Motivated by a representation for the least squares estimator, we propose a class of weighted jackknife variance estimators for the least squares estimator by deleting any fixed number of observations at a time. They are unbiased for homoscedastic errors and a special case, the delete-one jackknife, is almost unbiased for heteroscedastic errors. The method is extended to cover nonlinear parameters, regression $M$-estimators, nonlinear regression and generalized linear models. Interval estimators can be constructed from the jackknife histogram. Three bootstrap methods are considered. Two are shown to give biased variance estimators and one does not have the bias-robustness property enjoyed by the weighted delete-one jackknife. A general method for resampling residuals is proposed. It gives variance estimators that are bias-robust. Several bias-reducing estimators are proposed. Some simulation results are reported.

A new family of estimators of a bivariate survival function based on censored vectors is obtained from a decomposition of the bivariate survival function. These estimators are uniformly consistent under bivariate censoring and are self-consistent under univariate censoring.

The renewal function is a basic tool used in many probabilistic models and sequential analysis. Based on a random sample of size $n$, a nonparametric estimator of the renewal function is introduced. Asymptotic properties of the estimator such as consistency and asymptotic normality are developed. A discussion of an application to warranty analysis is also provided.

It is shown that Page's stopping time is optimal for the detection of changes in distributions, in a well defined sense. This work is a generalization of an existing result where it was shown that Page's stopping time is optimal asymptotically.

Given that an item is of age $t$, the expected value of the random remaining life is called the mean residual life (MRL) at age $t$. We propose two new nonparametric classes of life distributions for modeling aging based on MRL. The first class of life distributions consists of those with "increasing initially, then decreasing mean residual life" (IDMRL). The IDMRL class models aging that is initially beneficial, then adverse. The second class, "decreasing, then increasing mean residual life" (DIMRL), models aging that is initially adverse, then beneficial. We propose two testing procedures for $H_0$: constant MRL (i.e., exponentiality) versus $H_1$: IDMRL, but not constant MRL (or $H'_1$: DIMRL, but not constant MRL). The first testing procedure assumes the turning point, $\tau$, from IMRL to DMRL is specified by the user or is known. The second procedure assumes knowledge of the proportion, $\rho$, of the population that "dies" at or before the turning point (knowledge of $\tau$ itself is not assumed).

The relative curvature measures of nonlinearity proposed by Bates and Watts (1980) are extended to an arbitrary subset of the parameters in a normal, nonlinear regression model. In particular, the subset curvatures proposed indicate the validity of linearization-based approximate confidence intervals for single parameters. The derivation produces the original Bates-Watts measures directly from the likelihood function. When the intrinsic curvature is negligible, the Bates-Watts parameter-effects curvature array contains all information necessary to construct curvature measures for parameter subsets.

Because the likelihood ratio statistic is invariant under reparameterization, it is possible to make a large-sample expansion of the statistic itself and of its expectation in terms of invariants. In particular, the Bartlett adjustment factor can be expressed in terms of invariant combinations of cumulants of the first two log-likelihood derivatives. Such expansions are given, first for a scalar parameter and then for vector parameters. Geometrical interpretation is given where possible and some special cases discussed.

We derive an explicit formula for the first term in an unconditional Edgeworth-type expansion of coverage probability for the nonparametric bootstrap technique applied to a very broad class of "Studentized" statistics. The class includes sample mean, $k$-sample mean, sample correlation coefficient, maximum likelihood estimators expressible as functions of vector means, etc. We suggest that the bootstrap is really an empiric one-term Edgeworth inversion, with the bootstrap simulations implicitly estimating the first term in an Edgeworth expansion. This view of the bootstrap is reinforced by our discussion of the iterated bootstrap, which inverts an Edgeworth expansion to arbitrary order by simulating simulations.

We make two points about the number, $B$ of bootstrap simulations needed to construct a percentile-$t$ confidence interval based on an $n$ sample from a continuous distribution: (i) The bootstrap's reduction of error of coverage probability, from $O(n^{-1/2})$ to $O(n^{-1})$, is available uniformly in $B$, provided nominal coverage probability is a multiple of $(B + 1)^{-1}$. In fact, this improvement is available even if the number of simulations is held fixed as $n$ increases. However, smaller values of $B$ can result in longer confidence intervals. (ii) In a large sample, the simulated statistic values behave like random observations from a continuous distribution, unless $B$ increases faster than any power of sample size. Only if $B$ increases exponentially quickly with $n$ is there a detectable effect due to discreteness of the bootstrap statistic.

An Edgeworth expansion with remainder $o(N^{-1})$ is established for a $U$-statistic with a kernel $h$ of degree 2. The assumptions involved appear to be very mild; in particular, the common distribution of the summands $h(X_i, X_j)$ is not assumed to be smooth.

Motivated by the construction of independent scale-free spacings for the exponential distribution, analogous quantities are defined for an arbitrary distribution, and their joint asymptotic behaviour studied under suitable tail conditions. These scale-free spacings provide a unified approach to the problem of consecutive discordancy testing when location and scale parameters are unknown. A normal example is discussed and the accuracy of the approximations assessed via some Monte Carlo studies.

Twenty years have elapsed since the Shapiro-Wilk statistic $W$ for testing the normality of a sample first appeared. In that time a number of statistics that are close relatives of $W$ have been found to have a common (known) asymptotic distribution. It was assumed, therefore, that $W$ must have that asymptotic distribution. We show this to be the case and examine the norming constants that are used with all the statistics. In addition the consistency of the $W$ test is established.

Let $X_1, \cdots, X_n$ be i.i.d. r.v.'s having common distribution belonging to a family $\mathscr{F} = \{F_\theta: \theta \in \Theta \subset R\}$ indexed by a parameter $\theta$. $\mathscr{F}$ is said to be reproducible if there exists a sequence $\{\alpha(n)\}$ such that $\mathscr{L}(\alpha(n)\sum^n_{i=1} X_i) \in \mathscr{F}$ for all $\theta \in \Theta$ and $n = 1, 2, \cdots$. This property is investigated in connection with linear exponential families of order 1 and its intimate relationship to such families having a power variance function is demonstrated. Moreover, the role of such families is examined, in a unified approach, with respect to properties relative to infinite divisibility, steepness, convolution, stability, self-decomposability, unimodality, and cumulants.

Latent variable models represent the joint distribution of observable variables in terms of a simple structure involving unobserved or latent variables, usually assuming the conditional independence of the observable variables given the latent variables. These models play an important role in educational measurement and psychometrics, in sociology and in population genetics, and are implicit in some work on systems reliability. We study a broad class of latent variable models, namely the monotone unidimensional models, in which the latent variable is a scalar, the observable variables are conditionally independent given the latent variable and the conditional distribution of the observables given the latent variable is stochastically increasing in the latent variable. All models in this class imply a new strong form of positive dependence among the observable variables, namely conditional (positive) association. This positive dependence condition may be used to test whether any model in this class can provide an adequate fit to observed data. Various applications, generalizations and a numerical example are discussed.

Anderson (1955) established the monotonicity of the integral of a symmetric, unimodal density over translates of a symmetric, convex set. A similar result is developed for integrals of elliptically contoured, unimodal densities over translates of an asymmetric, convex set in certain directions related to the set. This result is used to establish some monotonicity properties of the power functions of the likelihood ratio tests for determining whether or not a vector of normal means satisfies a specified ordering.

Suppose the $(k \times 1)$ vectors $\mathbf{x}$ and $\mathbf{y}$ are independent with $\mathbf{x} \sim N(\mu, \Sigma)$ and $\mathbf{y} \sim N(\eta, \Sigma), \Sigma$ positive definite. If for a positive scalar $c, \mathbf{\eta} = c\mu$, we find the posterior of $c$, using noninformative priors, given the data $\{\mathbf{x}_i\}^{N_1}_1, \{\mathbf{y}_j\}^{N_2}_1$. The $\mathbf{x}_i$ are $N_1$ independent observations on $\mathbf{x}$, and independent of the $\mathbf{y}_j$, which are $N_2$ independent observations on $\mathbf{y}$. The constant $c$ is called the magnitudinal effect, and the posterior of $c$ turns out to involve a truncated Student-$t$ kernel. We also discuss the situation in which we wish to examine the truth of the statement $\mathbf{\eta} = c\mathbf{\mu}$, and proceed as follows. We first note that the matrix $\Lambda = (\lambda_{ij})$, where $\lambda_{11} = N_1\mathbf{\mu'}\Sigma^{-1}\mathbf{\mu}, \lambda_{12} = \lambda_{21} = \sqrt{N_1N_2}\mathbf{\mu'}\Sigma^{-1} \mathbf{\eta}$, and $\lambda_{22} = N_2\mathbf{\eta'}\Sigma^{-1}\mathbf{\eta}$, has a zero eigenroot if and only if $\mathbf{\eta} = c\mathbf{\mu}$, for some $c$. Hence, we are motivated to find the (joint) posterior distribution of $\omega_1, \omega_2$, the roots of $\Lambda$, where $\omega_1 > \omega_2 \geq 0$. Then, by integration with respect to $\omega_1$ over the region $\omega_1 > \omega_2$, we may find the marginal of $\omega_2$, and use it to examine the statement $\mathbf{\eta} = c\mathbf{\mu}$. The posterior of $(\omega_1, \omega_2)$ involves the multivariate hypergeometric function $_1F_^{(2)}1$, which in practice creates computational difficulties. Accordingly, some numerical considerations are discussed for computing of the posterior of $\omega_2$, and an example using real data is given.

Despite their lack of optimality in unbalanced normally distributed models, the ANOVA estimates of components of variance are convenient and widely used. The hierarchal (nested) design is well suited to this estimation scheme. In this paper the nonnormal, unbalanced hierarchal design is considered and mild conditions for a sequence of such designs are specified so that the vector of normalized ANOVA estimates converges to a multivariate normal distribution. The nested structure permits an expression of the estimates in terms of a sum of independent quadratic forms in mean zero random variables plus smaller order remainder, and a theorem of Whittle (1960) establishes the Liapounov criterion. Distinguishing features of this paper are the limit theory of nonnormal unbalanced models and the allowance that some variances other than the error variance may be null.

Easily implemented asymptotic off-line procedures for the change-point Poisson process with $\lambda(t)$, the intensity at time $t$, equal to $\lambda_1$ if $t \leq \tau$ and to $\lambda_2$ if $t > \tau$, are developed. They may also be applied to a problem of estimation of the location of a discontinuity in density discussed by Chernoff and Rubin (1956). A test for change is noted, a test of the hypothesis that $\tau = \tau_0$ is proposed, and point and interval estimates of $\tau, \lambda_1$, and $\lambda_2$ are provided. The small-sample performance of the proposed procedures is studied using simulation, and an example is given.

Sufficient conditions are given in order to ensure the existence of a sequence of strongly consistent estimators of unknown parameters in a nonlinear regression model. The primary difference between this and earlier work is in the generality of the parameter space. Indeed, the parameter space is assumed to be any separable, completely regular topological space; in particular, this includes all separable metric spaces.

Many authors have considered the problem of estimating a distribution function when the observed data is subject to random truncation. A prominent role is played by the product limit estimator, which is the analogue of the Kaplan-Meier estimator of a distribution function under random censoring. Wang and Jewell (1985) and Woodroofe (1985) independently proved consistency results for this product limit estimator and showed weak convergence to a Gaussian process. Both papers left open the exact form of the covariance structure of the limiting process. Here we provide a precise description of the asymptotic behavior of the product limit estimator, including a simple explicit form of the asymptotic covariance structure, which also turns out to be the analogue of the covariance structure of the Kaplan-Meier estimator. Some applications are briefly discussed.

A class of random processes whose likelihood functions are of exponential type is considered. A necessary and sufficient condition for a stopping time to be efficient (in the Cramer-Rao sense) is proved. This result is obtained after proving a characterization theorem, which asserts that after a suitable random-time transformation such processes become processes with stationary independent increments, by applying the solution of the problem of efficient sequential estimation in the case of processes with stationary independent increments.

In order to improve the rate of decrease of the IMSE for nonparametric kernel density estimators with nonrandom bandwidth beyond $O(n^{-4/5})$ all current methods must relax the constraint that the density estimate be a bona fide function, that is, be nonnegative and integrate to one. In this paper we show how to achieve similar improvement without relaxing any of these constraints. The method can also be applied for orthogonal series, adaptive orthogonal series, spline, jackknife, and other density estimators, and assures an improvement of the IMSE for each sample size.

In this paper Ferguson's univariate normal results for detection of outliers with variance slippage are extended to the multivariate elliptically symmetric case with dispersion slippage. The locally optimum test we derive possesses all three robustness properties, optimality, null and nonnull, and is based on Mardia's multivariate kurtosis statistic.

The problem of finding classes of estimators which dominate the usual estimator $X$ of the mean vector $\mu$ of a $p$-variate normal distribution $(p \geq 3)$ under general quadratic loss is analytically difficult in cases where the covariance matrix is unknown. Estimators of $\mu$ in this case depend upon $X$ and an independent Wishart matrix $W$. In the present paper, integration-by-parts methods for both the multivariate normal and Wishart distributions are combined to yield unbiased estimates of risk difference (versus $X$) for certain classes of estimators, defined indirectly through a "seed" function $h(X, W)$. An application of this technique produces a new class of minimax estimators of $\mu$.

Suppose that $X$ has a binomial distribution $B(n, p)$, with known $p \in (0, 1)$ and unknown $n \in \{1, 2, \cdots\}$. A natural estimator for $n$ is given by $T(0) = 1, T(x) = x/p, x = 1, 2, \cdots$. This estimator is shown to be inadmissible under quadratic loss. It is shown that modifying $T(0)$ to $T(0) = -(1 - p)/(p \ln p)$ results in an admissible estimator. For $p \geq \frac{1}{2}$ it is further shown that this is the only admissible modification of $T(0)$. A partial result is also obtained for $p < \frac{1}{2}$.