Statisticians know that one-parameter exponential families have very nice properties for estimation, testing, and other inference problems. Fundamentally this is because they can be considered to be "straight lines" through the space of all possible probability distributions on the sample space. We consider arbitrary one-parameter families $\mathscr{F}$ and try to quantify how nearly "exponential" they are. A quantity called "the statistical curvature of $\mathscr{F}$" is introduced. Statistical curvature is identically zero for exponential families, positive for nonexponential families. Our purpose is to show that families with small curvature enjoy the good properties of exponential families. Large curvature indicates a breakdown of these properties. Statistical curvature turns out to be closely related to Fisher and Rao's theory of second order efficiency.

Several invariant tests for uniformity of a distribution on the circle, the sphere and the hemisphere have been proposed by Rayleigh, Watson, Bingham, Ajne, Beran and others. In this paper a class of invariant tests for uniformity on compact Riemannian manifolds containing many of the known ones is presented and studied (the asymptotic theory as well as some local optimality properties for this class of tests are given). The examples include two new tests, one for the sphere and the other for the hemisphere. Let $X$ be a compact Riemannian manifold, $\mu$ the normalized volume element (the uniform distribution of $X$) and $\nu_n$ the empirical distribution corresponding to a sequence of i.i.d. $X$-valued random variables. The statistics in which these tests are based are just convergent weighted sums of the squares of the Fourier coefficients of $\nu_n(\omega) - \mu$ with respect to any orthonormal basis of $L_2(X, \mu)$ consisting of eigenfunctions of the Laplacian. An additional condition is imposed on the weights, namely that weights corresponding to coefficients of eigenfunctions in the same eigenspace of the Laplacian be equal (this condition is essential for the invariance of the tests). These statistics are related to Sobolev norms and so, the tests are called Sobolev tests. In connection with Sobolev statistics, it is interesting to note that the Sobolev norms of index $-s, s > (\dim X)/2$, metrize the weak-star topology of $\mathscr{P}(X)$, the space of Borel probability measures on $X$. A theorem about weak convergence of empirical distributions on compact manifolds, useful in proving some of the asymptotic results for Sobolev statistics, is also included. One of the sections (Section 2) is almost entirely devoted to give a short review of the facts needed in the paper about Riemannian manifolds, the Laplacian and Sobolev spaces.

This paper presents asymptotic formulae, lower and upper bounds for the expected sample size of certain sequential tests of the parameter of an exponential family of distributions. The tests involved are tests of power one based on mixture-type stopping rules and tests for the detecting of change in the underlying distribution. Analysis for incorrect assumptions of the underlying distribution yields asymptotic formulae for such cases, showing robustness of the original formulae. Monte Carlo results indicate the validity of asymptotic formulae for sample sizes one would expect in practical applications.

The autoregressive process with moving average residuals is a stationary process $\{y_t\}$ satisfying $\sum^p_{s = 0} \beta_sy_{t - s} = \sum^q_{j = 0} \alpha_j\nu_{t - j}$, where the sequence $\{\nu_t\}$ consists of independently identically distributed (unobservable) random variables. The distribution of $y_1,\cdots, y_T$ can be approximated by the distribution of the $T$-component vector $\mathbf{y}$ satisfying $\sum^p_{s = 0} \beta_s\mathbf{K}_s\mathbf{y} = \sum^q_{j = 0} \alpha_j\mathbf{J}_j\mathbf{v}$, where $\mathbf{v}$ has covariance matrix $\sigma^2\mathbf{I}, \mathbf{K}_s = \mathbf{J}_s = \mathbf{L}^s$, and $\mathbf{L}$ is the $T \times T$ matrix with 1's immediately below the main diagonal and 0's elsewhere. Maximum likelihood estimates are obtained when $\mathbf{v}$ has a normal distribution. The method of scoring is used to find estimates defined by linear equations which are consistent, asymptotically normal, and asymptotically efficient (as $T\rightarrow \infty$). Several special cases are treated. It is shown how to calculate the estimates.

Let $\{X_1, \cdots, X_N\}$ be $N$ observations on an $m$-state Markov chain with stationary transition probability matrix $\mathbf{P} = (p_{ij}), p_{ij} > 0, i,j = 1,\cdots, m$, where $N$ is a random variable. For any parametric function of $\mathbf{P}$, the information inequality gives a lower bound on the variance of an unbiased estimator; attaining the lower bound depends on whether the sampling plan or stopping rule $S$, the estimator $f = f(X_1,\cdots, X_N)$, and the function $E(f) = g(\mathbf{P})$ are "efficient". All "efficient triples" $(S, f, g)$ are characterized for the Markov chain in which $p_{ij}$ and $p_{i'j'} (i' \neq i)$ are not related functionally. It is also shown that efficient triples do not exist if $m > 2$ and $g$ is a function of two or more rows of $\mathbf{P}$. For the case $m = 2$, efficient triples in which $g$'s are functions of both rows are characterized.

Assume $X = (X_1,\cdots, X_p)^t$ has a $p$-variate density, with respect to Lebesgue measure, of the form $f((x - \theta)^t\not\sum^{-1}(x - \theta))$. Here $\not\sum$ is a known positive definite $p \times p$ matrix and $p \geqq 3$. Assume either (i) $f$ is completely monotonic, or (ii) there exist $\alpha > 0$ and $K > 0$ for which $h(s) = f(s)e^{\alpha s}$ is nondecreasing and nonzero if $s > K$. Then for estimating $\theta$ under a known quadratic loss, classes of minimax estimators are found.

Tne maximum likelihood estimate of a probability density function based on a random sample does not exist in the nonparametric case. For this reason and others based on heuristic Bayesian considerations Good and Gaskins suggested adding a penalty term to the likelihood. They proposed two penalty terms; however they did not establish existence or uniqueness of their maximum penalized likelihood estimates. Good and Gaskins also suggested an alternate approach for calculating the maximum penalized likelihood estimate which avoids the nonnegativity constraint on the estimate. In the present work the existence and uniqueness of both of Good's and Gaskins' maximum penalized likelihood estimates are rigorously demonstrated. Moreover, it is shown that one of these estimates is a positive exponential spline with knots only at the sample points and that in this case the alternate approach leads to the correct estimate; however in the other case the alternate approach leads to the wrong estimate. Finally, it is shown that a well-known class of reproducing kernel Hilbert spaces leads very naturally to maximum penalized likelihood estimates which are polynomial splines with knots at the sample points.

In the statistical literature Le Cam's third lemma (cf. Hajek and Sidak (1967), page 208) is extensively used in order to get asymptotic normality of a statistic $S_n$ under contiguous alternatives from asymptotic normality of $S_n$ under the nullhypothesis. Since Le Cam's lemma utilizes the joint asymptotic normality of $S_n$ and $\log$-likelihood-ratio $\log L_n$, which is a sufficient but in general not a necessary condition for contiguity, it is not possible to get asymptotic normality of $S_n$ for all contiguous alternatives from this lemma. On the other hand one is interested in the limiting distribution of $S_n$ under all contiguous alternatives in order to get general power and efficiency results for the respective tests. In this paper we utilize a truncation method in order to prove asymptotic normality under all contiguous alternatives from asymptotic normality under the nullhypothesis for sums of independent random variables which are interesting in rank test theory, since they often are asymptotically equivalent to certain rank statistics under the nullhypothesis, and thus under contiguous alternatives, too.

A transfer function relating a time series $y_t$ to present and past values of a series $x_t$ need not possess an inverse. When $(x_t, y_t)$ is a covariance stationary process, it is shown that noninvertibility in this transfer function has the effect of reducing the error variance of the minimum mean-square-error predictor of $y_t$ one or more steps ahead. In deriving these results a "dual" series to $x_t$ is constructed, which has univariate stochastic structure identical to that of $x_t$ itself, and an associated dual transfer function relating it to $y_t$ which is invertible.

This note shows that, under appropriate conditions, if a function $A(\theta; t)$ of an unknown parameter $\theta$ and a real variable $t$ has an infinite series expansion and if there is a function $B(S; t)$ of the sufficient statistic $S$ which is an unbiased estimator of $A$ for every $t$ and which also has an infinite series expansion, then the coefficients of the power of $t$ in the expansion of $B$ are the proper estimators for the coefficients of the corresponding powers in the expansion of $A$. This result is applied to estimate two functions of the normal parameters, $\mu$ and $\sigma^2$, which arise in the derivation of expressions for the removal of transformation bias.

It is assumed that a random sample of size $n$ is drawn from a bivariate distribution $f(t, u)$ which possesses a monotone likelihood ratio (MLR). However, before the sample values are observed, the pairs are `broken' into components $t$ and $u$. Therefore, the original sample pairings are unknown, and it is desired to optimally re-pair $t$- and $u$-values in order to reconstruct the original bivariate sample. It is observed that for the maximum likelihood pairing (MLP) to be the `natural' pairing for all $t$- and $u$-values, it is necessary that $f$ has MLR. It is shown that if it is desired to maximize the expected number of correct matches, then the class of procedures $\Phi_{1, n}$, which result in pairing the largest $t$ with the largest $u$ and the smallest $t$ with the smallest $u$, is a complete class. A sufficient condition under which the MLP maximizes the expected number of correct matches is also obtained.