Optimal collapsing of mixture distributions in robust recursive estimation

Several authors have discussed Kalman filtering procedures using a mixture of normals as a model for the distributions of the noise in the observation and/or the state space equations. Under this model, resulting posteriors involve a mixture of normal distributions, and a “collapsing method” must be found in order to keep the recursive procedure simple. We prove that the Kullback-Leibler distance between the mixture posterior and that of a single normal distribution is minimized when we choose the mean and variance of the single normal distribution to be the mean and variance of the mixture posterior. Hence, “collapsing by moments” is optimal in this sense. We then develop the resulting optimal algorithm for “Kalman filtering” for this situation, and illustrate its performance with an example.     

Variance estimation for integrated population models

State-space models are widely used in ecology. However, it is well known that in practice it can be difficult to estimate both the process and observation variances that occur in such models. We consider this issue for integrated population models, which incorporate state-space models for population dynamics. To some extent, the mechanism of integrated population models protects against this problem, but it can still arise, and two illustrations are provided, in each of which the observation variance is estimated as zero. In the context of an extended case study involving data on British Grey herons, we consider alternative approaches for dealing with the problem when it occurs. In particular, we consider penalised likelihood, a method based on fitting splines and a method of pseudo-replication, which is undertaken via a simple bootstrap procedure. For the case study of the paper, it is shown that when it occurs, an estimate of zero observation variance is unimportant for inference relating to the model parameters of primary interest. This unexpected finding is supported by a simulation study.     

Influence Functions for Sliced Inverse Regression

Abstract.  Sliced inverse regression (SIR) is a dimension reduction technique that is both efficient and simple to implement. The procedure itself relies heavily on estimates that are known to be highly non-robust and, as such, the issue of robustness is often raised. This paper looks at the robustness of SIR by deriving and plotting the influence function for a variety of contamination structures. The sample influence function is also considered and used to highlight that common outlier detection and deletion methods may not be entirely useful to SIR. The asymptotic variance of the estimates is also derived for the single index model when the explanatory variable is known to be normally distributed. The asymptotic variance is then compared for varying choices of the number of slices for a simple model example.     

Bias in the last observation carried forward method under informative dropout

Subject dropout is an inevitable problem in longitudinal studies. It makes the analysis challenging when the main interest is the change in outcome from baseline to endpoint of study. The last observation carried forward (LOCF) method is a very common approach for handling this problem. It assumes that the last measured outcome is frozen in time after the point of dropout, an unrealistic assumption given any time trends. Though existence and direction of the bias can sometimes be anticipated, the more important statistical question involves the actual magnitude of the bias and this requires computation. This paper provides explicit expressions for the exact bias in the LOCF estimates of mean change and its variance when the longitudinal data follow a linear mixed-effects model with linear time trajectories. General dropout patterns are considered that may depend on treatment group, subject-specific trajectories and follow different time to dropout distributions. In our case studies, the magnitude of bias for mean change estimators linearly increases as time to dropout decreases. The bias depends heavily on the dropout interval. The variance term is always underestimated.     

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.     

A Note on Sufficient Conditions for Valid Unmodified t Testing in Correlation Analysis with Autocorrelated and Heteroscedastic Sample Data

Traditionally, sphericity (i.e., independence and homoscedasticity for raw data) is put forward as the condition to be satisfied by the variance–covariance matrix of at least one of the two observation vectors analyzed for correlation, for the unmodified t test of significance to be valid under the Gaussian and constant population mean assumptions. In this article, the author proves that the sphericity condition is too strong and a weaker (i.e., more general) sufficient condition for valid unmodified t testing in correlation analysis is circularity (i.e., independence and homoscedasticity after linear transformation by orthonormal contrasts), to be satisfied by the variance–covariance matrix of one of the two observation vectors. Two other conditions (i.e., compound symmetry for one of the two observation vectors; absence of correlation between the components of one observation vector, combined with a particular pattern of joint heteroscedasticity in the two observation vectors) are also considered and discussed. When both observation vectors possess the same variance–covariance matrix up to a positive multiplicative constant, the circularity condition is shown to be necessary and sufficient. “Observation vectors” may designate partial realizations of temporal or spatial stochastic processes as well as profile vectors of repeated measures. From the proof, it follows that an effective sample size appropriately defined can measure the discrepancy from the more general sufficient condition for valid unmodified t testing in correlation analysis with autocorrelated and heteroscedastic sample data. The proof is complemented by a simulation study. Finally, the differences between the role of the circularity condition in the correlation analysis and its role in the repeated measures ANOVA (i.e., where it was first introduced) are scrutinized, and the link between the circular variance–covariance structure and the centering of observations with respect to the sample mean is emphasized.     

Generalized linear models and transformations for unreplicated factorial experiments in the presence of dispersion effects

For the analysis of replicated designs, many different methods have been suggested. These allow for the estimation of functional dependencies between mean and variance as well as possible dispersion effects within the same model framework. However, in the situation of unreplicated designs, most methods known so far rely on the assumption of constant variances, or a functional relationship between mean and variance as the only source of heteroscedasticity. In this paper, we propose two methods for dealing with unreplicated data, when dispersion effects might also be of importance. One of these is an extension of the Box–Cox-method [Box, G.E.P., Cox, D.R., 1964. An analysis of transformations. Journal of the Royal Statistical Society B 26, 211–252], the other is based on double generalized linear models. Both these methods turn out to yield approximately equivalent results in the case of comparable assumptions, whereas the double generalized linear model is the more general one and allows further extensions. If this class of models is assumed, consistency, asymptotic efficiency and normality of the resulting estimates are shown.     

Functional quasi-likelihood regression models with smooth random effects

Summary. We propose a class of semiparametric functional regression models to describe the influence of vector-valued covariates on a sample of response curves. Each observed curve is viewed as the realization of a random process, composed of an overall mean function and random components. The finite dimensional covariates influence the random components of the eigenfunction expansion through single-index models that include unknown smooth link and variance functions. The parametric components of the single-index models are estimated via quasi-score estimating equations with link and variance functions being estimated nonparametrically. We obtain several basic asymptotic results. The functional regression models proposed are illustrated with the analysis of a data set consisting of egg laying curves for 1000 female Mediterranean fruit-flies (medflies).     

Bounds for how much influence an observation can have

That outliers or influential observations can affect the results in a regression is well-known. But it is not clear how much influence a specific observation can have on other statistics. In time series, especially in predictive situations, the effect of additional observations is of singular importance. We here examine bounds for the effect of an additional observation on the mean, variance, Mahalanobis distance, product moment correlation, and coefficients of linearity and monotonicity.     

COMPONENTS OF VARIANCE MODELS WITH TRANSFORMATIONS

Power transformations are a popular way to improve the agreement between the observations and the assumptions in a statistical model. In this paper it is assumed that the data, after appropriate power transformation Λ, satisfies a variance components model, with independent Gaussian components. The focus is on inference for quantities which have an interpretation regardless of the choice of Λ (Carroll & Ruppert, 1981) – in particular the intraclass correlation coefficient ρ, the predicted probability of a new observation being less than a specified value and the predicted quantile. It is shown that, in the case Λ= 0, the asymptotic variance of ρ is the same, whether or not one treats Δ as estimated or as known. This supports an empirical conjecture of Solomon (1985). For predicted probabilities and predicted quantiles the variance when A is estimated is shown to be only slightly greater than the variance assuming Δ is known, except in the tails of the distribution where there can be substantial difference between the two variances.     

Posterior bimodality in the balanced one-way random-effects model

Summary. Although some researchers have examined posterior multimodality for specific richly parameterized models, multimodality is not well characterized for any such model. The paper characterizes bimodality of the joint and marginal posteriors for a conjugate analysis of the balanced one-way random-effects model with a flat prior on the mean. This apparently simple model has surprisingly complex and even bizarre mode behaviour. Bimodality usually arises when the data indicate a much larger between-groups variance than does the prior. We examine an example in detail, present a graphical display for describing bimodality and use real data sets from a statistical practice to shed light on the practical relevance of bimodality for these models.     

On using the jackknife to estimate quantile variance

We show that the jackknife technique fails badly when applied to the problem of estimating the variance of a sample quantile. When viewed as a point estimator, the jackknife estimator is known to be inconsistent. We show that the ratio of the jackknife variance estimate to the true variance has an asymptotic Weibull distribution with parameters 1 and 1/2. We also show that if the jackknife variance estimate is used to Studentize the sample quantile, the asymptotic distribution of the resulting Studentized statistic is markedly nonnormal, having infinite mean. This result is in stark contrast with that obtained in simpler problems, such as that of constructing confidence intervals for a mean, where the jackknife-Studentized statistic has an asymptotic standard normal distribution.     

Sum of a Random Number of Correlated Random Variables that Depend on the Number of Summands

The mean and variance of a sum of a random number of random variables are well known when the number of summands is independent of each summand and when the summands are independent and identically distributed (iid), or when all summands are identical. In scientific and financial applications, the preceding conditions are often too restrictive. Here, we calculate the mean and variance of a sum of a random number of random summands when the mean and variance of each summand depend on the number of summands and when every pair of summands has the same correlation. This article shows that the variance increases with the correlation between summands and equals the variance in the iid or identical cases when the correlation is zero or one.     

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.     

NON-SYMMETRICAL CORRESPONDENCE ANALYSIS WITH CONCATENATION AND LINEAR CONSTRAINTS

Correspondence analysis is a popular statistical technique used to identify graphically the presence, and structure, of association between two or more cross-classified categorical variables. Such a procedure is very useful when it is known that there is a symmetric (two-way) relationship between the variables. When such a relationship is known not to exist, non-symmetrical correspondence analysis is more appropriate as a method of establishing the source of association. This paper highlights some tools that can be used to explore the behaviour of asymmetric categorical variables. These tools consist of confidence regions, the link between non-symmetrical correspondence analysis and the analysis of variance of categorical variables, and the effect of imposing linear constraints. We also explore the application of non-symmetrical correspondence analysis to three-way contingency tables.     

Confidence intervals on the total variance in an unbalanced two-fold nested classification with equal subsampling

A problem of interest in a variance component analysis is the construction of a confidence interval on the variance of a single observation. This article considers an unbalanced two-fold nested classification with equal subsampling and compares two methods for constructing this interval . Computer simulations indicate that one of these methods in general will provide an interval that has an achieved confidence coefficient at least as great as the stated value.     

Optimal two-point designs for the michaelis-menten model with heteroscedastic errors

We construct D-optimal designs for the Michaelis-Menten model when the variance of the response depends on the independent variable. However, this dependence is only partially known. A Bayesian approacn is used to find an optimal design by incorporating the prior lnformation about the variance structure. We demonstrate the method for a class of error variance structures and present efficiencies of these optimal designs under prior mis-specifications. In particular, we show that an erroneous assumption on the variance structure for the Michaelis-Menten model can have serious consequences.     

RANKED SET SAMPLING FROM LOCATION-SCALE FAMILIES OF SYMMETRIC DISTRIBUTIONS

Statistical inference based on ranked set sampling has primarily been motivated by nonparametric problems. However, the sampling procedure can provide an improved estimator of the population mean when the population is partially known. In this article, we consider estimation of the population mean and variance for the location-scale families of distributions. We derive and compare different unbiased estimators of these parameters based on rindependent replications of a ranked set sample of size n.Large sample properties, along with asymptotic relative efficiencies, help identify which estimators are best suited for different location-scale distributions.     

Distribution-dependent and distribution-free confidence intervals for the variance

Finding an interval estimation procedure for the variance of a population that achieves a specified confidence level can be problematic. If the distribution of the population is known, then a distribution-dependent interval for the variance can be obtained by considering a power transformation of the sample variance. Simulation results suggest that this method produces intervals for the variance that maintain the nominal probability of coverage for a wide variety of distributions. If the underlying distribution is unknown, then the power itself must be estimated prior to forming the endpoints of the interval. The result is a distribution-free confidence interval estimator of the population variance. Simulation studies indicate that the power transformation method compares favorably to the logarithmic transformation method and the nonparametric bias-corrected and accelerated bootstrap method for moderately sized samples. However, two applications, one in forestry and the other in health sciences, demonstrate that no single method is best for all scenarios.