This paper is about random coefficient trigonometric regression models and their use in gait analysis. Here gait analysis means free-speed walking on a level surface. Our study focuses on bootstrap-based prediction regions for the angular rotation curves of test children, when the relevant training data are gathered from normal children of comparable ages. Considerations that led to our choice of model and use of the bootstrap are given. Prediction regions and empirical bootstrap distributions are displayed, as is their application to several test cases. Also included is a study of the almost sure asymptotic behavior of the theoretical bootstrap probability of the prediction regions.

The theory of semialgebraic sets is used to generate new Vapnik-Chervonenkis (VC) classes of positivity sets. It is also shown that certain analytic families of positivity sets are VC.

We develop the complete bootstrapped parallel to the asymptotic theory of weighted empirical and quantile processes. Utilizing this parallel theory, we present a general body of techniques to establish the asymptotic validity of the bootstrap method of constructing confidence bands for statistical functions. These techniques are demonstrated to be applicable to the construction of asymptotic bootstrap confidence bands for a variety of concrete functions.

The asymptotic performance of the bootstrap in linear regression models is studied. Edgeworth expansions show that asymptotically, the bootstrap is always at least as good as, and in some cases better than, the classical normal approximation. The performances of both the bootstrap and the normal approximation depend on the rate of increase in the elements of the design matrix.

Asymptotic validity of the bootstrap is established for the least squares estimate of the parameter of an explosive first-order autoregressive process. It is noted that nonnormal limit distributions are obtained for both the traditional and the bootstrap estimates. The theoretical bootstrap validity results are supported by appropriate simulation.

This paper discusses several lower bound results for the asymptotic performance of estimators of smooth functionals in i.i.d. models. The key idea is to look at a set of local limiting distributions of an estimator sequence, rather than to impose regularity conditions, or to consider limits of maximum risk. Special attention is paid to situations where the tangent cone is not a linear space. As an example, the local asymptotic minimax risk in mixture models is computed.

It is assumed that we observe one realization of an $r$-dimensional counting process with intensities that are products of a predictable weight process, a common function of time and parameters $\beta_i, i = 1, \cdots, r$, which distinguish the components. Provided the realization observed brings increasing information on $\beta$ as the observed time grows, strong consistency of a partial ML estimator is proved. For such realizations it is also proved that the estimate, after applying a random normalization, is asymptotically standard normal.

For the nonparametric estimation of a survival function when censoring is present, the Kaplan-Meier estimator is often used. The admissibility of this estimator and other related maximum likelihood estimators is demonstrated. This is done by reducing the problem to one involving just the multinomial distribution and then using the stepwise Bayes technique to prove admissibility.

In this paper we shall investigate some nonasymptotic properties of the Grenander estimator of a decreasing density $f$. This estimator is defined as the slope of the smallest concave majorant of the empirical c.d.f. It will be proved that its risk, measured with $\mathbb{L}^1$-loss, is bounded by some functional depending on $f$ and the number $n$ of observations. For classes of uniformly bounded densities with a common compact support, upper bounds for the functional are shown to agree with older results about the minimax risk over these classes. The asymptotic behavior of the functional as $n$ goes to infinity is also in accordance with the known asymptotic performances of the Grenander estimator.

Bayes solutions for i.i.d. sampling from a decreasing density on the half line and for i.i.d. sampling from a symmetric unimodal density with an unknown mode are given. Posterior quantities are obtained as finite sums; Monte Carlo methods based on sampling finite Markov chains are developed for their evaluation.

A data dependent method of estimating a regression function is proposed here. The model is allowed to be heteroscedastic. Applications to piecewise polynomial, spline, orthogonal series, kernel and nearest neighbor methods are discussed.

Let $X_1, \cdots, X_n$ be random vectors that take values in a compact set in $R^d, d = 1, 2$. Let $Y_1, \cdots, Y_n$ be random variables (the responses) which conditionally on $X_1 = x_1, \cdots, X_n = x_n$ are independent with densities $f(y \mid x_i, \theta(x_i)), i = 1, \cdots, n$. Assuming that $\theta$ lies in a sup-norm compact space $\Theta$ of real-valued functions, an $L_1$-consistent estimator (of $\theta$) is constructed via empirical measures. The rate of convergence of the estimator to the true parameter $\theta$ depends on Kolmogorov's entropy of $\Theta$.

This paper considers the problem of minimizing the maximum asymptotic bias of regression estimates over $\varepsilon$-contamination neighborhoods for the joint distribution of the response and carriers. Two classes of estimates are treated: (i) $M$-estimates with bounded function $\rho$ applied to the scaled residuals, using a very general class of scale estimates, and (ii) bounded influence function type generalized $M$-estimates. Estimates in the first class are obtained as the solution of a minimization problem, while estimates in the second class are specified by an estimating equation. The first class of $M$-estimates is sufficiently general to include both Huber Proposal 2 simultaneous estimates of regression coefficients and residuals scale, and Rousseeuw-Yohai $S$-estimates of regression. It is shown than an $S$-estimate based on a jump-function type $\rho$ solves the min-max bias problem for the class of $M$-estimates with very general scale. This estimate is obtained by the minimization of the $\alpha$-quantile of the squared residuals, where $\alpha = \alpha(\varepsilon)$ depends on the fraction of contamination $\varepsilon$. When $\varepsilon \rightarrow 0.5, \alpha(\varepsilon) \rightarrow 0.5$ and the min-max estimator approaches the least median of squared residuals estimator introduced by Rousseeuw. For the bounded influence class of $GM$-estimates, it is shown the "sign" type nonlinearity yields the min-max estimate. This estimate coincides with the minimum gross-error sensitivity $GM$-estimate. For $p = 1$, the optimal $GM$-estimate is optimal among the class of all equivariant regression estimates. The min-max $S$-estimator has a breakdown point which is independent of the number of carriers $p$ and tends to 0.5 as $\varepsilon$ increases to 0.5, but has a slow rate of convergence. The min-max $GM$-estimate has the usual rate of convergence, but a breakdown point which decreases to zero with increasing $p$. Finally, we compare the min-max biases for both types of estimates, for the case where the nominal model is multivariate normal.

Simplified conditions are given for consistency and asymptotic normality of $M$-estimates derived by maximization of averages of independent identically distributed random concave functions. Applications are made to maximum likelihood estimation.

We discuss the relation between $S$-estimators and $M$-estimators of multivariate location and covariance. As in the case of the estimation of a multiple regression parameter, $S$-estimators are shown to satisfy first-order conditions of $M$-estimators. We show that the influence function IF $(\mathbf{x; S}, F)$ of $S$-functionals exists and is the same as that of corresponding $M$-functionals. Also, we show that $S$-estimators have a limiting normal distribution which is similar to the limiting normal distribution which is similar to the limiting normal distribution of $M$-estimators. Finally, we compare asymptotic variances and breakdown point of both types of estimators.

Let $D_{p,m}$ be the determinant of the sample covariance matrix for $m + p + 1$ observations from a $p$-variate normal population having identity covariance matrix. We give bounds for the distribution of $D_{p,m}$ in terms of various chi-squared distribution functions. Let $F(\cdot \mid \nu)$ denote the chi-squared distribution function on $\nu$ degrees of freedom. We bound $P\{p(D_{p,m})^{1/p} > t\}$ above by $1 - F(t \mid p(m + 1) + \frac{1}{2}(p - 1)(p - 2))$ and below by $1 - F(t \mid p(m + 1))$. We give two more bounds involving chi-squared distributions. The proofs use a stochastic analog to the Gauss multiplication theorem.

The structure of the strict collapsibility model in a three-dimensional contingency table is investigated with the goal of developing more powerful tests. After showing that strict collapsibility can be written as a finite union of distinct models, an intersection union test is proposed which is at least as powerful asymptotically as the likelihood ratio test. Intersection union tests for strong collapsibility and marginal independence are also suggested.

Bounds are obtained on the limiting size of the level-$\alpha \chi^2$-test of independence in a contingency table, as the sample size increases. The situations considered include (a) sampling with one or both sets of marginal totals random, (b) performing the test with or without the continuity correction and (c) with or without conditioning on the event $\mathscr{E}_k$ that the minimum estimated expected cell count is greater than a given $k \geq 0$. Bounds for both the unconditional and conditional (on $\mathscr{E}_k$) size are derived. It is shown, for example, that the limiting conditional size of the test is unity for all $\alpha$ if the continuity correction is used with $k = 0$ and sampling is done with both margins random. The same conclusion holds if sampling is done with one set of margins fixed and the dimensions of the table are not too small.

For decomposable covariance selection models, stochastic inequalities which relate the null distribution of the log-likelihood ratio statistic to its asymptotic $\chi^2$ distribution are obtained. The implications are twofold: First, the null distribution of the log-likelihood ratio statistic is seen to be stochastically larger than its asymptotic $\chi^2$ distribution. Extremely large samples apart, for the $\chi^2$ approximation to be valid, a deflation of the log-likelihood ratio statistic is then necessary. Second, a simple adjustment to the log-likelihood ratio statistic, similar in spirit to the Bartlett adjustment, yields a conservative test.

If the distribution of $X$ belongs to a natural exponential family on the positive real line, this note studies the expectation of the reciprocal of $X$ as a function of the expectation $m$ of $X$ and characterizes the cases where this function is an affine function of $m^{-1}$ as gamma, inverse-Gaussian, Ressel or Abel families.

A rigorous proof is presented that global attainment of the Cramer-Rao bound is possible only if the underlying family of distributions is exponential. The proof is placed in the context of $\mathbb{L}_r(P_\vartheta)$-differentiability, requiring strong differentiability in $\mathbb{L}_r(P_\vartheta)$ of the $r$th root of the likelihood ratio relative to $P_\vartheta$.

Asymptotic normality of the maximum likelihood estimator for the parameters of a long range dependent Gaussian process is proved. Furthermore, the limit of the Fisher information matrix is derived for such processes which implies efficiency of the estimator and of an approximate maximum likelihood estimator studied by Fox and Taqqu. The results are derived by using asymptotic properties of Toeplitz matrices and an equicontinuity property of quadratic forms.

Let $(X_j)^\infty_{j = 1}$ be a stationary, mean-zero Gaussian process with covariances $r(k) = EX_{k + 1} X_1$ satisfying $r(0) = 1$ and $r(k) = k^{-D}L(k)$ where $D$ is small and $L$ is slowly varying at infinity. Consider the two-parameter empirical process for $G(X_j),$ $\bigg\{F_N(x, t) = \frac{1}{N} \sum^{\lbrack Nt \rbrack}_{j = 1} \lbrack 1\{G(X_j) \leq x\} - P(G(X_1) \leq x) \rbrack; // -\infty < x < + \infty, 0 \leq t \leq 1\bigg\},$ where $G$ is any measurable function. Noncentral limit theorems are obtained for $F_N(x, t)$ and they are used to derive the asymptotic behavior of some suitably normalized von Mises statistics and $U$-statistics based on the $G(X_j)$'s. The limiting processes are structurally different from those encountered in the i.i.d. case.

This paper proves the weak convergence of the residual empirical process in an explosive autoregression model to the Brownian bridge. As an application the Kolmogorov-Smirnov goodness-of-fit test for testing that the errors have a specified distribution is shown to be asymptotically distribution-free.

This paper consists of two parts. An easy proof is given for the weak consistency of Pickands' estimate for the main parameter of an extreme-value distribution. Moreover, further natural conditions are given for strong consistency and for asymptotic normality of the estimate. Next a large quantile of a distribution is estimated by a combination of extreme or intermediate order statistics. This leads to an asymptotic confidence interval.

We extend Hill's well-known estimator for the index of a distribution function with regularly varying tail to an estimate for the index of an extreme-value distribution. Consistency and asymptotic normality are proved. The estimator is used for high quantile and endpoint estimation.

A probability assessor or forecaster is a person who assigns subjective probabilities to events which will eventually occur or not occur. There are two purposes for which one might wish to compare two forecasters. The first is to see who has given better forecasts in the past. The second is to decide who will give better forecasts in the future. A method of comparison suitable for the first purpose may not be suitable for the second and vice versa. A criterion called calibration has been suggested for comparing the forecasts of different forecasters. Calibration, in a frequency sense, is a function of long run (future) properties of forecasts and hence is not suitable for making comparisons in the present. A method for comparing forecasters based on past performance is the use of scoring rules. In this paper a general method for comparing forecasters after a finite number of trials is introduced. The general method is proven to include calculating all proper scoring rules as special cases. It also includes comparison of forecasters in all simple two-decision problems as special cases. The relationship between the general method and calibration is also explored. The general method is also translated into a method for deciding who will give better forecasts in the future. An example is given using weather forecasts.

We develop a method for grouping the $2^k - 1$ factorial effects in a 2-level factorial design into mutually exclusive sets of the form $(s, t, st)$, where $st$ is the generalized interaction of effects $s$ and $t$. As an application, we construct orthogonal arrays $OA(2^k, 2^m4^n, 2)$ of size $2^k, m$ constraints with 2 levels and $n$ constraints with 4 levels in the construction cannot be further improved. In this sense our grouping scheme is optimal. We discuss the advantages of the present approach over other construction methods.

We present a simple but universal technique for the construction of $\pi PS$ sampling designs. A tool that is used in the construction consists of playing a game in which objects are removed from $N$ boxes, $n$ at a time, and at most one from each box at a time. Necessary and sufficient conditions on $N, n$ and the contents of the boxes are established such that all boxes can be emptied by this process. It is shown that every $\pi PS$ design can be derived from such a game. Sampling designs with additional properties are obtained through additional restrictions on emptying the boxes. Various rigorous methods are presented, complemented by numerous suggestions. The emphasis is on controlling sample selection probabilities and inequalities for the first- and second-order inclusion probabilities. The method is very adaptive to computer use.

In this paper we consider the problem of $A$-optimal weighing designs for $n$ objects in $N$ weighings on a chemical balance when $N \equiv 3(\operatorname{mod} 4)$. Let $D(N, n)$ denote the class of $N \times n$ design matrices $X_d$ whose elements are $+1$ and $-1$. It is shown that if $X_d$ is such that $X'_dX_d$ is a block matrix having a specified block structure, then $X_d$ is $A$-optimal in $D(N, n)$. It is found that in some cases the $A$-optimal design in $D(N, n)$ is not unique. A larger class of chemical balance weighing designs is $D^0(N, n)$, where $X_d$ may have some elements equal to zero. It is observed that the designs which are $A$-optimal in $D(N, n)$ are not necessarily $A$-optimal in $D^0(N, n)$.