Partial prior knowledge is quantified by an interval $I(L, U)$ of $\sigma$-finite prior measures $Q$ satisfying $L(A) \leq Q(A) \leq U(A)$ for all measurable sets $A$, and is interpreted as acceptance of a family of bets. The concept of conditional probability distributions is generalized to that of conditional measures, and Bayes theorem is extended to accommodate unbounded priors. According to Bayes theorem, the interval $I(L, U)$ of prior measures is transformed upon observing $X$ into a similar interval $I(L_x, U_x)$ of posterior measures. Upper and lower expectations and variances induced by such intervals of measures are obtained. Under weak regularity conditions, as the amount of data increases, these upper and lower posterior expectations are strongly consistent estimators. The range of posterior expectations of an arbitrary function $b$ on the parameter space is asymptotically $b_N \pm \alpha\sigma_N + o(\sigma_N)$ where $b_N$ and $\sigma^2_N$ are the posterior mean and variance of $b$ induced by the upper prior measure $U$, and where $\alpha$ is a constant determined by the density of $L$ with respect to $U$ reflecting the uncertainty about the prior.

The classical nonparametric hypotheses of symmetry and equality of distribution functions are extended to hypotheses of approximate symmetry and approximate equality, by allowing for gross errors, which is indispensable for practical applications. It is shown that for these hypotheses one- and two-sample rank statistics maintain their distribution freeness, which now refers to their stochastically extreme laws. These laws are evaluated asymptotically, also under similarly extended alternatives, in the author's previous local framework, which has not yet covered rank statistics due to a subtle asymptotic fine structure of infinitesimal neighborhoods. Consequences on asymptotic maximum size, minimum power and relative efficiency of rank tests are drawn. In particular, it is shown that if the scores are unbounded, then rank tests fail completely; and by suitable truncation of the classically optimal scores, an asymptotic maximin rank test is obtained.

For a particular pseudoloss function, local asymptotic minimaxity and admissibility in the sense of Hajek and Le Cam are studied when probability measures are replaced by certain capacities ($\epsilon$-contamination, total variation). A minimax bound for arbitrary estimator sequences is established, admissibility of minimax estimators is proved, and it is shown that minimax estimators must necessarily have an asymptotic expansion in terms of a truncated logarithmic derivative.

In a general variance component model, nonnegative quadratic estimators of the components of variance are considered which are invariant with respect to mean value translations and have minimum bias, analogously to estimation theory of mean value parameters. Here the minimum is taken over an appropriate cone of positive semidefinite matrices, after having made a reduction by invariance. Among these estimators, which always exist, the one of minimum norm is characterized. This characterization is achieved by systems of necessary and sufficient conditions, and by a nonlinear cone-restricted pseudoinverse. A representation of this pseudoinverse is given, that allows computation without consideration of the boundary. In models where the decomposing covariance matrices span a commutative quadratic subspace, a representation of the considered estimator is derived that requires merely to solve an ordinary convex quadratic optimization problem. As an example, we present the two-way nested classification random model. In the case that unbiased nonnegative quadratic estimation is possible, this estimator automatically becomes the "nonnegative MINQUE". Besides this, a general representation of the MINQUE is given, that involves just one matrix pseudoinversion in the reduced model.

The existence of unbiased nonnegative definite quadratic estimates for linear combinations of variance covariance components is characterized by means of the natural parameter set in a residual model. In the presence of a quadratic subspace condition the following disjunction for nonnegative estimability is derived: either standard methods suffice, or the concepts of unbiasedness and nonnegative definiteness are incompatible. For the case of a single variance component it is shown that unbiasedness and nonnegative definiteness always entail a reduction to a trivial model in which the variance component under investigation is the sole remaining parameter. Several examples illustrate these results.

An asymptotic lower bound is obtained for the integrated relative squared error of autoregressive spectral estimate when the order of autoregression is selected. The bound is attained in the limit by the same selection as has been proposed for prediction.

In this work we prove the asymptotic efficiency of a certain time series recursion for the parameters of an ARMAX time series model.

It is well known that Wald's SPRT for testing simple hypotheses based on i.i.d. observations minimizes the expected sample size both under the null and under the alternative hypotheses among all tests with the same or smaller error probabilities and with finite expected sample sizes under the two hypotheses. In this paper it is shown that this optimum property can be extended, at least asymptotically as the error probabilities tend to 0, to invariant SPRTs like the sequential $t$-test, the Savage-Sethuraman sequential rank-order test, etc. In fact, not only do these invariant SPRTs asymptotically minimize the expected sample size, but they also asymptotically minimize all the moments of the sample size distribution among all invariant tests with the same or smaller error probabilities. Modifications of these invariant SPRTs to asymptotically minimize the moments of the sample size at an intermediate parameter are also considered.

Kernel estimation of $f(0)$ is considered where $f$ is a density in some class $\mathscr{F}$ of $d$-dimensional densities, described in terms of a Taylor series expansion. A sequence of kernels which asymptotically minimizes the maximum mean square error of estimation over $\mathscr{F}$ is given. The shape of the kernel is fixed, the size of the window depends on $f(0)$, and an easily computed estimate is obtained to efficiently adapt the sequence to the unknown value of $f(0)$.

This paper studies some asymptotic properties of density estimates $\hat{f}$ of $f$ based on $d$-variate delta sequences. The mean-square consistency, almost sure consistency, and asymptotic normality of $\hat{f}$ have been obtained as corollaries to the $L_1$ convergence properties of these delta sequences. Estimators based on kernel functions, orthogonal series, and some histogram methods can be obtained as special cases of $\hat{f}$.

It is suggested that problems in a reliability context may be handled by a Bayesian nonparametric approach. A stochastic process is defined whose sample paths may be assumed to be increasing hazard rates by properly choosing the parameter functions of the process. The posterior distribution of the hazard rates is derived for both exact and censored data. Bayes estimates of hazard rates and $\operatorname{cdf's}$ are found under squared error type loss functions. Some simulation is done and estimates graphed to better understand the estimators. Finally, estimates of the hazard rate from some data in a paper by Kaplan and Meier are constructed.

Some plausible models for violations of homoscedasticity and linearity in the general linear model have been proposed by Tukey and Anscombe. Bickel has provided robust tests for such violations. In this paper Bickel's tests are shown to be asymptotically optimal as well.

When the error terms in two different regression equations are correlated, Zellner proposed an alternative estimator for the coefficients of each equation based on an estimated covariance matrix between the two error terms. However, since an estimated covariance matrix is used, the OLSE seems better than Zellner's estimator when the correlation of the two equations is close enough to zero. This paper considers the problem of testing the independence between two regression equations and derives a locally best invariant test for a one-sided alternative hypothesis and a locally best unbiased and invariant test for a two-sided alternative.

The class of distributions on $R^m, 1 \leq m < \infty$ which are the $m$-dimensional marginal distributions of orthogonally invariant distributions on $R^{m + n}$ is characterized. This result is then used to provide a partial answer to the following question: given a symmetric distribution on $R^1$ and an integer $n \geq 2$, under what conditions will there exist a random vector $X \in R^n$ such that $a'X$ has the given distribution (up to a positive scale factor) for all $a \neq 0, a \in R^m$.

In this paper it is shown that some distributions of the matrix variates and latent roots arising in the multivariate Behrens-Fisher discriminant problem can be explicitly expressed in terms of the invariant polynomials with two matrix arguments, due to A. W. Davis, extending the zonal polynomials of matrix argument.

Main result: $X_1, X_2, \cdots, X_n$ are independent random variables valued in Euclidean spaces $E_1, E_2, \cdots, E_n$ such that $P\lbrack X_j = 0 \rbrack = 0$ for all $j$. Denote $R = \lbrack \sum^n_{j = 1} \|X_j\|^2 \rbrack^{1/2}$. Suppose that $(R^{-1}X_1, R^{-1}X_2, \cdots, R^{-1}X_n)$ is uniformly distributed on the sphere of $\oplus^n_{j = 1} E_j$. Then the $X_j$ are normal if $n \geq 3$. The case $n = 2$ and the case of Hilbert spaces are also studied.

The partitioning of the complete parametric vector from the experimenter's viewpoint for the general mixed factorial gives rise to the usual four exhaustive cases, (i), (ii), (iii), and (iv). Cases (i) and (iv) may be viewed as special cases of cases (ii) and (iii) respectively in connection with the problem of spectrum invariance under the group of level permutations and the problem of unbiased estimation under a uniform randomized design. Srivastava, Raktoe and Pesotan (1976) resolved these problems for case (ii) and this paper does the same for case (iii) so that the study is now complete.

A technique is introduced to prove limit theorems for simple linear rank statistics by means of an approximating statistic based on sequential ranks. This approximation is shown to be close enough to prove asymptotic normality of simple linear rank statistics under the null hypothesis and to obtain a bound on the rate of convergence to normality when the score function is unbounded. In addition, a law of the iterated logarithm and an invariance principle are given for simple linear rank statistics.

The convolution result of Block and Savits is generalized.

In the context of simple linear regression we investigate a class of estimators for the parameters which contains the Brown-Mood estimators. We derive the asymptotic efficiency properties of the class of slope estimators and find the most efficient slope estimator. This estimator is shown to have efficiency properties analogous to those of other median estimators for slope found in the literature. The equations which define the optimal estimators are median analogues of the least squares normal equation.

An estimator for a monotone regression function was proposed by Brunk. He has shown that if the underlying regression function has positive slope at a point, then, based on $r$ observations, the difference of the regression function and its estimate at that point has a nondegenerate limiting distribution if this difference is multiplied by $r^{1/3}$. To understand how the behavior of the regression function at a point influences the asymptotic properties of the estimator at that point, we have generalized Brunk's result to points at which the regression function does not have positive slope. If the first $\alpha - 1$ derivatives of the regression function are zero at a point and the $\alpha$th derivative is positive there, then the norming constants are of order $r^{\alpha/(2\alpha + 1)}$.

The estimator proposed by Brunk for the indefinite integral of a regression function defined on the unit cube in $\beta$ dimensional Euclidean space is studied. It is shown to be strongly uniformly consistent if the errors satisfy a first moment type of condition and an almost sure rate of convergence of order $O((n/\log_2n)^{-1/2})$ is obtained.

For an integer $p \geq 0$, Singh has proposed a class of kernel estimators $\hat{f}^{(p)}$ of the $p$th order derivative $f^{(p)}$ of a density $f$. This paper examines the detailed asymptotic behavior of these estimators. In particular, asymptotically equivalent expressions for the bias $(E\hat{f}^{(p)} - f^{(p)})$, the mean squared error $E(\hat{f}^{(p)} - f^{(p)})^2$ and the error $(\hat{f}^{(p)} - f^{(p)})$ are obtained, which in turn give exact rates of convergence of these terms to zero.

Asymptotically universally efficient estimators of the parameters in the general linear hypothesis are proposed. These estimators are based on ranks of residuals (or their absolute values); they are analogous to the linearized rank estimators proposed by Kraft and van Eeden in the sense that they are obtained by replacing, in these estimators, the function generating the scores by certain estimators of this function. Finally, it is shown that estimators of the score function, satisfying the conditions used, exist.

Fortus' generalization of asymptotic shapes of optimal testing regions for composite hypotheses does away with the restriction to exponential families originally imposed by us. Here we survey his work critically, and suggest some improvements that may be crucial for its practical applicability to parametric problems, and point out its shortcomings for nonparametric ones.