Huge data sets from the teletraffic industry exhibit many nonstandard characteristics such as heavy tails and long range dependence. Various estimation methods for heavy tailed time series with positive innovations are reviewed. These include parameter estimation and model identification methods for autoregressions and moving averages. Parameter estimation methods include those of Yule-Walker and the linear programming estimators of Feigin and Resnick as well estimators for tail heaviness such as the Hill estimator and the qq-estimator. Examples are given using call holding data and interarrivals between packet transmissions on a computer network. The limit theory makes heavy use of point process techniques and random set theory.

We study what we call "dyadic CART" --a method of nonparametric regression which constructs a recursive partition by optimizing a complexity penalized sum of squares, where the optimization is over all recursive partitions arising from midpoint splits. We show that the method is adaptive to unknown degrees of anisotropic smoothness. Specifically, consider the anisotropic smoothness classes of Nikoľskii, consisting of bivariate functions $$ whose finite difference of distance h in direction i is bounded in $$ norm by $$. We show that dyadic CART, with an appropriate complexity penalty parameter $$, is within logarithmic terms of minimax over every anisotropic smoothness class $$.

The proof shows that dyadic CART is identical to a certain adaptive best-ortho-basis algorithm based on the library of all anisotropic Haar bases. Then it applies empirical basis selection ideas of Donoho and Johnstone. The basis empirically selected by dyadic CART is shown to be nearly as good as a basis ideally adapted to the underlying f. The risk of estimation in an ideally adapted anisotropic Haar basis is shown to be comparable to the minimax risk over anisotropic smoothness classes.

Underlying the success of this argument is harmonic analysis of anisotropic smoothness classes. We show that, for each anisotropic smoothness class, there is an anisotropic Haar basis which is a best orthogonal basis for representing that smoothness class; the basis is optimal not just within the library of anisotropic Haar bases, but among all orthogonal bases of $$.

We introduce interpolation methods that enable nonlinear wavelet estimators to be employed with stochastic design, or nondyadic regular design, in problems of nonparametric regression. This approach allows relatively rapid computation, involving dyadic approximations to wavelet-after-interpolation techniques. New types of interpolation are described, enabling first-order variance reduction at the expense of second-order increases in bias. The effect of interpolation on threshold choice is addressed, and appropriate thresholds are suggested for error distributions with as few as four finite moments.

The Kolmogorov-Smirnov distance is an important tool for constructing confidence sets and tests in univariate problems. In multivariate settings, an analogous role is played by the halfspace distance, which has the merit of being invariant under linear transformations. However, the evaluation of the halfspace distance between two samples is a computationally very intensive combinatorial problem even in moderate dimensions, which severely restricts the use of the halfspace distance, especially in resampling procedures. To approximate this distance in a fast and data-dependent way, the notion of a dual measure is introduced. Based on geometric concepts, it will be shown how the above problem can be put as a density estimation problem using Monte Carlo sampling in a certain dual space. A central limit theorem for the empirical halfspace distance is derived and used as a gauge to compare the new procedure with a traditional random search.

Numerous statistical applications require the evaluation of the probability content of a convex polyhedron. We demonstrate for a given polyhedron in $$ that there is a depth d inclusion-exclusion identity for its indicator function, which is a linear combination of indicator functions of intersections of at most d half-spaces. Terms in the identity are determined by the incidence of the facets of the polyhedron, which can be found using linear programming. This identity can be truncated at any depth to give a lower or upper bound. In addition, the resulting inequalities lead to importance sampling schemes for evaluating the probability content, and these methods tend to be more efficient than the naive hit-or-miss Monte Carlo method.

These results arise in a more general setting which we introduce. An abstract tube consists of a pair $$ where $$ is a collection of sets, $$ is a simplicial complex, and where each subcomplex $$ is contractible whenever $$. The notion presented here is stronger than the one introduced earlier by Naiman and Wynn. Several examples are given and key consequences are demonstrated. In particular, arrangements of points and half-spaces in $$ give rise to abstract tubes via Voronoi decompositions and their associated Delauney dual complexes. Every abstract tube is shown to give rise to an inclusion-exclusion identity for $$, and upper and lower bounds are obtained by truncating the identity at an even or an odd depth. This property is analogous to the truncation inequality property of the classical inclusion-exclusion identity, which may be viewed as a special case. The notion of an abstract subtube is introduced, and it is shown that if $$ is a subtube of $$ then the truncation inequality gotten from the depth m truncation for $$ is at least as sharp as the corresponding inequality from $$. As a consequence, the generalized inclusion-exclusion inequalities are always at least as sharp as their classical counterparts.

This paper presents two approaches for qualitative, quantitative and comparative concepts of skewness to be defined with respect to the spatial median for multivariate distributions. They extend the known quantile-based notions defined for real distributions. The main tool for such extensions consists of a family of central parts that provide suitable generalizations of the real interquantile intervals.

A family of trimmed regions is introduced for a probability distribution in Euclidean d-space. The regions decrease with their parameter $$, from the closed convex hull of support (at $$) to the expectation vector (at $$). The family determines the underlying distribution uniquely. For every $$ the region is affine equivariant and continuous with respect to weak convergence of distributions. The behavior under mixture and dilation is studied. A new concept of data depth is introduced and investigated. Finally, a trimming transform is constructed that injectively maps a given distribution to a distribution having a unique median.

Uhlig proposes two conjectures. The first concerns the Jacobian of the transformation $$ where B is the matrix $$ and m and X, Y belong to the class of positive semidefinite matrices of the order of $$ of rank $$. The second is concerned with the singular multivariate Beta distribution. This article seeks to prove the two conjectures. The latter result is then extended to the case of the singular multivariate F distribution, and the respective density functions are located for the nonzero positive eigenvalues of the singular Beta and F matrices.

In 1994 Tyler proposed a modification of the MAD in order to achieve the maximal possible finite sample breakdown point for the Stahel-Donoho estimator based upon the median and the modified MAD. A proof is not given, however, and as far as we know has not been published yet. Moreover there seems to be an error concerning the exact definition of the modified MAD which is needed to achieve this goal. For the sake of theoretical completeness and in order to correct this error we calculate the finite sample breakdown point of modifications of the MAD and point out which of these modifications lead to a Stahel-Donoho estimator with maximal finite sample breakdown point.

Fitting a circle to a set of data points on a plane is very common in engineering and science. An important practical problem is how to choose the locations of measurement on a circular feature. So far little attention has been paid to this design issue and only some simulation results are available. In this paper, for Berman's bivariate four-parameter model, $$-optimality is defined and shown to be equivalent to all $$-criteria with $$. Then $$-optimal exact designs on a circle or a circular arc are derived for any sample size and sampling range. As a by-product, $$-optimal approximate designs are also obtained. These optimal designs are used to evaluate the efficiency of the equidistant sampling method widely used in practice. These results also provide guidelines for users on sampling method and sample size selection.

We study the construction of orthogonal arrays with three levels and index 1 and the existence of orthogonal arrays with three levels and index 2. For strength greater than or equal to 2, we show that orthogonal arrays with three levels and index 1 are unique, and we establish the maximum number of factors for orthogonal arrays with three levels and index 2.

A central limit theorem is given for certain weighted partial sums of a covariance stationary process, assuming it is linear in martingale differences, but without any restriction on its spectrum. We apply the result to kernel nonparametric fixed-design regression, giving a single central limit theorem which indicates how error spectral behavior at only zero frequency influences the asymptotic distribution and covers long-range, short-range and negative dependence. We show how the regression estimates can be Studentized in the absence of previous knowledge of which form of dependence pertains, and show also that a simpler Studentization is possible when long-range dependence can be taken for granted.

This paper studies the method of empirical likelihood in models with weakly dependent processes. In such cases, if the likelihood function is formulated as if the data process were independent, obviously empirical likelihood fails. We propose to use empirical likelihood of blocks of observations to solve this problem in a nonparametric manner. This method of "blockwise empirical likelihood" preserves the dependence of data, and the resulting likelihood ratios can be used to construct asymptotically valid confidence intervals. We consider general estimating equations, for which an efficient estimator is derived by maximizing blockwise empirical likelihood. We also introduce "blocks-of-blocks empirical likelihood" to conduct inference for parameters of the infinite dimensional joint distribution of data; the smooth function model is used for such cases. We show that blockwise empirical likelihood of the smooth function model with weakly dependent processes is Bartlett correctable. A wide variety of problems, such as time series regressions and spectral densities, can be treated using our methodology.

We consider a bandit problem consisting of a sequence of n choices from an infinite number of Bernoulli arms, with $$. The objective is to minimize the long-run failure rate. The Bernoulli parameters are independent observations from a distribution F. We first assume F to be the uniform distribution on (0, 1) and consider various extensions. In the uniform case we show that the best lower bound for the expected failure proportion is between $$ and $$ and we exhibit classes of strategies that achieve the latter.

Suppose $$ are iid random variables with distribution $$, and $$ are are iid with distributed $$. The change point $$ is unknown. The problem is to raise an alarm as soon as possible after the distribution changes from $$ to $$ (detect the change), but to avoid false alarms.

Pollak found a version of the Shiryayev-Roberts procedure to be asymptotically optimal for the problem of minimizing the average run length to detection over all stopping times which satisfy a given constraint on the rate of false alarms. Here we find that this procedure is strictly optimal for a slight reformulation of the problem he considered.

Explicit formulas are developed for the calculation of the average run length (both before and after the change) for the optimal stopping time.

This paper considers the problem of sequential point estimation of the mean of a stable autoregressive process with unknown scale and autoregressive parameters. The construction of a sequential procedure makes use of special stopping rules and some modifications of least-squares estimates. The procedure enables estimating the mean with prescribed mean-square accuracy uniformly in nuisance parameters. The uniform asymptotic normality and the asymptotic minimaxity of the sequential estimate are established. The asymptotic formula for the mean sample size is obtained.

A new sequential sampling scheme is proposed in which, after an initial batch sample, sampling is continued in batches of data-dependent sizes (at most k such batches), and then one-at-a-time with a data-dependent stopping rule. This new scheme requires about the same sample size as the fully sequential Anscombe-Chow-Robbins (ACR) sampling scheme but substantially fewer sampling operations. The problem of constructing fixed-width confidence intervals for the mean of a normal population with unknown variance is used as an illustration.

The realized error of an estimate is determined not only by the efficiency of the estimator, but also by chance. For example, suppose that we have observed a bivariate normal vector whose expectation is known to be on a circle. Then, intuitively, the longer that vector happens to be, the more accurately its angle is likely to be estimated. Yet this chance, though its information is contained in the data, cannot be accounted for by the variance of the estimate. One way to capture it is by the direct estimation of the realized error. In this paper, we will demonstrate that the squared error of the maximum likelihood estimate, to the extent to which it can be estimated, can be most accurately estimated by the inverse of the observed Fisher information. In relation to this optimality, we will also study the properties of several other estimators, including the inverse of the expected Fisher information, the sandwich estimators, the jackknife and the bootstrap estimators. Unlike the observed Fisher information, these estimators are not optimal.

Although the cumulative distribution function may be differentiated to obtain the corresponding density function, whether or not a similar relationship exists between their asymptotic expansions remains a question. We provide a rigorous argument to prove that Lugannani and Rice's asymptotic expansion for the cumulative distribution function of the mean of a sample of i.i.d. observations may be differentiated to obtain Daniels's asymptotic expansion for the corresponding density function. We then apply this result to study the relationship between the truncated versions of the two series, which establishes the derivative of a truncated Lugannani and Rice series as an alternative asymptotic approximation for the density function. This alternative approximation in general does not need to be renormalized. Numerical examples demonstrating its accuracy are included.

An error bound in the normal approximation to the distribution of the double-indexed permutation statistics is derived. The derivation is based on Stein's method and on an extension of a combinatorial method of Bolthausen. The result can be applied to obtain the convergence rate of order $$ for some rank-related statistics, such as Kendall's tau, Spearman's rho and the Mann-Whitney-Wilcoxon statistic. Its applications to graph-related nonparametric statistics of multivariate observations are also mentioned.

A class of confidence sets with constant coverage probability for the mean of a p-variate normal distribution is proposed through a pseudo-empirical-Bayes construction. When the dimension is greater than 2, by combining analytical results with some exact numerical calculations the proposed sets are proved to have a uniformly smaller volume than the usual confidence region. Sufficient conditions for the connectedness of the proposed confidence sets are also derived. In addition, our confidence sets could be used to construct tests for point null hypotheses. The resultant tests have convex acceptance regions and hence are admissible by Birnbaum. Tabular results of the comparison between the proposed region and other confidence sets are also given.

Estimation of $$, location parameters of a distribution of a p-dimensional random vector $$ is considered under quadratic loss. Explicit estimators which are better than the best invariant one are given for a sign-invariantly distributed random vector $$. The results depend only on the second and the third moments of $$. The generalizations to concave loss functions and to n observations are also considered. Additionally, if the scale is unknown, we investigate the estimators of the location parameters when the observation contains a residual vector.