derived. The proposed tests are generalizations of the corresponding tests for continuous observations. Some results from a simulation study for evaluating the proposed tests are presented and an illustrative example from a clinical trial is discussed.