Most studies of robust estimators of location parameters have relied upon mathematical theory, computer simulated data, or a combination of these. This paper presents a comparison of the performances of eleven estimators using real data sets. Twenty sets of data from 1761 determinations of the parallax of the sun, from 1798 measurements of the mean density of the earth, and from circa 1880 measurements of the speed of light, are employed in the study, with the current values of these physical constants being compared with the estimators' realized values. We find that light trimming provides some improvement over the sample mean, but that the sample mean itself compares favorably with many recent proposals. The bias and nonnormality of the data sets is considered, and the data sets are presented and discussed in an appendix.

Two methods of ranking $K$ samples for rank tests comparing $K$ populations are considered. The first method ranks the $K$ samples jointly; the second ranks the $K$ samples pairwise. These procedures were first suggested by Dunn (1964), and Steel (1960), respectively. It is shown that both ranking procedures are asymptotically equivalent for rank-sum tests satisfying certain nonrestrictive conditions. The problem is formulated in terms of multiple comparisons, but is applicable to other nonparametric procedures based on $K$-sample rank statistics.

For independent random variables distributed symmetrically around an unknown location parameter, aligned rank order statistics are constructed by using an estimator of the location parameter based on suitable rank statistics. The sequence of these aligned rank order statistics is then incorporated in the construction of suitable stochastic processes which converge weakly to some Gaussian functions, and, in particular, to tied-down Wiener processes in the most typical cases. The results are extended for contiguous alternatives and then applied in two specific problems in nonparametric inference. First, the problem of testing for shift at an unknown time point is treated, and then, some sequential type asymptotic nonparametric tests for symmetry around an unknown origin are considered.

Frequency tables are often encountered which cannot be directly observed. Examples occur in gene frequency estimation problems, latent structure analysis, epidemiology, and studies of group interactions. A class of models is proposed for these various applications. Maximum likelihood equations are derived, together with methods for their solution. Large-sample properties of these estimates are studied.

In the case of frequency data, traditional discussions such as Rao (1973, pages 355-363, 391-412) consider asymptotic properties of maximum likelihood estimates and chi-square statistics under the assumption that all expected cell frequencies become large. If log-linear models are applied, these asymptotic properties may remain applicable if the sample size is large and the number of cells in the table is large, even if individual expected cell frequencies are small. Conditions are provided for asymptotic normality of linear functionals of maximum-likelihood estimates of log-mean vectors and for asymptotic chi-square distributions of Pearson and likelihood ratio chi-square statistics.

A person wishes to determine which, if any, of $n = \Pi^k_{j=1} a_j$ i.i.d. random variables, $X(i_1,\cdots, i_k), i_j = 1,\cdots, a_j$, lie in some specified set $A$. Such observations will be called unsafe. It is assumed that the density of the $X$'s is known and that $Y_j(i_1,\cdots, i_j)$, the sum of all the $X$'s whose first $j$ indices are $i_1,\cdots, i_j$, can be measured as easily as the individual $X$'s. In this paper, search procedures of the following form are studied. The person first measures $Y_0$, the sum of all the $X$'s. On the basis of $Y_0$, he decides whether to stop, and classify all the $X$'s as safe, or to continue and measure $Y_1(1),\cdots, Y_1(a_1 - 1)$ (and hence know $Y_1(a_1) = Y_0 - \sum^{a_1-1}_{i=1} Y_1(i))$. If he has decided to continue, he measures $Y_1(j)$. For each of $(Y_0, Y_1(j))$, he must decide whether to stop and classify as safe all $X$'s whose first index is $j$, or to continue and measure $Y_2(j, 1),\cdots, Y_2(j, a_2 - 1)$ (and hence know $Y_2(j, a_2))$. He continues in this fashion until each $X$ has either been classified safe or has been observed. Unlike most group testing problems, he is not restricted to procedures that will locate all the unsafe observations. Instead there is a loss function $L(x)$ measuring the loss if $X(i_1,\cdots, i_k) = x$ and is not observed. Let $V_1$ be the expected loss of a procedure (summed over all the $X$'s), and let $V_2$ be the expected number of measurements. For each $0 \leqq p \leqq 1$, a class of rules $D(p)$ is defined such that if a procedure is in $D(p)$, it minimizes $pV_1 + (1 - p)V_2$, and conversely, if a procedure minimizes $pV_1 + (1 - p)V_2$, then there is a rule in $D(p)$ that leads to the same decisions a.e. The union of the $D(p)$ is shown to be an essentially complete class of rules. A simpler form for the rules in $D(p)$ is derived for the case where the loss function is nondecreasing. More specific calculations are given for the case where the $X$'s are normally distributed, and $L(x)$ is the indicator function for the set $\{x \geqq d\}$.

Let the random variable $X$ have a distribution depending on a parameter $\theta \in \Theta$. Consider the problem of testing the hypothesis $H: \Theta_0 \subseteqq \Theta$ based on a sequence of observations on $X$. The likelihood ratio test for $H$ is constructed by selecting a model for the unknown distribution of $X$. In this paper the asymptotic performance of the likelihood ratio test is studied when the model is incorrect, that is, when the probability distribution of $X$ is not a member of the model from which the likelihood ratio test is constructed. Exact and approximate measures of the asymptotic efficiency of the likelihood ratio test when the model is incorrect are proposed.

Let $X_1, X_2,\cdots, X_m$ be a random sample of size $m$ from a normal population with mean $\theta$ and variance $\sigma_x^2$. Let $Y_1, Y_2,\cdots, Y_n$ be a random sample of size $n$ from a normal population with mean $\theta$ and variance $\sigma_y^2$. The $X$-sample and $Y$-sample are independent. Note that this model is appropriate in balanced incomplete blocks designs. We consider various hypotheses testing problems concerned with $\theta$ and obtain the following results: (1) For testing $H_0: \theta = 0 \mathrm{vs} H_1: \theta \neq 0$ (and the usual variants of these hypotheses), the usual $t$-test based on only one sample is proven to be admissible. This is somewhat surprising in light of results obtained in point and confidence estimation. (2) For testing $H_0: \theta = 0 \mathrm{vs} H_1 \neq 0$, suppose it is assumed that $\sigma_x \geqq \mathbf{B}$, where $\mathbf{B}$ is any positive constant. Then a similar test is found, which is better than the $t$-test based only on the $X$-sample, if $m \geqq 3$ and $n \geqq 4$. (3) Let $\delta = \theta/\sigma_x$ and test $H_0: \delta = 0 \mathrm{vs} H_1: \delta_0 \leqq \delta \leqq \delta_1$. Here $\delta_0 > 0$ can be determined by the sample sizes and size of the $t$-test and $\delta_1$ is arbitrarily large. For this separated hypothesis a test, based on improved estimators of $\theta$, is found which is better than the usual $t$-test for $m \geqq 2, n \geqq 6$. (4) Let $\sigma_x^2$ be known and test $H_0: \theta = 0 \mathrm{vs} H_1 \theta \neq 0$. It is shown in this case that the test which rejects if $|\bar{X}| > C$, is admissible.

This paper shows that the UMP or UMPU tests for serial correlation, derived under the assumption of a normal distribution, are quite robust against departure from normality. In fact, the tests are still UMP or UMPU in much broader classes of distributions and the null distributions remain unchanged under these classes. The results will be applied to a linear model.

A new characterization of monotonic dependence is given here proceeding in a natural way from the consideration of a type of dependence weaker than quadrant dependence. More precisely, each bivariate distribution of $(X, Y)$ is transformed onto a pair of functions $^\mu{X, Y}$ and $^\mu{Y, X}$ defined on the interval $0 < p < 1$ and taking values from [-1, 1], with $^\mu{X, Y}(p)$ being a suitably normalized expected value of $X$ under the condition that $Y$ exceeds its $p$th quantile. The usefulness of these functions as a kind of measures of the strength of monotonic dependence as well as their close relation to regression functions is demonstrated. It is also suggested that these functions and their sample analogues could serve as useful tools in modelling and solving some statistical decision problems.

Some results from the theory of total positivity and Schur convexity are applied in deriving inequalities for multivariate normal probabilities having a certain convariance matrix. The result is applied to determine an optimal experimental design in an analysis of covariance model when selection of the best treatment is desired.

Sufficient conditions are provided to ensure that the Bayes estimator with squared error loss for the natural parameter of an exponential family distribution is a linear function of the sample value, given that various contractions of the Bayes estimator are also Bayes estimators, generalizing a result of Meeden (1976).

It is shown that within the class of all multivariate distributions depending on a location parameter (and satisfying certain smoothness conditions) and with a weighted norm constraint on the covariance matrix, the one with minimum Fisherian information is the Gaussian distribution. This result is then used in obtaining a tight upper bound on the error of estimating an unknown random vector observed in additive Gaussian noise under quadratic loss.

In this paper characterization theorems for Bross' ridits are proved with the help of functional equations by assuming intuitively reasonable postulates.

The intrinsic geometry of a space of distributions is represented by means of an embedding in Hilbert space, which has been previously studied.

For the first order moving average we consider a proposal by Walker (Biometrika, 1961) to use $k$ sample autocorrelations $(1 < k < T, T$ sample size), to estimate the first autocorrelation of the model, and hence its basic parameter. When $k = k_T \rightarrow \infty$ as $T \rightarrow \infty$, the estimator is proved consistent and asymptotically normal and efficient, the latter provided $k_T$ dominates $\log T$ and is dominated by $T^\frac{1}{2}$. An alternative form of the estimator facilitates the calculations and the analysis of the role of $k$, without changing the asymptotic properties.

An unknown density function $f(x)$, its derivatives, and its characteristic function are estimated by means of Hermite functions $\{h_j\}$. The estimates use the partial sums of series of Hermite functions with coefficients $\hat{a}_{jn} = (1/n) \sum^n_{i=1} h_j(X_i)$ where $X_1\cdots X_n$ represent a sequence of i.i.d. random variables with the unknown density function $f$. The integrated mean square rate of convergence of the $p$th derivative of the estimate is $O(n^{(p/r) + (5/6r)-1})$. The same is true for the Fourier transform of the estimate to the characteristic function. Here the assumption is made that $(x - D)^r f \in L^2$ and $p < r$. Similar results are obtained for other conditions on $f$ and uniform mean square convergence.