EVVMA and cusum control charts in the presence of correlation

The statistical properties of control charts are usually evaluated under the assumption that the observations from the process are independent. For many processes however, observations which are closely spaced in time will be correlated. This paper considers EWMA and CUSUM control charts for the process mean when the observations are from an AR(1) process with additional random error. This simple model may be a reasonable model for many processes encountered in practice. The ARL and steady state ARL of the EWMA and CUSUM charts are evaluated numerically using an integral equation approach and a Markov chain approach. The numerical results show that correlation can have a significant effect on the properties of these charts. Tables are given to aid in the design of these charts when the observations follow the assumed model.     

Effect of correlation on the estimation of a mean in the presence of spurious observations

This paper examines the effect of correlation of observations on estimators of a mean which are designed to guard against the possibility of spurious observations (that is, observations generated in a manner not intended). The mean squared error, premium and protection of these estimators are evaluated and discussed for some specific correlation structures.     

The probability of degenerate maximum-likelihood estimates with randomly truncated survival data

We study the nonparametric maximum-likelihood estimator of the marginal distribution function in bivariate data with truncated sum (Woodroofe 1985), and calculate the probability that it will assign zero probability to a range of observed values. We show that in a simple artificial model with uniform intensity in both variables the probability may be expressed using recursive formulae, and that large-sample approximations valid under weak regularity conditions show good agreement with as few as 20 observations. In the uniform-intensity model, the probability of degenerate estimates is rather small, being just over 5% with a dataset of 20 observations and falling below 1% with 125 observations. However, in a model of transfusion-associated AIDS in 125 children and adults (Kalbfleisch and Lawless 1989), the probability of degenerate estimates is much larger—about 30% with 20 observations and 3% with the actual 125 observations. This is largely because both the infection and incubation density are increasing over time. Such a pattern makes the condition for degeneracy, i.e., relatively high observed intensity at extreme values of each variable, much more likely.     

Nonparametric change point detection for periodic time series

We consider detection of multiple changes in the distribution of periodic and autocorrelated data with known period. To account for periodicity we transform the sequence of vector observations by arranging them in matrices and thereby producing a sequence of independently and identically distributed matrix observations. We propose methods of testing the equality of matrix distributions and present methods that can be applied to matrix observations using the E-divisive algorithm. We show that periodicity and autocorrelation degrade existing change detection methods because they blur the changes that these procedures aim to discover. Methods that ignore the periodicity have low power to detect changes in the mean and the variance of periodic time series when the periodic effects overwhelm the true changes, while the proposed methods detect such changes with high power. We illustrate the proposed methods by detecting changes in the water quality of Lake Kasumigaura in Japan. The Canadian Journal of Statistics 48: 518–534; 2020 © 2020 Statistical Society of Canada     

The precision of an estimator due to ryan

Ryan (1974) presents a method of estimating the mean of a symmetric distribution when observations are recorded in classes of common width h, and when some of the observations may be repeats of earlier observations (the extent of which is unknown). (In other words, if a class contains two observations, the investigator is unsure as to whether they arose from two distinct subjects or the same subject twice.) Ryan’s results, as presented, do not permit one to construct asymptotic confidence intervals for the mean. In this note, we solve this problem.     

A comparative study on detection of influential observations in linear regression

A large number of statistics are used in the literature to detect outliers and influential observations in the linear regression model. In this paper comparison studies have been made for determining a statistic which performs better than the other. This includes: (i) a detailed simulation study, and (ii) analyses of several data sets studied by different authors. Different choices of the design matrix of regression model are considered. Design A studies the performance of the various statistics for detecting the scale shift type outliers, and designs B and C provide information on the performance of the statistics for identifying the influential observations. We have used cutoff points using the exact distributions and Bonferroni's inequality for each statistic. The results show that the studentized residual which is used for detection of mean shift outliers is appropriate for detection of scale shift outliers also, and the Welsch's statistic and the Cook's distance are appropriate for detection of influential observations.     

Leverages and Influential Observations in a Regression Model with Autocorrelated Errors

This article deals with the general form of the hat matrix and the DFBETA measure to detect the influential observations and the leverages in the linear regression model with more than one regressor when the errors are from AR(1) and AR(2) processes. Previous studies dealing with the influential observations and the leverages in the constant mean model and regression through the origin model are obtained as special cases. To demonstrate the utility of the hat matrix and the DFBETA measure, two numerical examples based on the ice cream consumption data with AR(1) errors and the Fox-Hartnagel data with AR(2) errors are analyzed. The results show that the parameter of the autoregressive process affects the influential and leverage points.     

Estimation of the parameters of a general student’s t distribution

We have observations for a t distribution with unknown mean, variance, and degrees of freedom, each of which we wish to estimate. The major problem lies in the estimate of the degrees of freedom. We show that a relatively efficient yet very simple estimator is a given function of the ratio of percentile estimates. We derive the appropriate estimator, provide equations for transformation and standard errors, contrast this with other estimators, and give examples.     

Bootstrapping Spatial Median for Location Problems

In multivariate location problems, the sample mean is most widely used, having various advantages. It is, however, very sensitive to outlying observations and inefficient for data from heavy tailed distributions. In this situation, the spatial median is more robust than the sample mean and could be a reasonable alternative. We reviewed several spatial median based testing methods for multivariate location and compared their significance level and power through Monte Carlo simulations. The results show that bootstrap method is efficient for the estimation of the covariance matrix of the sample spatial median. We also proposed bootstrap simultaneous confidence intervals based on the spatial median for multiple comparisons in the multi-sample case.     

Optimal allocations in sequential tests involving two populations with covariates

We consider the problem of testing which of two normally distributed treatments has the largest mean, when the tested populations incorporate a covariate. From the class of procedures using the invariant sequential probability ratio test we derive an optimal allocation that minimizes, in a continuous time setting, the expected sampling costs. Simulations show that this procedure reduces the number of observations from the costlier treatment and categories while maintaining an overall sample size closer to the “pairwise” procedure. A randomized trial example is given.     

Estimating the population median using only a central subset of observations

In estimating the population median, it is common to encounter estimators which are linear combinations of a small number of central observations. Sample medians, sample quasi medians, trimmed means, jackknifed (and delete‐d jackknifed) medians and jackknifed quasi medians are all familiar examples. The objective of this paper is to show that within this class the quasi medians turn out to have the best asymptotic mean squared error.     

High-correlated residuals improved estimation in the high-dimensional SUR model

This article focuses on estimating regression coefficients in the high-dimensional seemingly unrelated regression model. When the number of equations exceeds that of the observations, both the maximum likelihood estimator and Zellner’s two-stage estimator do not exist. As an alternative, we propose a two-stage conditional expectation improved estimator. The new estimator is further improved by the high-correlated residuals, and the high correlation is determined by hypothesis testings. Simulations show that the new estimator outperforms the ordinary least-squares estimator in terms of mean square errors, especially when high-correlated residuals exist between the equations.     

An Evaluation of the Double Exponentially Weighted Moving Average Control Chart

In this article we perform a careful investigation of the double exponentially weighted moving average (DEWMA) chart performance for monitoring the process mean. We compare the performance of this chart to the usual EWMA control chart based on zero-state and worst-case average run length (ARL) measures. We also evaluate the signal resistance measure of the DEWMA chart and compare its maximum value to that of the EWMA chart. We show that the superiority of the DEWMA chart over the simpler standard EWMA chart based on zero-state ARL performance disappears when the smoothing constant of the EWMA chart is chosen to give weights to past observations closer to those given by the DEWMA chart. Moreover, our results show that the standard EWMA chart has much better performance than the DEWMA chart in terms of worst-case ARL values, especially when small smoothing constants are used. We also demonstrate using an illustrative example that the DEWMA chart can build up an exceedingly large amount of inertia when used to monitor the process mean.     

Nonparametric estimation of the derivatives of a density by the method of wavelet for mixing sequences

The problem of estimation of the derivative of a probability density f is considered, using wavelet orthogonal bases. We consider an important kind of dependent random variables, the so-called mixing random variables and investigate the precise asymptotic expression for the mean integrated error of the wavelet estimators. We show that the mean integrated error of the proposed estimator attains the same rate as when the observations are independent, under certain week dependence conditions imposed to the {X _i }, defined in {Ω, N, P}.     

Spuriosity and outliers in directional data

This paper discusses inference regarding the mean direction and the concentration parameters based on data from the von Mises distribution from a Bayesian point of view, when k(k < n/2) of the n observations are spurious, that is, are from a von Mises population with a shifted mean direction. The Bayesian analysis for this spuriosity case provides both detection, identification, and estimation for the mean direction and the concentration parameter when indeed spurious observations are present, possibly giving rise to outliers.     

Influence functions applied to the estimation of mean rain rate

In this paper we illustrate the usefulness of influence functions for studying properties of various statistical estimators of mean rain rate using space-borne radar data. In Martin (1999), estimators using censoring, minimum chi-square, and least squares are compared in terms of asymptotic variance. Here, we use influence functions to consider robustness properties of the same estimators. We also obtain formulas for the asymptotic variance of the estimators using influence functions, and thus show that they may also be used for studying relative efficiency. The least squares estimator, although less efficient, is shown to be more robust in the sense that it has the smallest gross-error sensitivity. In some cases, influence functions associated with the estimators reveal counterintuitive behaviour. For example, observations that are less than the mean rain rate may increase the estimated mean. The additional information gleaned from influence functions may be used to understand better and improve the estimation procedures themselves.     

Missing observations in time series

A method based on forecasting techniques is proposed to estimate missing observations in time series. Using mean squares, this method is compared to the minimum mean square estimate.     

Chi-Square quantile-based multivariate variance monitoring for individual observations

Under the normality assumption, some statistics for monitoring a multivariate process variance for individual observations can be used to detect a variance shift, but the distribution of their in-control run length has a high variance as well as the median that is extremely smaller than the mean, which leads to many false alarms in the in-control process. In this paper, we propose a chi-square quantile-based monitoring statistic which is free of the problems. The numerical experiments show that the proposed monitoring statistics outperform the existing monitoring statistics in terms of the detection of a shift for the variance.     

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.     

Nonparametric estimation of the mean function of a stochastic process with missing observations

In an attempt to identify similarities between methods for estimating a mean function with different types of response or observation processes, we explore a general theoretical framework for nonparametric estimation of the mean function of a response process subject to incomplete observations. Special cases of the response process include quantitative responses and discrete state processes such as survival processes, counting processes and alternating binary processes. The incomplete data are assumed to arise from a general response-independent observation process, which includes right- censoring, interval censoring, periodic observation, and mixtures of these as special cases. We explore two criteria for defining nonparametric estimators, one based on the sample mean of available data and the other inspired by the construction of Kaplan-Meier (or product-limit) estimator [J. Am. Statist. Assoc. 53 (1958) 457] for right-censored survival data. We show that under regularity conditions the estimated mean functions resulting from both criteria are consistent and converge weakly to Gaussian processes, and provide consistent estimators of their covariance functions. We then evaluate these general criteria for specific responses and observation processes, and show how they lead to familiar estimators for some response and observation processes and new estimators for others. We illustrate the latter with data from an recently completed AIDS clinical trial.