Two zs are Better than One

Given only a random sample of observations, the usual estimator for the population mean is the sample mean. If additional information is provided it might be possible in some situations to obtain a better estimator. The situation considered here is when the variable whose mean is sought is composed of factors that are themselves observable. In the basic case, the variable can be expressed as the product of two, independent, more basic variables, but we also consider the case of more than two, the effect of correlation, and when there are observation costs.     

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.     

Robust estimation of location in samples of size three

Youden (1953) discussed the practice of averaging the two most concordant observations in sets of three measurements as a method of estimating location. Distributional results for this estimator can be found in Seth (1950) and Lieblein (1952). It follows from their work that the sample median has smaller variance for normal and uniform populations. In this paper it is shown that themedian stochastically dominates the average of the two closest observations for uniform, normal, double–exponential and Cauchy populations and thus is the superior resistant estimator in these cases for a broad class of loss functions. However, an example is given in which, for a particular contaminaion model and loss function, the mean of the closest two observations has smaller risk than the median.     

Best linear unbiased estimates in ranked-set sampling with particular reference to imperfect ordering

SUMMARY Ranked-set sampling is a widely used sampling procedure when sample observations are expensive or difficult to obtain. It departs from simple random sampling by seeking to spread the observations in the sample widely over the distribution or population. This is achieved by ranking methods which may need to employ concomitant information. The ranked-set sample mean is known to be more efficient than the corresponding simple random sample mean. Instead of the ranked-set sample mean, this paper considers the corresponding optimal estimator: the ranked-set best linear unbiased estimator. This is shown to be more efficient, even for normal data, but particularly for skew data, such as from an exponential distribution. The corresponding forms of the estimators are quite distinct from the ranked-set sample mean. Improvement holds where the ordering is perfect or imperfect, with this prospect of improper ordering being explored through the use of concomitants. In addition, the corresponding optimal linear estimator of a scale parameter is also discussed. The results are applied to a biological problem that involves the estimation of root weights for experimental plants, where the expense of measurement implies the need to minimize the number of observations taken.     

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.     

The Slippage Problem for the Circular Normal Distribution

The slippage problem occurs when an unspecified observation in a given random sample is from a distribution other than that for all the remaining observations. This paper studies the problem in terms of the 'slip' in the mean direction of a circular normal distribution. The slippage problem is first treated as a multiple decision problem with a prior which is invariant under the permutations of the hypotheses. The probabilities of accepting the various hypotheses for the Bayes rule with respect to this prior are explicitly obtained. The likelihood ratio tests for this slippage problem, for the cases when the mean directions are both known and unknown, are shown to be easily computable. The tests are illustrated through two well-known datasets. The performances of a range of tests are compared using extensive simulation.     

Estimation of Correlation for Continuous Semimartingales

Abstract. In this study we are concerned with inference on the correlation parameter ρ of two Brownian motions, when only high‐frequency observations from two one‐dimensional continuous Itô semimartingales, driven by these particular Brownian motions, are available. Estimators for ρ are constructed in two situations: either when both components are observed (at the same time), or when only one component is observed and the other one represents its volatility process and thus has to be estimated from the data as well. In the first case it is shown that our estimator has the same asymptotic behaviour as the standard one for i.i.d. normal observations, whereas a feasible estimator can still be defined in the second framework, but with a slower rate of convergence.     

A control chart for variance based on squared ranks

A nonparametric control chart for variance is proposed. The chart is constructed following the change-point approach through the recursive use of the squared ranks test for variance. It is capable of detecting changes in the behaviour of individual observations with performance similar to a self-starting CUSUM chart for scale when normality is assumed, and a relatively better power when assessing nonnormal observations. A comparison is also made with two equivalent nonparametric charts based on Mood and Ansari-Bradley statistics. When dealing with symmetrical distributions, the proposed chart shows smaller (better) out-of-control average run length (ARL), and a competing performance otherwise. In addition, sensitivity to changes in mean and variance at the same time was tested. Extensive Monte Carlo simulation was used to measure performance, and a practical example is provided to illustrate how the proposed control chart can be implemented in practice.     

Inferences for the ratio: Fieller’s interval,log ratio,and large sample based confidence intervals

In sample surveys and many other areas of application, the ratio of variables is often of great importance. This often occurs when one variable is available at the population level while another variable of interest is available for sample data only. In this case, using the sample ratio, we can often gather valuable information on the variable of interest for the unsampled observations. In many other studies, the ratio itself is of interest, for example when estimating proportions from a random number of observations. In this note we compare three confidence intervals for the population ratio: A large sample interval, a log based version of the large sample interval, and Fieller’s interval. This is done through data analysis and through a small simulation experiment. The Fieller method has often been proposed as a superior interval for small sample sizes. We show through a data example and simulation experiments that Fieller’s method often gives nonsensical and uninformative intervals when the observations are noisy relative to the mean of the data. The large sample interval does not similarly suffer and thus can be a more reliable method for small and large samples.     

Estimation of the linear-plateau segmented regression model in the presence of measurement error

It is well known that when the true values of the independent variable are unobservable due to measurement error, the least squares estimator for a regression model is biased and inconsistent. When repeated observations on each x_i are taken, consistent estimators for the linear-plateau model can be formed. The repeated observations are required to classify each observation to the appropriate line segment. Two cases of repeated observations are treated in detail. First, when a single value of y_i is observed with the repeated observations of x_i the least squares estimator using the mean of the repeated x_i observations is consistent and asymptotically normal. Second, when repeated observations on the pair (x_i, y_i ) are taken the least squares estimator is inconsistent, but two consistent estimators are proposed: one that consistently estimates the bias of the least squares estimator and adjusts accordingly; the second is the least squares estimator using the mean of the repeated observations on each pair.     

Monitoring processes with data censored owing to competing risks by using exponentially weighted moving average control charts

In industry, process monitoring is widely employed to detect process changes rapidly. However, in some industrial applications observations are censored. For example, when testing breaking strengths and failure times often a limited stress test is performed. With censored observations, a direct application of traditional monitoring procedures is not appropriate. When the censoring occurs due to competing risks, we propose a control chart based on conditional expected values to detect changes in the mean strength. To protect against possible confounding caused by changes in the mean of the censoring mechanism we also suggest a similar chart to detect changes in the mean censoring level. We provide an example of monitoring bond strength to illustrate the application of this methodology.     

Bayesian Point Estimation Using the Predictive Distribution

There is considerable question about how a Bayesian might provide a point estimate for a parameter when no loss function is specified. The mean, median, and mode of the posterior distribution have all been suggested. This article considers a natural Bayesian estimator based on the predictive distribution of future observations. It is shown that for the set of parameters that admit an unbiased estimate, this predictive estimate coincides with the posterior mean of the parameter. It is argued that this result provides some justification for use of the posterior mean as a Bayesian point estimate when there is no loss structure.     

On discrimination and classification with multivariate repeated measures data

We study the problem of classification with multiple q-variate observations with and without time effect on each individual. We develop new classification rules for populations with certain structured and unstructured mean vectors and under certain covariance structures. The new classification rules are effective when the number of observations is not large enough to estimate the variance–covariance matrix. Computational schemes for maximum likelihood estimates of required population parameters are given. We apply our findings to two real data sets as well as to a simulated data set.     

Testing for Spatial Isotropy Under General Designs

Spatial modeling is typically composed of a specification of a mean function and a model for the correlation structure. A common assumption on the spatial correlation is that it is isotropic. This means that the correlation between any two observations depends only on the distance between those sites and not on their relative orientation. The assumption of isotropy is often made due to a simpler interpretation of correlation behavior and to an easier estimation problem under an assumed isotropy. The assumption of isotropy, however, can have serious deleterious effects when not appropriate. In this paper we formulate a test of isotropy for spatial observations located according to a general class of stochastic designs. Distribution theory of our test statistic is derived and we carry out extensive simulations which verify the efficacy of our approach. We apply our methodology to a data set on longleaf pine trees from an oldgrowth forest in the southern United States.     

Small-sample performance of a modified least-failures sampling procedure for bernoulli subset selection

We describe two sequential sampling procedures for Bernoulli subset selection which were shown to exhibit desirable behavior for large-sample problems. These procedures have identical performance characteristics in terms of the number of observations taken from any one of the populations under investigation, but one of the procedures employs one-at-a-time sampling while theother allows observations to be taken in blocks during early stages of experimentation. In this paper, a simulation study of their behavior for small-sample cases (n > 25) reveals that they canresult in a savings (sometimes substantial) in the expected total number of observations requiredto terminate the experiment as compared to single-stage procedures. Hence they may be quite usefulto a practitioner for screening purposes when sampling is limited.     

Smoothing Time Series with Local Polynomial Regression on Time

Time series smoothers estimate the level of a time series at time t as its conditional expectation given present, past and future observations, with the smoothed value depending on the estimated time series model. Alternatively, local polynomial regressions on time can be used to estimate the level, with the implied smoothed value depending on the weight function and the bandwidth in the local linear least squares fit. In this article we compare the two smoothing approaches and describe their similarities. Through simulations, we assess the increase in the mean square error that results when approximating the estimated optimal time series smoother with the local regression estimate of the level.     

Influential subsets on the variable selection

When one or few observations are deleted in the multiple linear regression model, they can affect the variable selection. In this paper we derived the formula for the Mallows C_p criterion when k observations are deleted and express it as a functionn of basic building blocks such as residuals and leverages. Also, two real date sets are used to see how the selected model changes as few observations re deleted.     

Tests for a Multiple-Sample Problem in High Dimensions

In this article, we consider the problem of testing the hypothesis on mean vectors in multiple-sample problem when the number of observations is smaller than the number of variables. First we propose an independence rule test (IRT) to deal with high-dimensional effects. The asymptotic distributions of IRT under the null hypothesis as well as under the alternative are established when both the dimension and the sample size go to infinity. Next, using the derived asymptotic power of IRT, we propose an adaptive independence rule test (AIRT) that is particularly designed for testing against sparse alternatives. Our AIRT is novel in that it can effectively pick out a few relevant features and reduce the effect of noise accumulation. Real data analysis and Monte Carlo simulations are used to illustrate our proposed methods.     

THE EXACT BIAS OF THE LOG-PERIODOGRAM REGRESSION ESTIMATOR

The paper makes two contributions. First, we provide a formula for the exact distribution of the periodogram evaluated at any arbitrary frequency, when the sample is taken from any zero-mean stationary Gaussian process. The inadequacy of the asymptotic distribution is demonstrated through an example in which the observations are generated by a fractional Gaussian noise process. The results are then applied in deriving the exact bias of the log-periodogram regression estimator (Geweke and Porter-Hudak (1983), Robinson (1995)). The formula is computable. Practical bounds on this bias are developed and their arithmetic mean is shown to be accurate and useful.     

Testing hypotheses when there could be an initial effect

In many treatment-versus-control experiments, the observed random variables can be written as the product of a Bernoulli and a continuous random variable. The treatment can affect the distribution of the observations in two ways.

1. the probability that the observation is 0 could be altered.

2. the distribution of the nonzero observations could be changed.

We may also want to measure the combined effect of the treatment.

3. the expected value of control and treated units may differ.

A method is presented for testing for the presence of the combined effect when the general form of the distribution function of the continuous observations is known. For the case when this distribution function is from the family of gamma distributions, a previously proposed test criterion for the combined effect has poor power properties. In this paper, we discuss a test criterion that has improved power properties.