There are collected in this paper several observations and results more or less loosely related by their connections with the subject mentioned in the title. The discussion moves from the general to the specific, beginning with some remarks on minimization of convex functions subject to side conditions, and ending with a discussion of uniform consistency of estimators of linearly ordered parameters. Section 2 deals with one aspect of the problem of minimizing a function of several variables, subject to side conditions which specify that the variables must satisfy certain inequalities. It is frequently true in such problems that information as to which of the restricting sets contain the minimizing point on their boundaries is of great assistance in finding this point. Theorem 2.1 provides the basis for a stepwise procedure leading to this information when both the function to be minimized and the restricting sets are convex. It makes no contribution, however, to the problem of finding the minimizing point on a given boundary or intersection of boundaries. Brief mention is made in Section 3 of some examples of estimation problems for which the remark to which Section 2 is devoted is appropriate. Section 4 is concerned with a situation in which samples are taken from $k$ populations, each known to belong to a given one-parameter "exponential family". The problem is the maximum likelihood estimation of the $k$ parameters determining the populations, subject to certain restrictions. Methods are discussed of finding the minimizing point on a given intersection of boundaries of restricting sets. In the particular case when all populations belong to the same exponential family and when the restrictions on the parameters are order restrictions, it is observed that the maximum likelihood estimators (MLE's) of the means are independent of the particular exponential family. In Section 5 is discussed a property, related to sufficiency, of the MLE's discussed in Section 4. Let $y$ denote a vector representing a set of possible values of the MLE's, $E$ a Borel subset of the sample space, $\tau$ a parameter point, $S_0$ the intersection of the restricting sets. If $S_0$ is bounded by hyperplanes, there is a determination of the conditional probability $p\tau(E \mid y)$ which is independent of $\tau$ when $y$ is interior to $S_0$, and, when $y$ lies on a face, edge, or vertex of $S_0$, is independent of $\tau$ on the closure of that face, edge, or vertex. This result may be regarded as a generalization of a remark ([16], p. 77) to the effect that if $X$ and $Y$ are normally distributed random variables with unit standard deviation and means $\xi$ and $\eta$ respectively, and if $\xi$ and $\eta$ are known to satisfy a linear equation, then the foot of the perpendicular from the observation point $(x, y)$ to the line is a sufficient estimator. Section 6 is devoted to the same problem as are Sections 4 and 5, except that the parameters are linearly ordered, and that the populations need not belong to exponential families. Conditions are obtained for the strong uniform consistency of an estimator which is the MLE when the populations do belong to the same exponential family. An asymptotic lower bound is given for the probability of achieving a given precision uniformly.

Confidence region procedures for multidimensional quantities sometimes require prohibitive amounts of computation and the regions are difficult to represent in a useful way. Some approximate procedures are constructed by using regions obtained as the intersection of several regions, each much easier to construct. The procedures are applicable to the solution of simultaneous equations, whose coefficients are subject to random error. Approximations by convex polyhedra and by parallelepipeds are proposed. The procedures are illustrated for setting a confidence region for the location of the vertex of a quadratic regression surface.

An exact formula is presented for the probability of a specified frequency count of $m$-tuples $(m \geqq 1)$ in a sequence $X_1, X_2, \cdots, X_N$ from a Markov chain of order $m - 1$ having a denumerable number $a \leqq \infty$ of states. An exact expression is also obtained for the conditional probability of a specified $m$-tuple count, given the $n$-tuple count, when the chain is of order $n - 1 (n < m).$ If $a < \infty,$ then this conditional probability, when regarded as a statistic computed from the observed sequence, is shown to be asymptotically equivalent to the product of the probabilities (regarded as a statistic) associated with a corresponding set of $a^{n - 1}$ contingency tables with assigned marginals (each table having $a^{m - n}$ row and $a$ columns), where in each table the two attributes described by the table are independent. This fact leads to several simplified tests, related to standard tests of independence in contingency tables, for the null hypothesis $H_{n - 1}$ that the Markov chain is of order $n - 1$ within the alternate hypothesis $H_{m - 1}.$ Analogous results are also obtained for circular sequences.

In this paper a test based on the union-intersection principle is proposed for overall independence between $p$ variates distributed according to the multivariate normal law, and this is extended to the hypothesis of independence between several groups of variates which have a joint multivariate normal distribution. Methods used in earlier papers [3, 4] have been applied in order to invert these tests for each situation, and to obtain, with a joint confidence coefficient greater than or equal to a preassigned level, simultaneous confidence bounds on certain parametric functions. These parametric functions are, in case I, the moduli of the regression vectors: (a) of the variate $p$ on the variates $(p - 1), (p - 2), \cdots, 2, 1,$ or on any subset of the latter; (b) of the variate $(p - 1)$ on the variates $(p - 2), (p - 3), \cdots, 2, 1,$ or any subset of the latter, etc.; and finally, (c) of the variate 2 on the variate 1. For case II, parallel to each case considered above, there is an analogous statement in which the regression vector is replaced by a regression matrix, $\beta$, say, and the "modulus" of the regression vector is replaced by the (positive) square-root of the largest characteristic root of $(\beta\beta')$. Simultaneous confidence bounds on these sets of parameters are given. As far as the proposed tests of hypotheses of multiple independence are concerned they are offered as an alternative to another class of tests based on the likelihood-ratio criterion [5, 6] which has been known for a long time. So far as the confidence bounds are concerned it is believed, however, that no other easily obtainable confidence bounds are available in this area. One of the objects of these confidence bounds is the detection of the "culprit variates" in the case of rejection of the hypothesis of multiple independence, for the "complex" hypothesis is, in this case, the intersection of several more "elementary" hypotheses of two-by-two independence.

A general sequential sampling method is given for problems of comparing the variances of two or more normal populations in terms of ratios of variances. Sequential tests are given for a hypothesis specifying the ratio of two variances, including tests for variance components in the analysis of variance (Model II). Such tests provide savings in average required numbers of observations, relative to standard F tests, comparable to those typical of sequential probability ratio tests.

Let $(x_{nk}), k = 1, \cdots, k_n; n = 1, 2, \cdots$ be a double sequence of infinitesimal random variables which are rowwise independent (i.e., $\lim_{n\rightarrow\infty} \max_{1\leqq k \leqq k_n} P (| x_{nk} | > \epsilon) = 0$ for every $\epsilon > 0,$ and for each $n x_{n1}, \cdots, x_{nk_n}$ are independent). Let $S_n = x_{n1} + \cdots + x_{nk_n} - A_n$ where the $A_n$ are constants and let $F_n(x)$ be the distribution function of $S_n.$ Necessary and sufficient conditions for $F_n(x)$ to converge to a distribution function $F(x)$ are known, and in particular we know that $F(x)$ is infinitely divisible. In this paper we shall investigate the system of infinitesimal, rowwise independent random variables $(| x_{nk} | ^r), r \geqq 1.$ In particular we shall be interested in large values of $r$. Specifically, let $S^r_n = | x_{n1} | ^r + \cdots + | x_{n1} | ^r - B_n(r),$ where $B_n(r)$ are suitably chosen constants. Let $F_n^r(x)$ be the distribution function of $S^r_n.$ Necessary and sufficient conditions for $F_n^r(x)$ to converge $(n \rightarrow \infty)$ to a distribution function $F^r(x)$ are given, and also necessary and sufficient conditions for $F^r(x)$ to converge $(r \rightarrow \infty)$ to a distribution function $H(x)$ are given. The form that $H(x)$ must take is obtained and under rather general conditions it is shown that $H(x)$ is a Poisson distribution. In any case it is shown that $H(x)$ is the sum of two independent random variables, one Gaussian and the other Poisson (including their degenerate cases).

One of the most popular designs in experimental work is the randomized block. These designs can be put into three broad classes viz. complete block design, balanced incomplete block design, and the partially balanced incomplete block design. These designs are all special cases of the general two way classification with unequal numbers in the subclasses, but since the analysis of this general classification is quite complex, these special cases have evolved which are adequate to fit most needs and the analysis of these special designs is relatively easy. [1], [2], [6], [8]. However, most of the block designs considered to date have one feature in common--they require each block to contain an equal number of experimental units. The exceptions are given in [9], [10], where designs are considered in which the number of experimental units in blocks differ by one. The purpose of this paper is to extend the randomized block design to include the case where all blocks do not contain the same number of experimental units. We have called this the staircase design. Suppose an experimenter, wishing to run an experiment using $N$ treatments, decides to use a randomized block design, but after arranging his material into homogeneous groups he finds that he has blocks available which have varying number of experimental units. The experimenter has various courses open to him: (1) If enough blocks are available with $N$ or more experimental units he can discard the extra units in these blocks, discard all the blocks which have less than $N$ units, and use a randomized complete block design; (2) He can discard units in the blocks until he has enough units and blocks for a balanced incomplete block or a partially balanced incomplete block design; (3) He can use all the experimental units and use the staircase design proposed in this paper. For example, if an experimenter has $N$ treatments with which he wishes to experiment using a randomized block design, and if he has blocks of unequal size, then he must rank his $N$ treatments in the order of their importance, i.e., $T_1, T_2, \cdots, T_N,$ where he considers $T_1$ the most important and $T_N$ the least important. Now suppose he has at his disposal $b_1$ blocks which each contain $N$ experimental units. Then all $N$ treatments are randomized in each of the $b_1$ blocks. Suppose further that he has $b_2$ blocks which each contain $N_1$ experimental units $(N_1 < N).$ Then the first $N_1$ treatments are arranged at random in each of the $b_2$ blocks. This process is continued until all the blocks are used. A particular example where this would be useful is an experiment involving animals as experimental units where a block consists of litter mates. Let us suppose that we have two litters of size seven, three of size five, and one of size four. Using the staircase design we can include seven treatments and still have the four we are most interested in replicated six times.

In order to estimate the mean of a finite population from a random sample, the sample units may be selected in two ways. A pre-determined number $m$ of units may be selected with replacement or sampling with replacement may be continued till a desired number $n$ of distinct units is obtained. The first procedure is called sampling with replacement while the second may be called sampling without replacement. A comparison is generally made between the two procedures with $m = n.$ This comparison, however, is not fair since costs are usually proportional to the number of distinct units and in the first procedure this number would be less than or equal to $n.$ In the first procedure the population mean is usually estimated by the sample mean based on all the units in the sample including repetitions while in the second procedure the estimate is generally made to depend on the distinct units only. The object of this paper is to show that the estimate making use of only the distinct units is superior in either procedure. The following results are proved in this paper: (i) In sampling with replacement the estimate of the mean based on distinct units in the sample is superior to the estimate based on the total sample size when (a) the total sample size is fixed in advance, while the number of distinct units in the sample is a random variable, and, (b) when the total sample size is a random variable while the number of distinct units is fixed in advance. (ii) The same is true of ratio estimates. It is also shown that the bias is numerically less if the ratio estimate is based on distinct units regardless of whether these are fixed in advance or considered as random variables. (iii) Expressions for the estimation of the variances of the various estimates considered in this paper are given. (iv) The above results are extended to multistage sampling.

A solution for the problem of obtaining a distribution-free one-sided confidence interval for $p = \Pr \{Y < X\}$ has been proposed in [1]. At present a numerical procedure is given for computing the sample sizes needed for such a confidence interval with given width and confidence level.

This paper is concerned with inequalities connecting probabilities of hypotheses using Bayes' theorem (a posteriori probabilities), a priori probabilities, and Kullback-Leibler information-statistics in sampling from populations belonging to the exponential class of populations. As a corollary, it is shown that if it is known that the a priori probabilities are all equal, the choice of the hypothesis with the minimum Kullback-Leibler information-statistic is the same as the choice of the hypothesis with the maximum a posteriori probability, and conversely.

The c.d.f. of a $2 \times 2$ random determinant with mutually independent normally distributed entries is derived as an infinite series. Error functions that bound the tail of this series facilitate numerical calculation. Conditions are imposed on four variable quadratic forms for this distribution to apply. A normal approximation to the distribution is suggested.

The distinction between efficiency in the asymptotic sense originally introduced by Fisher ([2], 1925, p. 703), and the finite sample sense sometimes used by others has been recently stressed by various writers (e.g., Berkson [1]). The technique of proof used below was originally developed to provide a simple example where the maximum likelihood estimate of location, though asymptotically efficient, was not of minimum variance for any finite sample size whatever. The (symmetrical) double exponental distribution with known scale, where the sample median is the maximum likelihood estimator of location, could easily be shown to be such an example. (While this result is useful in deflating unwarranted views about minimum variance properties of maximum likelihood estimates, Fisher's ([2], p. 716) results about intrinsic accuracy in the same situation are of more basic interest.) On examination, however, the technique used to provide this rather isolated and special result was found capable of showing, for a class of distributions with suitable monotony properties (in particular all distributions for which $f'(y)/f(y)$ is monotone decreasing, and all normal, exponential, gamma and beta distributions), that the covariances of the order statistics in a sample of any chosen size are monotone in either index separately.

Suppose $X$ is a chance variable taking values in $k$-dimensional Euclidean space. That is, $X = (Y_1, \cdots, Y_k),$ where $Y_i$ is a univariate chance variable. The joint distribution of $(Y_1, \cdots, Y_k)$ has density $f(y_1, \cdots, y_k),$ say. We shall call a function $h(y_1, \cdots, y_k)$ "piecewise continuous" if it is everywhere bounded, and $k$-dimensional Euclidean space can be broken into a finite number of Borel-measurable subregions, such that in the interior of each subregion $h(y_1, \cdots, y_k)$ is continuous, and the set of all boundary points of all subregions has Lebesgue measure zero. We assume that $f(y_1, \cdots, y_k)$ is piecewise continuous. Let $h(y_1, \cdots, y_k)$ be some given nonnegative piecewise continuous function, and let $X_1, \cdots, X_n$ be independent chance variables, each with the density $f(y_1, \cdots, y_k).$ Choose a nonnegative number $t,$ and for each $i$, construct a $k$-dimensional sphere with center at $X_i = (Y_{i1}, \cdots, Y_{ik})$ and of $k$-dimensional volume $$\frac{th(Y_{i1}, \cdots, Y_{ik})}{n}.$$ Such a sphere will be called "of type $s$" if it contains exactly $s$ of the $(n - 1)$ points $X_1, \cdots, X_{i-1}, X_{i+1}, \cdots, X_n.$ Let $R_n(t; s)$ denote the proportion of the $n$ spheres which are of type $s$. For typographical simplicity, we denote the vector $(y_1, \cdots, y_k)$ by $y.$ Let $S(t; s)$ denote the multiple integral $$(t^s/s!) \int^\infty_{-\infty} \cdots \int^\infty_{-\infty} h^s(y)f^{s+1}(y) \exp \{-th(y)f(y)\} dy_1 \cdots dy_k.$$ It is shown that $R_n(t; s)$ converges stochastically to $S(t; s)$ as $n$ increases. This result is then used to construct a test of the hypothesis that the unknown density $f(y)$ is equal to a given density $g(y).$

The available data consists of a random sample $x(1) < \cdots < x(n)$ from a reasonably well-behaved continuous statistical population. The problem is to estimate the standard deviation of a specified $x(r)$ that is not in the tails of the sample. The estimates examined are of the form $$a\lbrack x(r + i) - x(r - i)\rbrack$$ and the explicit problem consists of determining suitable values for $a$ and $i.$ The solution $$a = (\frac{1}{2})(n + 1)^{-3/10}\{\lbrack r/(n + 1)\rbrack\lbrack 1 - r/(n + 1)\rbrack\}^{1/2}, i \doteq (n + 1)^{4/5}$$ appears to be satisfactory. Then the expected value of the estimate equals the standard deviation of $x(r)$ plus $O(n^{-9/10});$ also the standard deviation of this estimate is $O(n^{-9/10}).$ That is, the fixed and random errors for this point estimate are of the same order of magnitude with respect to $n.$ Solutions can be obtained which decrease the order of one of these types of error. However, these solutions increase the order of the other type of error, so that the overall error magnitude exceeds $O(n^{-9/10}).$

It is well known that if $X$ and $Y$ (or $1/X$ and $1/Y$) are independently normally distributed with mean zero and variance $\sigma^2,$ then $X/Y$ has a Cauchy distribution. It is the purpose of this note to show that the converse statement is not true. That is, the fact that the ratio of two independent, identically distributed, random variables $X$ and $Y$ follows a Cauchy distribution is not sufficient to imply that $X$ and $Y$ (or $1/X$ and $1/Y$) are normally distributed. This will be shown by exhibiting several counterexamples.