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In a graphical procedure for comparing k treatment means in a one-way ANOVA, one displays uncertainty intervals around the sample means and judges any pair to be significantly different if and only if their uncertainty intervals do not overlap. A graphical procedure is a Multiple Comparison Procedure (MCP) if and only if it controls the experimentwise error rate. In this paper we consider some new graphical MCP's for the unbalanced one-way ANOVA design. These procedures are based on different approximations to the Tukey-Kramer (TK) procedure (e.g., Kramer 1956). As such, they constitute alternatives to Gabriel (1978) (and its modification by Andrews, Snee, and Sarner 1980), which is based on approximating a less efficient MCP (the GT2 of Hochberg 1974). Two of the four procedures considered here are based on best and simple upper bounds to all the confidence-interval lengths of the TK method and hence must be conservative. The other two procedures are based on approximations (here too we have the best vs. the simple procedure), but simulations were used to find that their true experimentwise error rates are less than the nominal ones; that is, these procedures are still on the conservative side. The choice of a particular procedure will depend then on the relative importance of simplicity, efficiency, and the security of having a controlled experimentwise error rate. [ABSTRACT FROM AUTHOR]

This article is an expository paper to demonstrate the usefulness of Pearson curves in density estimation especially for those unaware of this early development in statistics. It is shown how to fit the curves and how very good approximate percentage points can be obtained for intractable distributions when the first four moments (or three moments and one endpoint) are known exactly (not estimated from sample data). The effectiveness of this method in density estimation is illustrated in three somewhat disparate contexts and reference is given to others. In general, the Pearson curves give an excellent approximation to the long tail of a distribution, the tail-most often needed in practical work. [ABSTRACT FROM AUTHOR]

The authors offer a response to the commentaries made on their article on statistical estimation and privacy. They discuss the issues raised by the discussants including the advantages and disadvantages of local notions of privacy as compared to classical differential and approximate differential privacy, the issue of interactivity, and the importance of local differential privacy. The goal of statistical inference is also mentioned.

Matching estimators are widely used in statistical data analysis. However, the large sample distribution of matching estimators has been derived only for particular cases. This article establishes a martingale representation for matching estimators. This representation allows the use of martingale limit theorems to derive the large sample distribution of matching estimators. As an illustration of the applicability of the theory, we derive the asymptotic distribution of a matching estimator when matching is carried out without replacement, a result previously unavailable in the literature. In addition, we apply the techniques proposed in this article to derive a correction to the standard error of a sample mean when missing data are imputed using the “hot deck,” a matching imputation method widely used in the Current Population Survey (CPS) and other large surveys in the social sciences. We demonstrate the empirical relevance of our methods using two Monte Carlo designs based on actual datasets. In these Monte Carlo exercises, the large sample distribution of matching estimators derived in this article provides an accurate approximation to the small sample behavior of these estimators. In addition, our simulations show that standard errors that do not take into account hot-deck imputation of missing data may be severely downward biased, while standard errors that incorporate the correction for hot-deck imputation perform extremely well. This article has online supplementary materials. [ABSTRACT FROM AUTHOR]

Kinetic analysis is used to extract metabolic information from dynamic positron emission tomography (PET) uptake data. The theory of indicator dilutions, developed in the seminal work of Meier and Zierler (1954), provides a probabilistic framework for representation of PET tracer uptake data in terms of a convolution between an arterial input function and a tissue residue. The residue is a scaled survival function associated with tracer residence in the tissue. Nonparametric inference for the residue, a deconvolution problem, provides a novel approach to kinetic analysis—critically one that is not reliant on specific compartmental modeling assumptions. A practical computational technique based on regularized cubic B-spline approximation of the residence time distribution is proposed. Nonparametric residue analysis allows formal statistical evaluation of specific parametric models to be considered. This analysis needs to properly account for the increased flexibility of the nonparametric estimator. The methodology is illustrated using data from a series of cerebral studies with PET and fluorodeoxyglucose (FDG) in normal subjects. Comparisons are made between key functionals of the residue, tracer flux, flow, etc., resulting from a parametric (the standard two-compartment of Phelps et al. 1979) and a nonparametric analysis. Strong statistical evidence against the compartment model is found. Primarily these differences relate to the representation of the early temporal structure of the tracer residence—largely a function of the vascular supply network. There are convincing physiological arguments against the representations implied by the compartmental approach but this is the first time that a rigorous statistical confirmation using PET data has been reported. The compartmental analysis produces suspect values for flow but, notably, the impact on the metabolic flux, though statistically significant, is limited to deviations on the order of 3%-4%. The general advantage of the nonparametric residue analysis is the ability to provide a valid kinetic quantitation in the context of studies where there may be heterogeneity or other uncertainty about the accuracy of a compartmental model approximation of the tissue residue. [ABSTRACT FROM AUTHOR]

This article deals with the status of approximation method in both statistical theory and practice. Approximation is ubiquitous in both statistical theory and practice. Many approximations routinely used in statistics can be derived by approximating certain integrals and are asymptotic in nature. That is, such approximations are typically developed under the assumption that some quantity connected to the amount of information available becomes infinitely large. In the case of approximating density and distribution functions, one example here is local linearization. This method of approximation has a long history, with roots that can be traced back to the time of Laplace and Gauss. In general, local linearization leads to approximations that are accurate to first order. Methods of higher-order asymptotic approximation, in principle more accurate than those of first order, include Edgeworth expansion, tilted Edgeworth expansion, and saddlepoint approximations for density and distribution functions. Laplace's method for approximating integrals can be viewed as a special case of a more general class of methods for approximating integrals known as saddlepoint methods. The saddlepoint method, and an important special case known as the method of steepest descent, are designed to approximate integrals of the Laplace type in which both the integrand and contour of integration are allowed to be complex valued.

We consider the problem of setting approximate confidence intervals for a single parameter theta in a multiparameter family. The standard approximate intervals based on maximum likelihood theory, theta +/- sigma z[sup (a)], can be quite misleading. In practice, tricks based on transformations, bias corrections, and so forth, are often used to improve their accuracy. The bootstrap confidence intervals discussed in this article automatically incorporate such tricks without requiring the statistician to think them through for each new application, at the price of a considerable increase in computational effort. The new intervals incorporate an improvement over previously suggested methods, which results in second-order correctness in a wide variety of problems. In addition to parametric families, bootstrap intervals are also developed for nonparametric situations. [ABSTRACT FROM AUTHOR]

The BC[sub a] bootstrap procedure (Efron 1987) for constructing parametric and nonparametric confidence intervals is considered. Like the bootstrap, this procedure can be applied to complicated problems in a wide range of situations. For models indexed by a scalar parameter theta with efficient estimator theta, the BC[sub a] procedure relies on the existence of a transformation g(.) such that g(O) is approximately normally distributed with standard deviation 1 + ag(theta), although explicit knowledge of g(.) is not required. In this article, we show how to construct this transformation by generalizing the one given by Efron (1987, sec. 10) for translation families. This construction consists of the composition of a variance-stabilizing transformation and a skewness-reducing transformation. It produces a new interval, the BC[sup 0, sub a] interval, that is asymptotically equivalent to the BC[sub a] interval and can be computed without bootstrap sampling. We also derive from this construction an accurate approximation to the bootstrap distribution of theta that also does not require bootstrap sampling. Both the new interval and the approximation require only n + 2 evaluations of the statistic. Like the BC[sub a] procedure, the BC[sup 0, sub a] interval can be extended to multiparameter and nonparametric problems. As an example, we compute the BC[sup 0, sub a] interval for Cox's partial likelihood estimator, a complicated statistic that is obtained by iterative solution of a score equation. [ABSTRACT FROM AUTHOR]