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Let $X_{nr}$ be the $r$th largest of a random sample of size $n$ from a distribution $F (x) = 1 - \sum_{i = 0}^\infty c_i x^{-\alpha - i \beta}$ for $\alpha > 0$ and $\beta > 0$. An inversion theorem is proved and used to derive an expansion for the quantile $F^{-1} (u)$ and powers of it. From this an expansion in powers of $(n^{-1}, n^{-\beta/\alpha})$ is given for the multivariate moments of the extremes $\{X_{n, n - s_i}, 1 \leq i \leq k \}/n^{1/\alpha}$ for fixed ${\bf s} = (s_1, ..., s_k)$, where $k \geq 1$. Examples include the Cauchy, Student $t$, $F$, second extreme distributions and stable laws of index $\alpha < 1$.

We give analytic methods for nonparametric bias reduction that remove the need for computationally intensive methods like the bootstrap and the jackknife. We call an estimate {\it $p$th order} if its bias has magnitude $n_0^{-p}$ as $n_0 \to \infty$, where $n_0$ is the sample size (or the minimum sample size if the estimate is a function of more than one sample). Most estimates are only first order and require O(N) calculations, where $N$ is the total sample size. The usual bootstrap and jackknife estimates are second order but they are computationally intensive, requiring $O(N^2)$ calculations for one sample. By contrast Jaeckel's infinitesimal jackknife is an analytic second order one sample estimate requiring only O(N) calculations. When $p$th order bootstrap and jackknife estimates are available, they require $O(N^p)$ calculations, and so become even more computationally intensive if one chooses $p>2$. For general $p$ we provide analytic $p$th order nonparametric estimates that require only O(N) calculations. Our estimates are given in terms of the von Mises derivatives of the functional being estimated, evaluated at the empirical distribution. For products of moments an unbiased estimate exists: our form for this "polykay" is much simpler than the usual form in terms of power sums.

We give the distribution of $M_n$, the maximum of a sequence of $n$ observations from a moving average of order 1. Solutions are first given in terms of repeated integrals and then for the case where the underlying independent random variables have an absolutely continuous density. When the correlation is positive, $$ P(M_n %\max^n_{i=1} X_i \leq x) =\ \sum_{j=1}^\infty \beta_{jx} \nu_{jx}^{n} \approx B_{x} \nu_{1x}^{n} $$ where %$\{X_i\}$ is a moving average of order 1 with positive correlation, and $\{\nu_{jx}\}$ are the eigenvalues (singular values) of a Fredholm kernel and $\nu_{1x}$ is the eigenvalue of maximum magnitude. A similar result is given when the correlation is negative. The result is analogous to large deviations expansions for estimates, since the maximum need not be standardized to have a limit. % there are more terms, and $$P(M_n

We give the distribution of $M_n$, the maximum of a sequence of $n$ observations from a moving average of order 1. Solutions are first given in terms of repeated integrals and then for the case where the underlying independent random variables are discrete. When the correlation is positive, $$ P(M_n \max^n_{i=1} X_i \leq x) = \sum_{j=1}^\infty \beta_{jx} \nu_{jx}^{n} \approx B_{x} r{1x}^{n} $$ where $\{\nu_{jx}\}$ are the eigenvalues of a certain matrix, $r_{1x}$ is the maximum magnitude of the eigenvalues, and $I$ depends on the number of possible values of the underlying random variables. The eigenvalues do not depend on $x$ only on its range., Comment: 13 pages. This version gives full solutions to the examples

We give the Jordan form and the Singular Value Decomposition for an integral operator ${\cal N}$ with a non-symmetric kernel $N(y,z)$. This is used to give solutions of Fredholm equations for non-symmetric kernels, and to determine the behaviour of ${\cal N}^n$ and $({\cal N}{\cal N^*})^n$ for large $n$., Comment: 12 A4 pages