Although regression problems have been considered by workers in all sciences for many years, until recently relatively little attention has been paid to the optimum design of experiments in such problems. At what values of the independent variable should one take observations, and in what proportions? The purpose of this paper is to develop useful computational procedures for finding optimum designs in regression problems of estimation, testing hypotheses, etc. In Section 2 we shall develop the theory for the case where the desired inference concerns just one of the regression coefficients, and illustrative examples will be given in Section 3. In Section 4 the theory for the case of inference on several coefficients is developed; here there is a choice of several possible optimality criteria, as discussed in [1]. In Section 5 we treat the problem of global estimation of the regression function, rather than of the individual coefficients. We shall now indicate briefly some of the computational aspects of the search for optimum designs by considering the problem of Section 2 wherein the inference concerns one of $k$ regression coefficients. For the sake of concreteness, we shall occasionally refer here to the example of polynomial regression on the real interval $\lbrack -1, 1\rbrack$, where all observations are independent and have the same variance. The quadratic case is rather trivial to treat by our methods, so we shall sometimes refer here to the case of cubic regression. In the latter case we suppose all four regression coefficients to be unknown, and we want to estimate or test a hypothesis about the coefficient $a_3$ of $x^3$. If a fixed number $N$ of observations is to be taken, we can think of representing the proportion of observations taken at any point $x$ by $\xi(x)$, where $\xi$ is a probability measure on $\lbrack -1, 1\rbrack$. To a first approximation (which is discussed in Section 2), we can ignore the fact that in what follows $N\xi$ can take only integer values. We consider three methods of attacking the problem of finding an optimum $\xi$: A. The direct approach is to compute the variance of the best linear estimator of $a_3$ as a function of the values of the independent variable at which observations are taken or, equivalently, as a function of the moments of $\xi$. Denoting by $\mu_i$ the $i$th moment of $\xi$, and assuming $\xi$ to be concentrated entirely on more than three points (so that $a_3$ is estimable), we find easily that the reciprocal of this variance is proportional to $$\frac{\mu^2_5(\mu^2_1 - \mu_2) + 2\mu_5(\mu^2_2 \mu_3 + \mu_3 \mu_4 - \mu_1 \mu^2_3 - \mu_1 \mu_2 \mu_4)\\- \mu^3_4 + \mu^2_4(\mu^2_2 + 2\mu_1 \mu_3) - 3\mu_4 \mu_2 \mu^2_3 + \mu^4_3}{\mu_4(\mu_2 - \mu^2_1) - \mu^2_3 - \mu^3_2 + 2\mu_1 \mu_2 \mu_3} + \mu_6$$ in the case of cubic regression. The problem is to find a $\xi$ on $\lbrack -1, 1\rbrack$ which maximizes this expression. Thus, this direct approach leads to a calculation which appears quite formidable. This is true even if one uses the remark on symmetry of the next paragraph and restricts attention to symmetrical $\xi$, so that $\mu_i = 0$ for $i$ odd. For polynomials of higher degree or for regression functions which are not polynomials, the difficulties are greater. B. The results of Section 2 yield the following approach to the problem: Let $c_0 + c_1x + c_2x^2$ be a best Chebyshev approximation to $x^3$ on $\lbrack -1, 1\rbrack$, i.e., such that the maximum over $\lbrack -1, 1\rbrack$ of $|x^3 - (c_0 + c_1x + c_2x^2)|$ is a minimum over all choices of the $c_i$, and suppose $B$ is the subset of $\lbrack -1, 1\rbrack$ where the maximum of this absolute value is taken on. Then $\xi$ must give measure one to $B$, and the weights assigned by $\xi$ to the various points of $B$ (there are four in this case) can be found either by solving the linear equations (2.10) or by computing these weights so as to make $\xi$ a maximum strategy for the game discussed in Section 2. Two points should be mentioned: (1) In the general polynomial case, where there are $k$ parameters ($k = 4$ here), the results described in [10], p. 42, or in Section 2 below imply that there is an optimum $\xi$ concentrated on at most $k$ points. Thus, even if we use this result with the approach of the previous paragraph, we obtain the following comparison in a $k$-parameter problem in Section 2: Method A: minimize a nonlinear function of $2k - 1$ real variables. Method B: solve the Chebyshev problem and then solve $k - 1$ simultaneous linear equations. The fact that the solution of the Chebyshev problem can often be found in the literature (e.g., [2]) makes the comparison of the second method with the first all the more favorable. (2) Although the computational difficulty cannot in general be reduced further, in the case of polynomial regression on $\lbrack -1, 1\rbrack$ there is present a kind of symmetry (discussed in Section 2) which implies that there is an optimum $\xi$ which is symmetrical about 0 and which is concentrated on four points; thus, in the case of cubic regression, this fact reduces the computation under Method A to a minimization in 3 variables, but Method B involves only the solution of a single linear equation. C. A third method, which rests on the game-theoretic results of Section 2, and which is especially useful when one has a reasonable guess of what an optimum $\xi$ is, involves the following steps: first guess a $\xi$, say $\xi^{\ast}$, and compute the minimum on the left side of (2.8); second, if this minimum is achieved for $c = c^{\ast}$, compute the square of the maximum on the right side of (2.9); then, if these two computations yield the same number, $\xi^{\ast}$ is optimum. If one has a guess of a class of $\xi$'s depending on one or several parameters, among which it is thought that there is an optimum $\xi$, then one can maximize over that class at the end of the first step and, the maximum being at $\xi^{\ast}$, go through the same analysis as above. This method is illustrated in Example 3.5 and Example 4. Of course, the remarks (1) and (2) of the previous paragraph can be used in applying Method C, as in these examples. In the example of cubic regression just cited, the optimum procedure turns out to be $\xi(-1) = \xi(1) = \frac{1}{6}, \xi(\frac{1}{2}) = \xi(-\frac{1}{2}) = \frac{1}{3}$. It is striking that any of the commonly used procedures which take equal numbers of observations at equally spaced points on $\lbrack -1, 1\rbrack$ requires over 38% more observations than this optimum procedure in order to yield the same variance for the best linear estimator of $a_3$ (see Example 3.1); the comparison is even more striking for higher degree regression. The unique optimum procedure in the case of degree $h$ is given by (3.3). The comparison of a direct computational attack, analogous to that of A above, with the methods developed in Sections 4 and 5 for the problems considered there, indicates even more the inferiority of the direct attack. In particular cases, e.g., Example 5.1, special methods may prove useful. Among recent work in the design of experiments we may mention the papers of Elfving [3], [4], Chernoff [5], Williams [11], Ehrenfeld [12], Guest [13], and Hoel [15]. Only Guest and Hoel explicitly consider computational problems of the kind discussed below. Our methods of employing Chebyshev and game theoretic results seem to be completely new. The results obtained in the examples below are also new, except for some slight overlap with results of [13] and [15], which is explicitly described below. We shall consider elsewhere some further problems of the type considered in this paper.

Suppose we are given $N$ objects to be weighed in $N$ weighings with a chemical balance having no bias. Let $x_{ij} = 1$ if the $j$th object is placed in the left pan in the $i$th weighing, $= -1$ if the $j$th object is placed in the right pan in the $i$th weighing, $= 0$ if the $j$th object is not weighed in the $i$th weighing. The $N$th order matrix $X = (x_{ij})$ is known as the design matrix. Also let $y_i$ be the result recorded in the $i$th weighing, $\epsilon_i$ be the error in this result and $w_j$ be the true weight of the $j$th object, so that we have the $N$ equations $$x_{i1}w_1 + x_{i2}w_2 + \cdots + x_{iN}w_N = y_i + \epsilon_i, i = 1, \cdots, N.$$ We assume $X$ to be a non-singular matrix. The method of Least Squares or theory of Linear Estimation gives the estimated weights $(\hat w_i)$ by the equation $$\hat w = (X'X)^{-1}X'Y,$$ where $Y$ is the column vector of the observations and $\hat w$ is the column vector of the estimated weights. If $\sigma^2$ is the variance of each weighing, then $$\operatorname{Var} (\hat w) = (X'X)^{-1}\sigma^2 = (c_{ij})\sigma^2,$$ where $(c_{ij})$ is the inverse matrix of $(X'X)$. An expository article reviewing the work done in weighing designs is given by Banerjee [2]. Kishen [4] treats the reciprocal of the increase in variance resulting from the adoption of any design other than the most efficient design, with mean variance $\sigma^2/N$, as the efficiency of the design. This efficiency can be measured by $$1/\sum^N_{i=1} c_{ii}.$$ Mood [5] gives an alternative definition for the best weighing design. In his view the best weighing design should give the smallest confidence region in the $\hat w_i(i = 1, \cdots, N)$ space for the estimates of the weights. Hence a design will be called best if the determinant of the matrix $(c_{ij})$ is minimised. In this paper we follow Kishen's definition in obtaining the best weighing designs. Hotelling [3] proved that for the best weighing design $c_{ii} = 1/N$ and $c_{ij} = 0 (i \neq j)$. The weighing designs for which $c_{ii} = 1/N$ and $c_{ij} = 0$ are best in the sense of both Kishen and Mood. Later Mood proved that the above property is satisfied by Hadamard matrices. Plackett and Burman [6] have constructed Hadamard matrices, $H_N$, up to and including $N = 100$, excepting $N = 92$. It may be remarked here that a necessary condition for the existence of $H_N$ is $N \equiv 0 (\mod 4)$, with the exception of $N = 2$. It is not known whether this condition is sufficient or not. In this paper, the best weighing designs are obtained in the cases (i) $N$ is odd and (ii) $N \equiv 2 (\mod 4)$ subject to the conditions: i) The variances of the estimated weights are equal; and ii) The estimated weights are equally correlated. The 2nd condition here is the same as that of Banerjee [1].

Two models are considered in detail which are the Models I and II of ANOVA in the terminology of Eisenhart [2]. The present paper, which deals with the one dimensional or univariate case, and its sequel, which will deal with the multidimensional or multivariate case, seek to give a unified general treatment, using matrix methods, of certain problems under the two models of ANOVA. Section 1 of each paper, which deals with Model I, is of the nature of a resume giving the main results of a general treatment discussed elsewhere [10, 11, 12] by one of the authors. Section 2 of each paper, which deals with Model II or variance components model, is self-contained, and presents a natural tie-up between the analyses under the two models for a $k$-way classification. Results in estimation, testing of hypotheses and confidence bounds are presented, although the main emphasis is on the results in confidence bounds (simultaneous and/or separate) on meaningful parametric functions which are physically natural and mathematically convenient measures of departure from customary null hypotheses. It will be seen that a mixed model, which would include both Models I and II as special cases, can be defined, and the associated problems can be studied by using methods which are, essentially, a combination of the methods given for the separate models in Sections 1 and 2, respectively, of this paper. Since nothing essentially new is involved in such a study, this paper does not explicitly discuss it. Unless otherwise stated, capital letters will denote matrices and small letters in boldface will denote column vectors. Such letters when primed denote transposes. For instance, $A(p \times q)$ denotes a matrix with $p$ rows and $q$ columns, $A'(q \times p)$ denotes the transpose of $A, \mathbf{a}(p \times 1)$ denotes a column vector with $p$ elements and $\mathbf{a}'(1 \times p)$, the transpose of $\mathbf{a}$, is a row vector. In particular, $I(p)$ will denote the identity matrix of order $p$ and $0(p \times 1)$ and $0(p \times q)$ will stand, respectively, for the null vector of order $p$ and the null matrix with $p$ rows and $q$ columns.

Unbiased estimators for functions of a location parameter $\theta$ and a scale parameter $\rho$ are expressed as unknown functions in integral equations of convolution type, and are then obtained by integral transform methods. An outline of the paper is contained in Section 3. The main results consist in the application of various derived expressions to the exponential distribution with parameters $\theta$ and $\rho$, the gamma and Weibull distributions with parameter $\rho$, and to general distributions with truncation parameter $\theta$. In the latter case, a simple formula is given for a minimum variance unbiased estimator of any absolutely continuous function of $\theta$; this extends slightly a result of Davis [3] concerning distributions of exponential type. Throughout the paper particular attention is paid to the estimation of the probability that a single observation will lie in a certain Borel set, when this probability is regarded as a function of the parameters $\theta$ and/or $\rho$. Extensions to sample points of $m$ observations and Borel sets in $m$-space are in most cases immediate.

Cramer ([6], p. 474 ff.), Darmois [8], Frechet [10], and Rao [14] derived independently a lower bound for the mean square error of an estimate $t$ of a parameter which appears in a frequency function of a specified form. This expression, alternately termed the Cramer-Rao inequality or the information limit, is \begin{equation*}\tag{1.1}]E(t - \alpha)^2 \geqq \lbrack E(t) - \alpha\rbrack^2 + \frac{\big\lbrack\frac{\partial E(t)}{\partial\alpha}\big\rbrack^2}{E\big(\frac{\partial\ln\phi} {\partial\alpha}\big)^2}\end{equation*} where $\phi$ is the likelihood of the sample. The expression $E(\partial \ln \phi/\partial\alpha)^2$ is called the information on $\alpha$ and is sometimes denoted by $I(\alpha)$. Under rather general conditions it can be shown equal to $E( - \partial^2 \ln \phi/\partial\alpha^2)$. The equality in (1.1) is reached if and only if, \begin{equation*}\tag{1.2}\phi = \phi_1e^{tV(\alpha) + W(\alpha)}\end{equation*} where $t$ and $\phi_1$ are functions of the observations alone and $V(\alpha)$ and $W(\alpha)$ are functions of $\alpha$ alone. By the results of Pitman [13] and Koopman [12], the form of (1.2) implies that $t$ must be a sufficient statistic. The fact that this form of the likelihood yields a minimum variance estimate was first pointed out by Aitken and Silverstone [1]. If we have $n$ observations which are independently and identically distributed, the frequency function of the underlying population must be of the so-called Pitman-Koopman form, \begin{equation*}\tag{1.3}f(x; \alpha) = (\alpha)h(x)e^{P(\alpha)g(x)}\end{equation*} and $t$ must be a function of $\sum^n_{i = 1} g(x_i)$ for the equality in (1.2) to hold. Several extensions of the basic inequality have been derived. Bhattacharyya [4] and Chapman and Robbins [5] have derived results which yield more stringent inequalities in certain instances. Wolfowitz [21] has extended the result to sequential sampling situations. Cramer [7], Darmois [8], and Barankin [2] have considered joint bounds on sets of estimates of parameters and Hammersley [11] has derived a lower bound of the mean square error of an estimate for the situation in which the parameter to be estimated can only assume discrete values. Barankin [3] has also considered lower bounds on the general absolute central moments of the estimate. All these results assume that the parameters involved are constants. Here we shall consider the case where the parameters are random variables. Thus the lower bound of the mean square error of an estimate will take into account the variability due to both the observations and the parameters involved. Necessary and sufficient conditions for equality of the extended inequality are derived. Most unfavorable distributions, i.e., distributions which maximize the lower bound, are defined, and several examples are given. Extensions analogous to those of Bhattacharyya [4] and Wolfowitz [21] are also considered. Finally, bounds on the variance of linear estimates of the mean of the parameter are derived.

If a variable $X$ has density function $f(x, \theta)$, then in many cases the Cramer-Rao bound or the Bhattacharyya bounds may be used to show that a function $d(x)$ is a uniformly minimum variance unbiased estimate of the real parameter $\theta$. In this paper it is shown that if $f(x, \theta)$ is a member of the family of densities of the Darmois-Koopman form, and if the variance of $d(x)$ achieves the $k$th Bhattacharyya bound, but not the $(k - 1)$th bound, then $f(x, \theta) = \exp\lbrack t(x)g(\theta) + g_0(\theta) + h(x)\rbrack$ and $d(x)$ is a polynomial in $t(x)$ of degree $k$. Further, the variance of any polynomial in $t(x)$ of degree $k$ will achieve the $k$th bound, so that if any such unbiased polynomial exists, it will necessarily be uniformly minimum variance unbiased. Some properties of these polynomial estimates are discussed.

The main purpose of this paper is to obtain the limiting distribution of certain statistics described in the title. It was suggested by the author in [1] that these statistics might be useful for testing the homogeneity hypothesis $H_1$ that $k$ random samples of real random variables have the same continuous probability law, or the goodness-of-fit hypothesis $H_2$ that all of them have some specified continuous probability law. Most tests of $H_1$ discussed in the existing literature, or at least all such tests known to the author before [1] in the case $k > 2$, have only been shown to have desirable consistency or power properties against limited classes of alternatives (see e.g., [2], [3], [4] for lists of references on these tests), while those suggested here are shown to be consistent against all alternatives and to have good power properties. Some test statistics whose distributions can be computed from known results are also listed.

Let $z_n$ denote the position at time $n$ of a particle describing a one-dimensional random walk, such that the increments $\zeta_n = z_n - z_{n-1} (n = 1, 2, \cdots)$ are independent random variables, assuming only the values +1 and -1, each with probability $\frac{1}{2}$. Of considerable importance in many applications is the conditional probability $$p_n(i, j, c) = P(z_n = j, 0 < z_m < c, m = 1, \cdots, n \mid z_0 = i);$$ here, $i, j, c, n$ denote positive integers. In section 1, an asymptotic development for $p_n(i, j, c)$ is given; for each positive integer $m$, it yields an approximation to $p_n(i, j, c)$ with error smaller than $Cn^{-m}$ where $C$ is independent of $i, j, c$ and $n$. As a simple application, an asymptotic development for the binomial coefficient $\binom{n}{s}$ is derived by letting $i, j, c$ tend to infinity in such a manner that $j - i = 2s - n$. As a second application, an asymptotic expansion is derived for the joint distribution of the extrema of the difference between the empirical distributions of two samples of size $n$. The above asymptotic development for $p_n(i, j, c)$ is obtained by applying the central Lemma 4 to an exact formula for $p_n(i, j, c)$. In Section 5, using this formula, an exact formula is obtained for the distribution of the range $R_n$ of the $n + 1$ numbers $z_0, \cdots, z_n$. Applying Lemma 4 to it, a complete asymptotic expansion for the distribution of $R_n$ is derived.

The purpose of this paper is to prove Theorem 1 stated in Section 1 below and Theorem 2 of Section 6 and the results of Section 7. These theorems are the generalizations to vector chance variables of Theorems 4 and 5 and Section 6 of [1], and state that the sample distribution function (d.f.) is asymptotically minimax for the large class of weight functions of the type described below. The main difficulties are embodied in the proof of Theorem 1 (Sections 2 to 5), where the loss function is a function of the maximum difference between estimated and true d.f. The proof utilizes the results of [2] and is not a straight-forward extension of the result of [1], because the sample d.f. is no longer "distribution free" (even in the limit), and hence it is necessary to prove the uniformity of approach, to its limit, of the d.f. of the normalized maximum deviation between sample and population d.f.'s (for a certain class of d.f.'s). The latter fact enables us essentially to infer the existence of a uniformly (with the sample number) approximately least favorable (to the statistician) d.f., by means of which the proof of the theorem is achieved. Theorem 2 (Section 6) considers loss functions of integral type, and more general loss functions are treated in Section 7.

A class of distributions is defined and studied which includes as particular cases (cf. Section 13) the ordinary $\beta$-distribution, the (univariate) triangular distribution, the uniform distribution over any nondegenerate simplex, and a continuous range of other distributions over such a simplex, called basic $\beta$-distributions (Section 6) and immediately analogous to the ordinary $\beta$-distribution. Our class also includes (Section 13 (vi)) various (univariate and other) distributions which arise in connection with the random division of an interval. The main results are given in Section 2 and further results for the univariate case are given in Section 8. This paper is exclusively concerned with the mathematical theory. One application may, however, be mentioned, which will be considered in more detail elsewhere. Suppose we wish to test the hypothesis $H_0$ that $n - 1$ numbers $y_1, \cdots, y_{n-1}$ (all lying between 0 and 1) were drawn independently from a rectangular distribution over (0, 1). Let $u_1, \cdots, u_n$ be the lengths of the $n$ intervals into which the $y_j$ divide the interval (0, 1). Then $H_0$ is equivalent to the hypothesis that the point with vector-coordinate $\mathbf{u}$ is distributed uniformly over a certain non-degenerate simplex $S$, and a useful set of alternative hypotheses is the set of basic $n$-dimensional $\beta$-distributions. Hence (using Section 4) this theory can be used to find the power-functions of certain tests of the hypothesis $H_0$.

Assume that a random sample of size $N$ has been drawn from a multinomial population with an unknown and perhaps countably infinite number of classes. Hence, if $X_j$ is the $j$th observation, and $M_i$ the $i$th class, then $$P\{X_j \varepsilon M_i\} = p_i \geqq 0\quad i = 1,2, \cdots; \text{for all} j,$$ and $\sum^\infty_1 p_i = 1$. If the number of classes is finite, then $p_i = 0$, for all $i > S$, where $S$ is the number of classes. We do not suppose the classes to have a natural ordering, since the classes may be species of insects, or chess openings. Let $n_r$ be the number of classes which occur exactly $r$ times in the sample. Then $$\sum^infty_{r = 0} rn_r = N \quad\text{and}\quad d = \sum^N_{r = 1} n_r$$ where $d$ is the number of distinct classes observed in the sample. It is the purpose of this paper to present some techniques to aid the experimenter in answering the following kinds of questions. 1) Prediction of the number of distinct classes that will be observed in a second sample of size $\alpha N, \alpha \geqq 1$. 2) Prediction of the number of additional classes that will be observed when the sample size is increased by $(\alpha - 1)N$ additional observations. 3) Estimation of the coverage of the sample, where coverage, denoted by $C$, is defined as follows: \begin{equation*}\tag{1}C = \sum_j p_j.\end{equation*} The sum is to be taken over those classes for which at least one representative has been observed. 4) Prediction of the coverage of a second sample of size $\alpha N$. 5) Prediction of the increased coverage to be obtained when the sample size is augmented by $(\alpha - 1)N$ additional observations. Let $d(\alpha)$ and $C(\alpha)$ denote the number of classes and coverage obtained from $\alpha N$ observations, either in the case of a second sample, or an augmented sample. Subsequently, we will show that \begin{equation*}\tag{2} Ed(\alpha) \backsim d + n_1 E\Big\{\frac{1 - e^{-(\alpha-1)x}}{x}\Big\}\end{equation*} and \begin{equation*}\tag{3}EC(\alpha) \backsim 1 - \frac{n_1}{N} + \frac{n_1}{N} E\{1 - e^{-(\alpha-1)x}\}\end{equation*} where the expectations on the right hand side of (2) and (3) are taken with respect to $F(x)$, a constructed cumulative distribution function on $\lbrack 0, \infty)$, which is unknown to the experimenter, but estimates of the moments of this distribution are available. It will be shown that \begin{equation*}\tag{4}\mu_r \backsim \frac{(r + 1)!(n_{r+1})}{n_1}\end{equation*} where $\mu_r = \int^\infty_0 x^r dF(x)$. Hence, a reasonable procedure is to compute upper (lower) predictors for $d(\alpha)$ and $C(\alpha)$ by computing the supremum (infimum) of (2) and (3) respectively, where the supremum (infimum) is taken over all cumulative distribution functions whose first $k$ moments are specified by (4). It will be shown that $$\varphi(x) = \frac{1 - e^{-(\alpha-1)x}}{x} \quad\text{and}\quad\psi(x) = 1 - e^{-(\alpha-1)x}$$ may be treated identically in the computation of extrema with the observation that the computation of the supremum of $E\{\varphi(x)\}$ will be identical with the computation of the infimum of $E\{\psi(x)\}$; and similarly for the computation of the infimum of $E\{\varphi(x)\}$. Thus, we will restrict the discussion to $\varphi(x)$. The solution will be of the form \begin{equation*}\tag{5}\sup(\inf) E\{\varphi(x)\} = \sum^r_{i = 1} \lambda_i\varphi(x_i)\end{equation*} where $\lambda_i \geqq 0, \sum^r_{i = 1} \lambda_i = 1$ and $x_i$ belongs to the extended non-negative real numbers. To determine the $x_i$ and $\lambda_i$, the following system of equations must be solved. \begin{equation*}\tag{6}\begin{split}\lambda_1x_1 + \lambda_2x_2 + \cdots + \lambda_rx_r = \mu_1\\ \lambda_1x^2_1 + \lambda_2x^2_2 + \cdots + \lambda_rx^2_r = \mu_2\\ ...........................\\ \lambda_1x^k_1 + \lambda_2x^k_2 + \cdots + \lambda_rx^k_r = \mu_k\end{split}\end{equation*} where $x_1 \leqq x_2 \leqq \cdots \leqq x_r$. If $k = 2c, c$ an integer, the supremum will be obtained by solving (6), with $x_1 = 0, r \leqq (k + 2)/2$, and $x_2, \cdots, x_r$ interior points of $\lbrack 0, \infty)$. If $k = 2c + 1$, the solution of (6) will coincide with that for $k - 1$ with the addition of a mass point at infinity, with mass tending to zero at a rate which will satisfy the $k$th moment constraint. Since $\varphi(\infty) = 0$, no change in (5) is obtained by the addition of the $k$th moment constraint. If $k = 2c + 1$, the infimum is obtained by solving (6) with $r \leqq (k + 1)/2$ and the $x_i$ are all interior points of $\lbrack 0, \infty)$. If $k = 2c, c > 0$, the solution of (6) is obtained from that for $k - 1$ by the addition of a mass point at infinity with mass tending to zero at a rate which will satisfy the last moment constraint. Explicit solutions are computed for $k \leqq 3$, and applied to several examples. In addition, the low order moments of $d, n_r$, and $C$ are computed, and the asymptotic sampling error of $\hat C(1) = 1 - (n_1/N)$ as an estimate of sample coverage is given. It will be convenient before proceeding to the general problem to compute the low order moments of $d, n_r, C$.

Chernoff [1] has called attention to a paper of Mann and Wald [5], which provides a general theory and a convenient notation for the derivation of theorems concerning stochastic limits and limit distributions. The present paper attempts to clarify the first of these topics, stochastic limits, by applying one form of the definition of convergence in probability to any event rather than just to convergence. As in the convergence case, most of the reasoning one is intuitively disposed to do in this connection is valid. Its justification is made more transparent but no more difficult by broadening the applicability of the definition. Thus broadened, "in probability" neither implies nor follows from "with probability one."

Let $\{x_\theta(t), \theta \varepsilon \Omega\}$ be a family of stochastic processes defined by their finite dimensional distributions; that is, $\{F_\theta\lbrack x(t_1), \cdots, x(t_n)\rbrack; \theta \varepsilon \Omega\}$ is given for all finite sets of time points $t_1, \cdots, t_n$. A general procedure for treating a statistical problem concerning this family has been to solve the problem for the finite dimensional families and then see what happens to the solution when limits are taken over suitably selected sets of time points. For example, if $\hat\theta\lbrack x(t_1), \cdots, x(t_n)\rbrack$ is an estimate of $\theta$ based on the finite dimensional family and it can be shown that the limit $$\hat\theta\lbrack x(t_1), \cdots, x(t_n)\rbrack \rightarrow \hat\theta\lbrack x(t)\rbrack$$ exists in some sense and is independent of the defining set $(t_1, t_2, \cdots)$, then this limit will usually provide an adequate estimate of $\theta$ for the process. Frequently the properties of the estimates $\hat\theta\lbrack x(t_1), \cdots, x(t_n)\rbrack$ can be extended to $\hat\theta\lbrack x(t)\rbrack$. An alternative approach to the problem is proposed by Grenander [1]. He introduces the likelihood ratio of two processes $P$ and $Q$ restricted to a finite number of time points, shows that it converges to a limit as the number of points goes to infinity and that this limit is the density of $P$ with respect to $Q$ if this density exists. He uses these results to derive numerous statistical results. The only criterion which he gives for the existence of the density is that the limit of the likelihood ratio be finite a.s. $P$. In applying this criterion he must always make use of some additional knowledge of the processes such as a.s. existence of certain integrals. In Section 2 these results are established very simply using the theory of martingales, and a criterion for the existence of the density is given which proves convenient in several applications. A condition also is given under which a density computed for a countable number of time points is valid for the continuous parameter process. Once the existence of the density is established standard statistical techniques can be applied directly. For example, sufficient statistics can often be found by inspection, or maximum likelihood methods can be used. Densities for a normal process with continuous covariance and unknown mean value function are derived in Section 3. Minimum variance unbiased estimates of regression coefficients are obtained. In [2] Cameron and Martin consider processes which are obtained from a Wiener process by linear transformations. They state conditions under which such a process is absolutely continuous with respect to the Wiener process, and they give a formula for computing the density. These results can be applied to the Ornstein Uhlenbeck process with covariance $\sigma^2e^{-\beta|s-t|}$ to obtain a family of densities for $2\beta\sigma^2 =$ constant. In Section 4 this family of densities is derived using the methods of Section 2. Using the same techniques it is shown that this family of Ornstein Uhlenbeck processes is mutually absolutely continuous. The maximum likelihood estimate of the correlation parameter $\beta$ is computed.

In a recent paper [1] Schumann and Bradley present a table of the ratio of central variance ratios for use in comparing the sensitivity of experiments. The purpose of this note is to point out some other applications of the Schumann-Bradley Table I [1] which stem from a variance components model.

The results reported by T. Austin, R. Fagen, T. Lehrer, and W. Penney [1] are extended to include a general recurrence relation for the factorial moments of the distribution. This recurrence relation is solved for the mean and second factorial moments, and it is shown that the method applied may also be used to obtain a general solution for any desired factorial moment of higher order.

$X_1, X_2, \cdots, X_n$ are independent chance variables, each with the same distribution. This common distribution assigns all the probability to the closed interval $\lbrack 0, 1\rbrack$, and has a density function $f(x)$ whose graph consists of any finite number of horizontal line segments. That is, there are $H$ non-degenerate subintervals $$I_1, I_2, \cdots, I_H, I_1 = \lbrack 0, z_1), I_2 = \lbrack z_1, z_2), \cdots, I_H = \lbrack z_{H-1}, 1 \rbrack,$$ and for each $x$ in $I_j, f(x) = a_j$. We assume that $a_j$ is positive for all $j$. Let $z_0$ denote zero, and $z_H$ denote unity. $M$ will denote $\min_ja_j, B$ will denote $$\sum_{j:a_j=M} (z_j - z_{j-1}),$$ and $S$ shall denote $\int^1_0 f^2(x) dx = \sum^H_{j=1} a^2_j(z_j - z_{j-1})$. Let $Y_1 \leqq Y_2 \leqq \cdots \leqq Y_n$ denote the ordered values of $X_1, \cdots, X_n$, and define $W_1 = Y_1, W_2 = Y_2 - Y_1, \cdots, W_n = Y_n - Y_{n-1}, W_{n + 1} = 1 - Y_n, U_n = \min (W_1, \cdots, W_{n + 1}), V_n = \max(W_1, \cdots, W_{n + 1})$. In [1] it is shown that if $f(x)$ is the uniform density function over $\lbrack 0, 1 \rbrack$, then $$\lim_{n\rightarrow\infty} P\big\lbrack U_n > \frac{u}{(n + 1)^2}, V_n < \frac{\log(n + 1) - \log v}{n + 1}\big\rbrack = \exp \{- (u + v)\},$$ for any positive numbers $u, v$. It is easy to see that the convergence must be uniform over any bounded rectangle in the space of $u$ and $v$. In this paper it is shown that if $f(x)$ is of the type described above, then $$\lim_{n\rightarrow\infty} P\Big\lbrack U_n > \frac{u}{(n + 1)^2}, V_n < \frac{\log(n + 1) + \log M - \log M - log v}{M(n + 1)}\Big\rbrack \\ \neq \exp \{-(Su + Bv)\},$$ for any positive values $u, v$. This result can be used to study the asymptotic power of various tests of fit based on $U_n$ and $V_n$ which have been proposed (see [1], p. 253).

Apart from pre- and post-multiplication by a fixed matrix and its transpose, the Wishart matrix $\mathbf{A}$ can be written as the product of a triangular matrix and its transpose, whose elements are independent normal and chi variables. Various applications of this representation are indicated. Examples are given concerning the diagonal elements of $\mathbf{A}^{-1}$, the sample ordinary and multiple correlation coefficient, the characteristic roots of $\mathbf{A}$ and the sphericity criterion in the bivariate case.