The most famous priority dispute in the history of statistics is that between Gauss and Legendre, over the discovery of the method of least squares. New evidence, both documentary and statistical, is discussed, and an attempt is made to evaluate Gauss's claim. It is argued (though not conclusively) that Gauss probably possessed the method well before Legendre, but that he was unsuccessful in communicating it to his contemporaries. Data on the French meridian arc are presented that could, conceivably, permit a definitive verification of the claim.

Let $M_0$ be a normal linear regression model and let $M_1,\cdots, M_K$ be distinct proper linear submodels of $M_0$. Let $\hat k \in \{0,\cdots, K\}$ be a model selection rule based on observed data from the true model. Given $\hat k$, let the unknown parameters of the selected model $M_{\hat k}$ be fitted by the maximum likelihood method. A loss function is introduced which depends additively on two parts: (i) a measure of the difference between the fitted model $M_{\hat k}$ and the true model; and (ii) a measure $C_{\hat k}$ of the "complexity" of the selected model. A natural model selection rule $\bar{k}$, which minimizes an empirical version of this loss, is shown to be admissible and very nearly Bayes.

Using the results of Barrow, et al., and Chow on the optimal placement of knots in the approximation of functions by piecewise polynomials, we show the uniqueness or "eventual uniqueness" of optimal designs for certain time series models considered by Sacks and Ylvisaker, and Wahba. In addition, the limiting behavior (as the sample size increases) of the variance of the BLUE of the regression coefficient is characterized in terms of the density defining the design, and the density for the asymptotically optimal design is given.

In this paper design techniques for continuous parameter time series regression analysis are employed to develop a general approach to optimal spacing selection for the linear estimation of location and scale parameters by sample quantiles from uncensored or censored samples. The spacings derived from this approach are asymptotically optimal in the sense that they result in near optimal asymptotic relative efficiencies for large values of $k$, the number of spacing elements. A comparison with the optimum efficiencies for several distribution types indicates that the asymptotically optimum spacings perform well for $k \geq 7$. The regression framework is also utilized to develop sufficient conditions for optimal spacing unicity and to obtain asymptotically optimal spacings for quantile estimation.

For a linear regression model, the necessary and sufficient condition for the asymptotic consistency of the least squares estimator is known. An analogous condition for the nonlinear model is considered in this paper. The condition is proved to be necessary for the existence of any weakly consistent estimator, including the least squares estimator. It is also sufficient for the strong consistency of the nonlinear least squares estimator if the parameter space is finite. For an arbitrary compact parameter space, its sufficiency for strong consistency is proved under additional conditions in a sense weaker than previously assumed. The proof involves a novel use of the strong law of large numbers in $C(S)$. Asymptotic normality is also established.

Assume that every probability measure $P$ in $\mathscr{P}$ of a statistical structure $(X, \mathscr{A}, \mathscr{P})$ has a density $p(x, P)$ w.r.t. a (not necessarily $\sigma$-finite) measure $m$. Let $\mathscr{B}$ be any subfield and suppose that the densities are factored as $p(x, P) = g(x, P)h(x)$ where $g$ is $\mathscr{B}$-measurable. Then $\mathscr{B}$ is pairwise sufficient and contains supports of $P$'s. Assume further that $m$ is locally localizable and $\mathscr{B}$ is pairwise sufficient and contains supports of $P$'s. Then the densities are factored as above. Two partial orders are introduced for pairwise sufficient subfields. Assuming that every $P$ has a support, a subfield is constructed which is the smallest with supports under the first partial order, and is the smallest under the second. This is used to give a simple proof of existence of the minimal sufficient subfield for the coherent case. In the (uncountable) discrete case it is proved that under the first partial order there are infinitely many minimal pairwise sufficient subfields and hence there is none that is smallest.

Let $Y_t$ satisfy the stochastic difference equation $Y_t = \sum^q_{i=1} \psi_{ti} \alpha_i + \sum^p_{j=1} \gamma_j Y_{t-j} + e_t,$ where the $\{\psi_{ti}\}$ are fixed sequences and (or) weakly stationary time series and the $e_t$ are independent random variables, each with mean zero and variance $\sigma^2$. The form of the limiting distributions of the least squares estimators of $\alpha_i$ and $\gamma_j$ depend upon the absolute value of the largest root of the characteristic equation, $m^p - \sum^p_{j=1} \gamma_jm^{p-j} = 0$. Limiting distributions of the least squares estimators are established for the situations where the largest root is less than one, equal to one, and greater than one in absolute value. In all three situations the regression $t$-type statistic is of order one in probability under mild assumptions. Conditions are given under which the limiting distribution of the $t$-type statistic is standard normal.

We consider sequential observation of independent random variables $X_1,\cdots, X_N$ whose distribution changes from $F$ to $G$ after the first $\lbrack N\theta \rbrack$ variables. The object is to detect the unknown change-point quickly without too many false alarms. A nonparametric control chart based on partial weighted sums of sequential ranks is proposed. It is shown that if the change from $F$ to $G$ is small, then as $N \rightarrow \infty$, the appropriately scaled and linearly interpolated graph of partial rank sums converges to a Brownian motion on which a drift sets in at time $\theta$. Using this, the asymptotic performance of the one-sided control chart is compared with one based on partial sums of the $X$'s. Location change, scale change and contamination are considered. It is found that for distributions with heavy tails, the control chart based on ranks stops more frequently and faster than the one based on the $X$'s. Performance of the two procedures are also tested on simulated data and the outcomes are compatible with the theoretical results.

Let $S$ be a set, $R$ the real line, and $M$ a real function on $R \times S$. Assume there exists a real function, $f$, on $S$ such that $(x - f(s))M(x, s) \geq 0$ for all $x$ and $s$. Initially neither $M$ nor $f$ are known. The goal is to estimate $f$. At time $n, s_n$ (a value in $S$) is observed, $x_n$ (a real number) is chosen, and an unbiased estimator of $M(x_n, s_n)$ is observed. This problem has applications, for example, to process control. In a previous paper the author proposed estimation of $f$ by a generalization of the Robbins-Monro procedure. Here that procedure is generalized and asymptotic distributions are studied.

The problem considered is that of optimizing a function of a finite number of linear functionals over an infinite dimensional convex set $S$. It is shown that under some reasonably general conditions the method of moment spaces can be used to reduce the problem to one of optimizing over a simple finite dimensional set (generally a set of convex combinations of extreme points of $S$). The results are applied to finding the maximum asymptotic variance of M-estimators over classes of distributions arising in the theory of robust estimation.

Let $X_1, \cdots, X_n$ be a sample from a population with density $f(x - \theta)$ such that $f$ is symmetric and positive. It is proved that the tails of the distribution of a translation-invariant estimator of $\theta$ tend to 0 at most $n$ times faster than the tails of the basic distribution. The sample mean is shown to be good in this sense for exponentially-tailed distributions while it becomes poor if there is contamination by a heavy-tailed distribution. The rates of convergence of the tails of robust estimators are shown to be bounded away from the lower as well as from the upper bound.

Tukey's jackknife estimate of variance for a statistic $S(X_1, X_2, \cdots, X_n)$ which is a symmetric function of i.i.d. random variables $X_i$, is investigated using an ANOVA-like decomposition of $S$. It is shown that the jackknife variance estimate tends always to be biased upwards, a theorem to this effect being proved for the natural jackknife estimate of $\operatorname{Var} S(X_1, X_2, \cdots, X_{n-1})$ based on $X_1, X_2, \cdots, X_n$.

Conditional exponential families of Markov processes are defined and a representation of the score function martingale is established for the important conditionally additive case. This result unifies those obtained separately for different examples and provides the key to asymptotic normality results for the maximum likelihood estimate.

We establish contiguity of certain families of probability measures indexed by $T$, as $T \rightarrow \infty$, for classes of stochastic processes with stationary, independent increments whose sample paths are discontinuous. Many important consequences pertaining to properties of tests and estimates then apply. A new expression for the Radon-Nikodym derivative of these processes is obtained.

This paper considers the two approaches for estimating the parameters specifying the spectral density of the counting process of a stationary point process, namely the frequency domain and the time domain approaches. The relation between the two is clarified; consistency and asymptotic normality of the estimates are established. Finally the special case of a rational spectral density is considered in some detail.

In this paper we consider a naive estimator of the failure rate function and smooth it using any band-limited window. We show that this smoothed estimate is equivalent to estimates obtainable from the sample hazard function, as in Rice and Rosenblatt (1976). We obtain the asymptotic distribution of the global deviation of the smoothed estimate from the failure rate function, which can then be used to construct uniform confidence bands.

Consider a general linear model for a column vector $y$ of data having $E(y) = X \alpha$ and $\operatorname{Var}(y) = \sigma^2H$, where $\alpha$ is a vector of unknown parameters and $X$ and $H$ are given matrices that are possibly deficient in rank. Let $b = Ty$, where $T$ is any matrix of maximum rank such that $TH = \phi$. The estimation of a linear function of $\alpha$ by functions of the form $c + a'y$, where $c$ and $a$ are permitted to depend on $b$, is investigated. Allowing $c$ and $a$ to depend on $b$ expands the class of unbiased estimators in a nontrivial way; however, it does not add to the class of linear functions of $\alpha$ that are estimable. Any minimum-variance unbiased estimator is identically [for $y$ in the column space of $(X, H)$] equal to the estimator that has minimum variance among strictly linear unbiased estimators.

Two measures of content of information in statistical experiments are considered. They are both based on Le Cam's notion of deficiency of one experiment with respect to another. The measures are: (i) The deficiency of the given experiment with respect to a totally informative experiment. (ii) The deficiency of a totally uninformative experiment with respect to the given experiment. We shall here discuss the interpretations of such measures, establish inequalities for them and related quantities, and study their behaviour under replications. If the parameter set is finite then closed expressions for exponential rates of convergence, as the number of replications increase, are given. In particular the exponential rate of the minimax probability of not covering the true values of the parameter by an $r$-point confidence set is expressed in terms of Hellinger transforms. If convergence to the totally informative experiment takes place at all, then the speed of convergence is necessary exponential. Examples are given indicating various possibilities.

It is proved that fine and tight comparative probability structures (where the set of events is assumed to be an algebra, not necessarily a $\sigma$-algebra) have agreeing probability measures. Although this was often claimed in the literature, all proofs the author encountered are not valid for the general case, but only for $\sigma$-algebras. Here the proof of Niiniluoto (1972) is supplemented. Furthermore an example is presented that reveals many misunderstandings in the literature. At the end a necessary and sufficient condition is given for comparative probability structures to have an almost agreeing probability measure.

A general approach to Pitman efficiency as a limit of the ratio of sample sizes is presented. The results can be used especially to derive the Pitman efficiency of tests based on asymptotically $\chi^2$-distributed statistics with different degrees of freedom.

A criterion is suggested for the size of a sample from a distribution of unknown form, and an upper bound for this quantity is obtained, involving only certain simple aspects of the prior measure.

Comparison of tests with respect to contiguous alternatives is mostly concerned with fixed levels. Properties of locally most powerful (LMP) tests in this sense are well-known in statistical literature. In this note the behaviour of LMP tests is studied for local (not necessarily contiguous) alternatives and vanishing levels of significance. It turns out that the shortcoming of the LMP test tends to zero at the rate $n^{-1} |\log \alpha_n|^{3/2}$.

Consider the problem of sequentially testing composite, contiguous hypotheses where the risk function is a linear combination of the probability of error in the terminal decision and the expected sample size. Assume that the common boundary of the closures of the null and the alternative hypothesis is compact. Observations are independent and identically distributed. We study properties of Bayes tests. One property is the exponential boundedness of the stopping time. Another property is continuity of the risk functions. The continuity property is used to establish complete class theorems as opposed to the essentially complete class theorems in Brown, Cohen and Strawderman.

It is pointed out that several results due to Walter and Blum do not hold strictly as they are stated. We find expansions for the mean square errors and mean integrated square errors of trigonometric series estimates of densities, and use them to compare the efficiencies of the estimates.

The maximum likelihood estimator $\hat{\mu}$ of a nondecreasing regression function has been studied in detail in the literature. However, little is known about its quadratic loss pointwise. This paper shows that the mean square error of $\hat{\mu}_i$ is less than that of the usual estimator $\bar{X}_i$ for each $i$ when $\bar{X}_1,\cdots, \bar{X}_k$ are independent normal variates.

This note considers, under minimal assumptions, the strong consistency of least squares estimates in regression with correlated errors.

Consider an ordinary estimation problem for an unknown parameter $\theta$. Let the estimator $\theta^\ast_n$ be the jackknife of a function of a $U$-statistic. Under mild assumptions, we demonstrate that $\sup_t |P\lbrack n^{1/2}(\theta^\ast_n - \theta)/S^\ast_n \leq t \rbrack - \Phi (t)| = O(n^{-p/2(p+1)})$, where $S^{\ast 2}_n$ is a jackknife estimator of the asymptotic variance of $n^{1/2}\theta^\ast_n, \Phi (t)$ is the standard normal distribution and $p$ is some positive number.