In the present paper the problem of point estimation is considered in terms of risk functions, without the customary restriction to unbiased estimates. It is shown that, whenever the loss is a convex function of the estimate, it suffices from the risk viewpoint to consider only nonrandomized estimates. For a number of specific problems the minimax estimates are found explicitly, using the squared error as loss. Certain minimax prediction problems are also solved.

This paper discusses the theory of certain probability distributions arising from points arranged in the form of lattices in two, three and higher dimensions. The points are of $k$ characters which for convenience are described as colors. A two-dimensional lattice will consist of $m \times n$ points in $m$ columns and $n$ rows. In a three-dimensional lattice there will be $l \times m \times n$ points in the form of a rectangular parallelopiped. Two situations arise for consideration. They are, to use the term of Mahalanobis, free and non-free sampling. In free sampling the color of each point is determined, on null hypothesis, independently of the color of the other points. The probabilities of the points belonging to the different colors, say black, white, etc. are $p_1, p_2 \cdots p_k$, such that $\sum_1^kp_r = 1$. In non-free sampling the number of points of each color is specified in advance, say $n_1, n_2 \cdots n_k$ so that $\sum_1^kn_r = mn$ or $lmn$ according as the lattice is two- or three-dimensional. Only the arrangements of these points in the lattice are varied. The distributions considered in this paper are the following:-- (i) the number of joins between adjacent points of the same color, say black-black joins, (ii) the number of joins between adjacent points of two specified colors, say black-white joins, and (iii) the total number of joins between points of different colors, along mutually perpendicular axes. The methods used here are the same as those developed by the author [3] for the linear case. All the distributions tend to the normal form when $l, m$ and $n$ tend to infinity, provided the $p$'s are not very small. Before considering the various distributions, we shall have a brief review of the work done on this topic by other people. For free sampling, Moran [5] and [6] has discussed the distribution of black-white and black-black joins for an $m \times n$ lattice of points of two colors. For a three-dimensional lattice, he has given the first and the second moments for the distribution of black-white joins. Levene [4] has announced some results closely allied to those of Moran for a square of side $N$ (with $N^2$ cells) each cell taking the characteristic $A$ or $B$ with probabilities $p$ and $q = 1 - p$ respectively. Bose [2] has found the expectation of $x =$ the number of black patches - the number of embedded white patches, for a square divided into $n^2$ small cells, having $p$ and $q = 1 - p$ as the probability of the cells being black or white. An embedded white patch is one that lies completely inside a black patch. The above review shows that the work done so far is confined entirely to the free sampling distributions, the points taking only two characters. As mentioned in the beginning of this article, we shall deal here with the free and non-free sampling distributions for points possessing $k$ characters or colors.

It is proved that the classical estimation procedures for the mean of a normal distribution with known variance are minimax solutions of properly formulated problems. A result of Stein and Wald [1] is an immediate consequence. Other such optimum properties follow. Sequential and non-sequential problems can be treated in this manner. Interval and point estimation are discussed.

The paper investigates certain asymptotic properties of the test of randomness based on the statistic $R_h = \sum^n_{i=1} x_ix_{i+h}$ proposed by Wald and Wolfowitz. It is shown that the conditions given in the original paper for asymptotic normality of $R_h$ when the null hypothesis of randomness is true can be weakened considerably. Conditions are given for the consistency of the test when under the alternative hypothesis consecutive observations are drawn independently from changing populations with continuous cumulative distribution functions. In particular a downward (upward) trend and a regular cyclical movement are considered. For the special case of a regular cyclical movement of known length the asymptotic relative efficiency of the test based on ranks with respect to the test based on original observations is found. A simple condition for the asymptotic normality of $R_h$ for ranks under the alternative hypothesis is given. This asymptotic normality is used to compare the asymptotic power of the $R_h$-test with that of the Mann $T$-test in the case of a downward trend.

Since the "unit" frequency differences (see (2.2) below) are dependent, the usual methods for establishing the normal character of the asymptotic distribution of the sum of random variables fail. However, the essential character of the distribution is disclosed by the integral functional relationship (3.6). From this it is possible to show that for large samples the distribution approximates "stability" in the normal sense ([2] and Lemma 2). Using the condition that the third logarithmic derivative of the characteristic function is uniformly bounded for all $n$ on a neighborhood of $t = 0$ one can prove that the asymptotic distribution exists and is normal.