The kind of calibration problem considered may be roughly described as follows: There are two related quantities $\mathscr{U}$ and $\mathscr{V}$ such that $\mathscr{U}$ is relatively easy to measure and $\mathscr{V}$ relatively difficult, requiring more effort or expense; furthermore the error in a measurement of $\mathscr{V}$ is negligible compared with that for $\mathscr{U}$. A distinguishing feature of the problem is, that from a single calibration experiment, where measurements are made on a number of pairs $(\mathscr{U}, \mathscr{V})$, we wish subsequently to estimate the unknown values of $\mathscr{V}$ corresponding to a very large number of measurements of $\mathscr{U}$. The problem is solved by a procedure of interval estimation, whose operating characteristic is expressed in terms of a reformulation of the law of large numbers. Some idea of the contents of the article may be obtained from the table of contents.

Consider independent identically distributed observations whose distribution depends on a parameter $\theta$. Measure the distance between two parameter points $\theta_1, \theta_2$ by the Hellinger distance $h(\theta_1, \theta_2)$. Suppose that for $n$ observations there is a good but not perfect test of $\theta_0$ against $\theta_n$. Then $n^{\frac{1}{2}}h(\theta_0, \theta_n)$ stays away from zero and infinity. The usual parametric examples, regular or irregular, also have the property that there are estimates $\hat{\theta}_n$ such that $n^{\frac{1}{2}}h(\hat{\theta}_n, \theta_0)$ stays bounded in probability, so that rates of separation for tests and estimates are essentially the same. The present paper shows that need not be true in general but is correct under certain metric dimensionality assumptions on the parameter set. It is then shown that these assumptions imply convergence at the required rate of the Bayes estimates or maximum probability estimates.

Hajek (1968) proved that under weak conditions the distribution of a simple linear rank statistic $S$ is asymptotically normal, centered at the mean $ES$. He left open the question whether under the same conditions the centering constant $ES$ may be replaced by a simpler constant $\mu$, as was found to be true in the two-sample case and under different conditions by Chernoff and Savage (1958) and Govindarajulu, LeCam and Rhagavachari (1966). In this paper it is shown that the replacement of $ES$ by $\mu$ is permissible if one of Hajek's conditions is slightly strengthened.

Let $X_1, X_2, \cdots$ be a sequence of independent and identically distributed random variables with the common distribution being uniform on [0, 1]. Let $Y_1, Y_2, \cdots$ be a sequence of i.i.d. variables with continuous $\operatorname{cdf}F(t)$ and with [0, 1] support. Let $F_n(t, \omega)$ denote the empirical distribution function based on $Y_1(\omega), \cdots, Y_n(\omega)$ and let $G_m(t, \omega)$ the empirical $\operatorname{cdf}$ pertaining to $X_1(\omega), \cdots, X_m(\omega)$. Let $\sup_{0\leqq t \leqq 1}|F(t) - t| = \lambda$ and $D_n = \sup_{0 \leqq t \leqq 1}|F_n(t, \omega) - t|$. The limiting distribution of $n^{\frac{1}{2}}(D_n - \lambda)$ is obtained in this paper. The limiting distributions under the alternative of the corresponding one-sided statistic in the one-sample case and the corresponding Smirnov statistics in the two-sample case are also derived. The asymptotic distributions under the alternative of Kuiper's statistic are also obtained.

It is shown that certain conditional distributions, obtained by conditioning on a sufficient statistic, can be used to transform a set of random variables into a smaller set of random variables that are identically and independently distributed with uniform distributions on the interval from zero to one. This result is then used to construct distribution-free tests of fit for composite goodness-of-fit problems. In particular, distribution-free chi-square goodness-of-fit tests are obtained for univariate normal, exponential, and normal linear regression model families of distributions.

After reviewing the asymptotic variance stabilizing transformations in one dimension, a generalization of these to multivariate cases is discussed. Results are given for the uniqueness of solutions when they exist, but unlike the one-dimensional case, covariance stabilizing transformations need not exist. In the two-dimensional case, a necessary and sufficient condition is given for the existence of solutions. It takes the form of a second-order partial differential equation that the elements of any square root of the inverse of the limiting covariance matrix must satisfy. This condition is applied to three examples with the conclusion that no covariance stabilizing transformation exists for the trinomial distribution. It is conjectured that this non-existence of solutions is true for the general multinomial.

We consider estimation (subject to quadratic loss) of the vector of coefficients of a multiple linear regression model in which the error vector is assumed to have 0 mean and covariance matrix $\sigma^2 I$ but is not assumed to take on a specific parametric form, e.g., Normal. The vector of coefficients is taken to be randomly distributed according to some unknown prior. Restricted minimax solutions are exhibited relative to equivalence classes on the space of all prior probability distributions which group distributions with the same specified moments. In the context of the classic Empirical Bayes formulation, we determine restricted asymptotically optimal estimators--i.e., decision functions whose Bayes risks converge to the risk of the restricted minimax decision at each component stage.

Asymptotic forms for the optimal payoff and optimal stop rule for a generalized class of "secretary" problems are obtained by the analysis of a related family of ordinary differential equations.

In the present work, we study characterizations of populations obtained by using regression properties of one statistic on another. We extend some of the results of Lukacs and Laha by considering the cubic regression (polynomial regression of order 3) of a cubic statistic $S$ on a linear one $L$. This assumption of cubic regression is used to derive a third order non-linear differential equation in the characteristic function $h(t)$ of a set of $n$ independently and identically distributed random variables. The coefficients in this differential equation represent certain fixed relationships between the coefficients of the statistic $S$ and the regression coefficients. For appropriate choices of the coefficients in the fundamental differential equation, the resulting equation can be solved to yield the characteristic function of a particular distribution. In this way, we are able to obtain a series of characterization theorems for each of a variety of populations including Normal, Gamma, Binomial, Poisson, Geometric and several others. Moreover, all of the results obtained by Lukacs and Laha for characterizing populations using quadratic and constant regression are shown to be special cases of the theorems obtained in the present work. Finally, in the last section, we present an outline of a technique which can be used to study any general $r \leqq m$th order polynomial regression of any $m$th order statistic on a linear one. The approach used to generate the $m$th order differential equation is indicated and a method for determining the appropriate conditions on the coefficients is discussed.

An admissible minimax estimate for the mean of a normal random vector with known covariance is derived for a generalized quadratic loss function. This loss function is quadratic in both the estimation error and the unknown mean. The estimate is derived using the method of least favorable prior distributions. The decision rule is linear, and the least favorable prior distribution for the unknown mean is normal with zero mean. The covariance of this least favorable normal distribution is determined by the solution of a certain nonlinear algebraic matrix equation.

One or more observations are made on a random vector, whose covariance matrix may be a linear combination of known symmetric matrices and whose mean vector may be a linear combination of known vectors; the coefficients of the linear combinations are unknown parameters to be estimated. Under the assumption of normality, equations are developed for the maximum likelihood estimates. These equations may be solved by an iterative method; in each step a set of linear equations is solved. If consistent estimates of $\sigma_0, \sigma_1, \cdots, \sigma_m$ are used to obtain the coefficients of the linear equations, the solution of these equations is asymptotically efficient as the number of observations on the random vector tends to infinity. This result is a consequence of a theorem that the solution of the generalized least squares equations is asymptotically efficient if a consistent estimate of the covariance matrix is used. Applications are made to the components of variance model in the analysis of variance and the finite moving average model in time series analysis.

Let $\mathbf{X}$ be a $p$-variate $(p \geqq 3)$ vector normally distributed with mean $\mathbf{\theta}$ and covariance matrix $\Sigma$, positive definite but unknown. Let $A$ be a $p \times p$ Wishart matrix with parameters $(n, \Sigma)$, independent of $\mathbf{X}$. To estimate $\mathbf{\theta}$ relative to quadratic loss function $(\hat{\mathbf{\theta}} - \mathbf{\theta})'\Sigma^{-1}(\hat{\mathbf{\theta}} - \mathbf{\theta})$, we obtain a family of minimax estimators $\mathbf{\delta}(\mathbf{X}, \mathbf{A})$ based on $\mathbf{X}$ and $\mathbf{A}$ through $\mathbf{X}$ and $\mathbf{X}'\mathbf{A}^{-1}\mathbf{X}$. It is shown that there are minimax estimators of the form $\mathbf{\delta}(\mathbf{X}, \mathbf{A})$ which are also generalized Bayes. A special case where $\Sigma = \sigma^2\mathbf{I}$ is also considered.

Any triangular array of row independent $\mathrm{rv}$'s having continuous $\mathrm{df}$'s can be transformed naturally so that the empirical and quantile processes of the resulting $\mathrm{rv}$'s are relatively compact. Moreover, convergence (to a necessarily normal process) takes place if and only if a simple covariance function converges pointwise. Using these results we derive the asymptotic normality of linear combinations of functions of order statistics of non-i.i.d. $\mathrm{rv}$'s in the case of bounded scores.

Bounds are provided for the rates of convergence in the central limit theorem and the strong law of large numbers for $U$-statistics. The results are obtained by establishing suitable bounds upon the moments of the difference between a $U$-statistic and its projection. Analogous conclusions for the associated von Mises statistical functions are indicated. Statistics considered for exemplification are the sample variance and the Wilcoxon two-sample statistic.

It is shown that under very mild assumptions Wald's large-sample test statistic (quadratic form based on unrestricted maximum likelihood estimators) converges to noncentral chi-square under a sequence of local alternatives of the order $n^{-\frac{1}{2}}$, when the family of distributions is assumed to be of exponential type. This eliminates, for these families, the necessity of invoking the strict regularity conditions of Wald for the purpose of justifying the asymptotic distribution.

A modification of the one-sided sign test, which was proposed by Putter (1955), is shown to be an asymptotic UMP test. In contrast to the usual practice, zeros are not omitted but taken into account.

The most powerful scale and location invariant test of normality against the double exponential alternative is derived by the technique of integrating with respect to the scale and location transformation group. The resultant test is asymptotically equivalent to the likelihood ratio test of this hypothesis and to Geary's test (i.e. mean deviation over standard deviation) for all three test statistics are shown to have the same asymptotic normal distribution when the sampling is from a symmetric, absolutely continuous distribution, whose density is continuous in the neighborhood of its median and whose fourth moment exists.

Interval estimation of the largest mean of $k$ normal populations $(k \geqq 1)$ with a common variance $\sigma^2$ is considered. When $\sigma^2$ is known the optimal fixed-width interval is given so that, to have the probability of coverage uniformly lower bounded by $\gamma$ (preassigned), the sample size needed is minimized. This optimal interval is unsymmetric for $k > 2$. When $\sigma^2$ is unknown a sequential procedure is proposed and its behavior is studied. It is shown that the confidence interval obtained, which is also unsymmetric for $k > 2$, behaves asymptotically as well as the optimal interval. This represents an improvement of the procedure of symmetric intervals considered by the author previously; the improvement is significant, especially when $k$ is large.

It is shown that two distributions both of which have a finite expectation are equal if and only if for every $n \geqq 1$ there exists $1 \leqq k \leqq n$ such that the $k$th order statistics from samples of size $n$ of each distribution have equal expectations. Similarly, it is shown that a distribution with finite expectation is symmetric about zero if and only if for every $n \geqq 0$ there exists $0 \leqq k \leqq 2n + 1$ such that the sum of the expectations of the $k$th smallest and the $k$th largest observations in a sample of size $2n + 1$ is zero.

Let $(X, \mathscr{A})$ be a measurable space and $\mathscr{P} \mid \mathscr{A}$ a family of probability measures. Given a sufficient sub-$\sigma$-field, we define the sufficiency operator by assigning to each integrable function a conditional expectation which is independent of $P \in \mathscr{P}$. The operator thus defined is $\mathscr{P}$-a.e. linear, monotone, idempotent, and expectation invariant for every $P \in \mathscr{P}$. Not all of these properties are necessary to make such an operator useful in statistical theory: For the most important applications, linearity may be replaced by homogeneity and translation invariance; idempotency may be relinquished. It was therefore suggested (Pfanzagl (1967), page 416) to study homogeneous, translation invariant, monotone, and expectation invariant operators as a possibly useful generalization of sufficiency-operators. The purpose of this paper is to show that there exists to any such operator a sufficient sub-$\sigma$-field whose sufficiency operator effects at least the same reduction. (Hence there is no real gain in introducing the idea of "reduction by homogeneous, translation invariant, monotone, and expectation invariant operators" into statistical theory.) This result generalizes Proposition 9 of LeCam (1964), page 1435, where, roughly speaking, a similar result was obtained for monotone and expectation invariant operators which are linear (rather than homogeneous and translation invariant). An ergodic lemma for homogeneous and translation invariant (but not necessarily linear) operators, needed here as a tool for the proof of the main theorem, may be of independent interest.

The use of cyclic incomplete block designs for factorial experiments is considered. Necessary and sufficient conditions are established under which a cyclic design for an $m \times n$ factorial experiment will give an orthogonal analysis of the main effects and interaction. Various properties, the construction and analysis of such designs are then considered.

The concept of resolvability and affine resolvability was generalized to $\mu$-resolvability and affine $\mu$-resolvability by Shrikhande and Raghavarao (1964). In this paper, a representation of parameters of an affine $\mu$-resolvable BIB design is given and necessary conditions for the existence of this design are derived. Some methods of constructing (affine) $\mu$-resolvable BIB designs are given and some inequalities for these designs are obtained. Finally, some information on the block structure of $\mu$-resolvable BIB designs is provided.

For a $\mu$-resolvable Balanced Incomplete Block Design (BIBD) with parameters $v, b = mt, r = \mu t, k$ and $\lambda$, Kageyama (1973) obtained an inequality $b \geqq v + t - 1$. The main purpose of this note is to improve $b \geqq v + t - 1$ to $b \geqq \max \{v + t - 1, (m^2\lambda + m)/\mu^2\}$. This inequality is also improved further for a $\mu$-resolvable BIBD which is not affine $\mu$-resolvable.