Showing papers in "Annals of Mathematical Statistics in 1946"
TL;DR: In this paper, several statistical techniques are proposed for economically analyzing large masses of data by means of punched-card equipment; most of these techniques require only a counting sorter, and they are designed especially for situations where data are inexpensive compared to the cost of analysis by using statistically efficient or powerful procedures.
Abstract: Several statistical techniques are proposed for economically analyzing large masses of data by means of punched-card equipment; most of these techniques require only a counting sorter. The methods proposed are designed especially for situations where data are inexpensive compared to the cost of analysis by means of statistically “efficient” or “most powerful” procedures. The principal technique is the use of functions of order statistics, which we call systematic statistics.
317 citations
TL;DR: In this paper, statistical test criteria are developed for testing equality of means, equality of variances and equality of covariances in a normal multivariate population of $k$ variables on the basis of a sample.
Abstract: In this paper statistical test criteria are developed for testing equality of means, equality of variances and equality of covariances in a normal multivariate population of $k$ variables on the basis of a sample. More specifically, three statistical hypotheses are considered: (i) $H_{mvc}$, the hypothesis that the means are equal, the variances are equal, and covariances are equal, (ii) $H_{vc}$, the hypothesis that variances are equal and covariances are equal, irrespective of the values of the means, and (iii) $H_m$, the hypothesis of equal means, assuming variances are equal and covariances are equal. Test criteria $L_{mvc}$, $L_{vc}$, and $L_m$ are developed by the Neyman-Pearson method of likelihood ratios for testing $H_{mvc}$, $H_{vc}$ and $H_m$ respectively. The exact moments of each of the three test criteria when the three corresponding hypotheses are true are determined for any number $k$ of variables and for any size, $n$, of the sample for which the distributions exist. The exact distributions of $L_{mvc}$ and $L_{vc}$ are determined for $k = 2$ and $k = 3$, and the exact distribution of $L_m$ is found for any $k$; these are all beta (Pearson Type I) distributions. Tables of 5% and 1% points of $L_{mvc}$, $L_{vc}$ and $L_m$, based on Thompson's tables of percentage points of the Incomplete Beta Function, are given for certain values of $k$ and $n$ (Tables I and II). Also tables of values of approximate 5% and 1% points of $-n \ln L_{mvc}, -n \ln L_{vc}$ and $-n(k - 1) \ln L_m$ for large values of $n$ are given (Table III), based on the fact that these three quantities are approximately distributed according to chi-square laws for large values of $n$ with $\frac{1}{2}k(k + 3) - 3, \frac{1}{2}k(k + 1) - 2$, and $k - 1$ degrees of freedom respectively. A table (Table IV) is given which shows how accurate the resulting approximate 5% and 1% points of $L_{mvc}, L_c$ and $L_m$ are. The paper is written in two parts. In Part I the problem of testing the three hypotheses is discussed and the mathematical results are presented together with an illustrative example. Part II is given for the reader who wishes to study the mathematical derivation of the results.
305 citations
Abstract: A type of population frequently encountered in extensive samplings is one in which the variance within a group of elements increases steadily as the size of the group increases. This class of populations may be represented by a model in which the elements are serially correlated, the correlation between two elements being a positive and monotone decreasing function of the distance apart of the elements. For populations of this type, the relative efficiencies are compared for a systematic sample of every $k$th element, a stratified random sample with one element per stratum and a random sample. The stratified random sample is always at least as accurate on the average as the random sample and its relative efficiency is a monotone increasing function of the size of the sample. No general result is valid for the relative efficiency of the systematic sample. In fact, there are populations in the class in which the systematic sample is more accurate than the stratified sample for one sampling rate, but is less accurate than the random sample for another sampling rate. If, however, the correlogram is in addition concave upwards, the systematic sample is on the average more accurate than the stratified sample for any size of sample. Some numerical results are given for the cases in which the correlogram is (i) linear (ii) exponential.
271 citations
TL;DR: In this article, it was shown that the symmetric estimate of the first moment is the best in the sense of having the least variance, and it was proved that among them, among all the non-symmetric estimates, the symmetry of the variance is the smallest.
Abstract: Let $F(P)$ be a real valued function defined on a subset $\mathscr{D}$ of the set $\mathscr{D}^\ast$ of all probability distributions on the real line. A function $f$ of $n$ real variables is an unbiased estimate of $F$ if for every system, $X_1, \cdots, X_n$, of independent random variables with the common distribution $P$, the expectation of $f(X_1 \cdots, X_n)$ exists and equals $F(P)$, for all $P$ in $\mathscr{D}$. A necessary and sufficient condition for the existence of an unbiased estimate is given (Theorem 1), and the way in which this condition applies to the moments of a distribution is described (Theorem 2). Under the assumptions that this condition is satisfied and that $\mathscr{D}$ contains all purely discontinuous distributions it is shown that there is a unique symmetric unbiased estimate (Theorem 3); the most general (non symmetric) unbiased estimates are described (Theorem 4); and it is proved that among them the symmetric one is best in the sense of having the least variance (Theorem 5). Thus the classical estimates of the mean and the variance are justified from a new point of view, and also, from the theory, computable estimates of all higher moments are easily derived. It is interesting to note that for $n$ greater than 3 neither the sample $n$th moment about the sample mean nor any constant multiple thereof is an unbiased estimate of the $n$th moment about the mean. Attention is called to a paradoxical situation arising in estimating such non linear functions as the square of the first moment.
250 citations
TL;DR: The problem of constructing tolerance limits for a normal universe is considered in this article, where the tolerance limits are required to be such that the probability is equal to a pre-assigned value (i.e., the probability that the tolerance limit includes at least a given proportion of the population).
Abstract: The problem of constructing tolerance limits for a normal universe is considered. The tolerance limits are required to be such that the probability is equal to a preassigned value $\beta$ that the tolerance limits include at least a given proportion $\gamma$ of the population. A good approximation to such tolerance limits can be obtained as follows: Let $\bar x$ denote the sample mean and $s^2$ the sample estimate of the variance. Then the approximate tolerance limits are given by $\bar x - \sqrt\frac{n}{\chi^2_{n,\beta}} rs \text{and} \bar x + \sqrt\frac{n}{\chi^2_{n,\beta}} rs$ where $n$ is one less than the number $N$ of observations, $\chi^2_{n,\beta}$ denotes the number for which the probability that $\chi^2$ with $n$ degrees of freedom will exceed this number is $\beta$, and $r$ is the root of the equation $\frac{1}{\sqrt{2\pi}} \int^{1/\sqrt{N} + r}_{1/\sqrt{N}-r} e^{-t^2/2} dt = \gamma.$ The number $\chi^2_{n,\beta}$ can be obtained from a table of the $\chi^2$ distribution and $r$ can be determined with the help of a table of the normal distribution.
214 citations
TL;DR: The non-central Wishart distribution as mentioned in this paper is the joint distribution of the sums of squares and cross-products of the deviations from the sample means when the observations arise from a set of normal multivariate populations with constant covariance matrix but expected values that vary from observation to observation.
Abstract: The non-central Wishart distribution is the joint distribution of the sums of squares and cross-products of the deviations from the sample means when the observations arise from a set of normal multivariate populations with constant covariance matrix but expected values that vary from observation to observation. The characteristic function for this distribution is obtained from the distribution of the observations (Theorem 1). By using the characteristic functions it is shown that the convolution of several non-central Wishart distributions is another non-central Wishart distribution (Theorem 2). A simple integral representation of the distribution in the general case is given (Theorem 3). The integrand is a function of the roots of a determinantal equation involving the matrix of sums of squares and cross-products of deviations of observations and the matrix of sums of squares and cross-products of deviations of corresponding expected values. The knowledge of the non-central Wishart distribution is applied to two general problems of multivariate normal statistics. The moments of the generalized variance, which is the determinant of sums of squares and cross-products multiplied by a constant, are given for the cases of the expected values of the variates lying on a line (Theorem 4) and lying on a plane (Theorem 5). The likelihood ratio criterion for testing linear hypotheses can be expressed as the ratio of two determinants or as a symmetric function of the roots of a determinantal equation. In either case there is involved a matrix having a Wishart distribution and another matrix independently distributed such that the sum of these two matrices has a non-central Wishart distribution. When the null hypothesis is not true the moments of this criterion are given in the non-central planar case (Theorem 6).
140 citations
TL;DR: In this paper, a necessary and sufficient condition that p be the unique unbiased estimate for p is given. But this condition is not satisfied, and it is not known whether more than one unbiased estimate is possible.
Abstract: We would like to call attention to a few problems raised by but not solved in this paper: 1) find a necessary and sufficient condition that \( \hat p\) be the unique unbiased estimate for p; 2) suggest criteria for selecting one unbiased estimate when more than one is possible; 3) evaluate the variance of \( \hat p\).
127 citations
TL;DR: In this article, the authors considered the chemical balance problem and the spring balance problem with bias and provided designs for a balance which has a bias and for one which has no bias.
Abstract: The paper contains some solutions of the weighing problems proposed by Hotelling [1]. The experimental designs are applicable to a broad class of problems of measurement of similar objects. The chemical balance problem (in which objects may be placed in either of the two pans of the balance) is almost completely solved by means of designs constructed from Hadamard matrices. Designs are provided both for a balance which has a bias and for one which has no bias. The spring balance problem (in which objects may be placed in only one pan) is completely solved when the balance is biased. For an unbiased spring balance, designs are given for small numbers of objects and weighing operations. Also the most efficient designs are found for the unbiased spring balance, but it is shown that in some cases these cannot be used unless the number of weighings is as large as the binomial coefficient $\binom {p}{\frac{1}{2}p}$ or $\binom {p}{\frac{1}{2}(p + 1)}$ where $p$ is the number of objects. It is found that when $p$ objects are weighed in $N \geq p$ weighings, the variances of the estimates of the weights are of the order of $\sigma^2/N$ in the chemical balance case $(\sigma^2$ is the variance of a single weighing), and of the order of $4\sigma^2/N$ in the spring balance case.
112 citations
TL;DR: In the Army demobilization point system as mentioned in this paper, each of the individuals compares or ranks or ranks $n$ things; the problem is to determine a numerical value for each of these things that will best represent the comparisons in some sense.
Abstract: Research for the Army demobilization point system evolved a new approach to paired comparisons and rank order. Each of $N$ individuals compares or ranks $n$ things; the problem is to determine a numerical value for each of the $n$ things that will best represent the comparisons in some sense. The new criterion adopted is that the numerical values be determined so as best to distinguish between those things judged higher and those judged lower for each individual. Least-squares is employed in the analysis, and the solution appears in the form of the latent vector associated with the largest root of a matrix obtained from the comparisons or rankings. This approach applies to the conventional problem of ordinary comparisons, the numerical solution being easily obtainable by simple iterations; the conventional use of hypothetical variables and unverified hypotheses is avoided. The Army point system is an example of a new and more complicated class of problems; the same principle for the solution applies here, only more details occur in the derivations and computations.
98 citations
86 citations
TL;DR: The enlargement principle provides techniques for inverting any nonsingular matrix by building the inverse upon the inverses of successively larger submatrices as discussed by the authors, and three different enlargement routines are outlined: first-order, second-order and geometric.
Abstract: The enlargement principle provides techniques for inverting any nonsingular matrix by building the inverse upon the inverses of successively larger submatrices. The computing routines are relatively easily learned since they are repetitive. Three different enlargement routines are outlined: first-order, second-order, and geometric. None of the procedures requires much more work than is involved in squaring the matrix.
TL;DR: In this paper, the sequential probability ratio (SPR) criterion was proposed to test the composite hypothesis of the binomial and the exponential distribution of the distributions of two populations, and it was shown that the power of this test is constant on a curve $h(\theta_1, \theta-2) =$ constant.
Abstract: Given two populations $\pi_1$ and $\pi_2$ each characterized by a distribution density $f(x, \theta)$ which is assumed to be known, except for the value of the parameter $\theta$. It is desired to test the composite hypothesis $\theta_1 \theta_2$ where $\theta_i$ is the value of the parameter in the distribution density of $\pi_i, (i = 1, 2)$. The criterion proposed for testing this hypothesis is based on the sequential probability ratio and consists of the following: Choose two positive constants $a$ and $b$ and two values of $\theta$, say $\theta^0_1$ and $\theta^0_2$. Take pairs of observations $x_{1\alpha}$ from $\pi_1$ and $x_{2\alpha}$ from $\pi_2, (\alpha = 1,2, \ldots)$, in sequence and compute $Z_j = \sum^j_{\alpha = 1} z_\alpha$ where $z_\alpha = \log \big\lbrack \frac{f(x_{2\alpha}, \theta^0_1)f(x_{1\alpha}, \theta^0_2)} {f(x_{2\alpha}, \theta^0_2)f(x_{1\alpha}, \theta^0_1)}\big\rbrack.$ The hypothesis tested is accepted or rejected depending on whether $Z_n \geq a$ or $Z_n \leq - b$ where $n$ is the smallest integer $j$ for which either one of these relationships is satisfied. The boundaries $a$ and $b$ are partly given in terms of the desired risks of making an erroneous decision. The values $\theta^0_1$ and $\theta^0_2$ define the magnitude of the difference between the values of $\theta$ in $\pi_1$ and in $\pi_2$ which is considered worth detecting. It is shown that the power of this test is constant on a curve $h(\theta_1, \theta_2) =$ constant. If $E\big(\log \frac{f(x, \theta^0_2)}{f(x, \theta^0_1)}\big)$ is a monotonic function of $\theta$, then the test is unbiased in the sense that all points $(\theta_1, \theta_2)$ which lie on the curve $h(\theta_1, \theta_2) =$ constant are such that either every $\theta_1 \theta_2$. For a large class of known distributions the quantity $h$ is shown to be an appropriate measure of the difference between $\theta_1$ and $\theta_2$ and the test procedure for this class of distributions is simple and intuitively sensible. For the case of the binomial, the exact power of this test as well as the distribution of $n$ is given.
TL;DR: In this article, it was shown that any distribution with a very long tail has the distributions of its sample means grow flatter and flatter as the sample size increases, thus the sample mean provides less information than a single value.
Abstract: It is shown that certain monomials in normally distributed quantities have stable distributions with index $2^{-k}$. This provides, for $k > 1$, simple examples where the mean of a sample has a distribution equivalent to that of a fixed, arbitrarily large multiple of a single observation. These examples include distributions symmetrical about zero, and positive distributions. Using these examples, it is shown that any distribution with a very long tail (of average order $\geq x^{-3/2}$) has the distributions of its sample means grow flatter and flatter as the sample size increases. Thus the sample mean provides less information than a single value. Stronger results are proved for still longer tails.
TL;DR: In this article, the problem of estimating from a sample a confidence region for the parameters of the distribution of maximum values is treated by setting up what are called "statistical estimation functions" suggested by the functional form of the probability distribution of the sample, and finding the moment generating function of these estimation "functions".
Abstract: The problem of estimating from a sample a confidence region for the parameters of the distribution of maximum values is treated by setting up what are called "statistical estimation functions" suggested by the functional form of the probability distribution of the sample, and finding the moment generating function of the probability distribution of these estimation "functions. Such an estimate by the method of maximum likelihood is also treated. A definition of "sufficiency" is proposed for "statistical estimation functions" analogous to that which applies to "statistics". Also the concept of "stable statistical estimation functions" is introduced. By means of a numerical illustration, four methods are discussed for setting up an approximate confidence interval for the estimated value of $x$ of the universe of maximum values which corresponds to a given cumulative frequency .99, for confidence level .95. Two procedures for solving this problem are recommended as practicable.
TL;DR: In this article, the authors present a comprehensive set of curves illustrating graphically the power of each test for the 5% significance level for the power curves of common tests of significance involving the $\chi^2, F, t,$ and normal distributions.
Abstract: Methods making possible quick calculation of operating characteristics of power curves of common tests of significance involving the $\chi^2, F, t,$ and normal distributions are presented. In addition, a comprehensive set of curves illustrating graphically the power of each test for the 5% significance level are included. We are interested in the power of: (1) the $\chi^2$-test to determine whether an unknown population standard deviation is greater or less than a standard value, (2) the $F$ test to determine whether one unknown population standard deviation is greater than another (one-sided alternative), and (3) the $t$-test and normal test to determine whether an unknown population mean differs from a standard or two unknown population means differ from each other. Such operating characteristics have application for the quality control engineer and statistician in the design of sampling inspection plans using variables where they may be used to determine the sample size that will guarantee a specified consumer's and producer's risk. On the other hand they are of use in displaying the power of a test if the sample size has already been set. Finally, they are a necessary adjunct to the proper interpretation of the common tests of significance.
TL;DR: In this paper, it was shown that if the population distribution is non-singular and if the existence of a sufficient number of moments is assumed, then an asymptotic expansion can be obtained with the appropriate remainder.
Abstract: It is well known that various statistics of a large sample (of size $n$) are approximately distributed according to the normal law. The asymptotic expansion of the distribution of the statistic in a series of powers of $n^{-\frac{1}{2}}$ with a remainder term gives the accuracy of the approximation. H. Cramer [1] first obtained the asymptotic expansion of the mean, and recently P. L. Hsu [2] has obtained that of the variance of a sample. In the present paper we extend the Cramer-Hsu method to Student's statistic. The theorem proved states essentially that if the population distribution is non-singular and if the existence of a sufficient number of moments is assumed, then an asymptotic expansion can be obtained with the appropriate remainder. The first four terms of the expansion are exhibited in formula (35).
TL;DR: In this paper, it was shown that the equality was never in error by as much as three-fourths of one percent, and other approximations were discussed, such as 1/2 and 1/1/2.
Abstract: It is shown that $\frac{1}{\sqrt{2\pi}} \int_{-x}^x e^{-\frac{1}{2}t^2} dt \geq \lbrack 1 - e^{-(2/\pi)x^2}\rbrack^\frac{1}{2}$ and that the equality is never in error by as much as three-fourths of one percent. Other approximations are discussed.
TL;DR: In this article, a new recursion formula is used to give the exact distribution of arrangements of $n$ numbers, no two alike, with runs up or down of length $p$ or more.
Abstract: Using the notation of Levene and Wolfowitz [1], a new recursion formula is used to give the exact distribution of arrangements of $n$ numbers, no two alike, with runs up or down of length $p$ or more. These are tabled for $n$ and $p$ through $n = 14$. An exact solution is given for $p \geq n/2$. The average and variance determined by Levene and Wolfowitz are presented in a simplified form. The fraction of arrangements of $n$ numbers with runs of length $p$ or more are presented for the exact distributions, for the limiting Poisson Exponential, and for an extrapolation from the exact distributions. Agreement among the tables is discussed.