Showing papers in "Annals of Mathematical Statistics in 1952"
TL;DR: In this paper, it was shown that the likelihood ratio test for fixed sample size can be reduced to this form, and that for large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample with the second test.
Abstract: In many cases an optimum or computationally convenient test of a simple hypothesis $H_0$ against a simple alternative $H_1$ may be given in the following form. Reject $H_0$ if $S_n = \sum^n_{j=1} X_j \leqq k,$ where $X_1, X_2, \cdots, X_n$ are $n$ independent observations of a chance variable $X$ whose distribution depends on the true hypothesis and where $k$ is some appropriate number. In particular the likelihood ratio test for fixed sample size can be reduced to this form. It is shown that with each test of the above form there is associated an index $\rho$. If $\rho_1$ and $\rho_2$ are the indices corresponding to two alternative tests $e = \log \rho_1/\log \rho_2$ measures the relative efficiency of these tests in the following sense. For large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample of size $en$ with the second test. To obtain the above result, use is made of the fact that $P(S_n \leqq na)$ behaves roughly like $m^n$ where $m$ is the minimum value assumed by the moment generating function of $X - a$. It is shown that if $H_0$ and $H_1$ specify probability distributions of $X$ which are very close to each other, one may approximate $\rho$ by assuming that $X$ is normally distributed.
3,760 citations
TL;DR: In this article, a general method for calculating the limiting distributions of these criteria is developed by reducing them to corresponding problems in stochastic processes, which in turn lead to more or less classical eigenvalue and boundary value problems for special classes of differential equations.
Abstract: The statistical problem treated is that of testing the hypothesis that $n$ independent, identically distributed random variables have a specified continuous distribution function $F(x)$. If $F_n(x)$ is the empirical cumulative distribution function and $\psi(t)$ is some nonnegative weight function $(0 \leqq t \leqq 1)$, we consider $n^{\frac{1}{2}} \sup_{-\infty
3,082 citations
2,735 citations
TL;DR: In this article, the authors give a scheme whereby, starting from an arbitrary point, one obtains successively $x_2, x_3, \cdots$ such that the regression function converges to the unknown point in probability as n \rightarrow \infty.
Abstract: Let $M(x)$ be a regression function which has a maximum at the unknown point $\theta. M(x)$ is itself unknown to the statistician who, however, can take observations at any level $x$. This paper gives a scheme whereby, starting from an arbitrary point $x_1$, one obtains successively $x_2, x_3, \cdots$ such that $x_n$ converges to $\theta$ in probability as $n \rightarrow \infty$.
2,141 citations
TL;DR: The chi square test of goodness of fit as discussed by the authors is intended for the student and user of statistical theory rather than for the expert, and it has been used extensively in the application of the test.
Abstract: This paper contains an expository discussion of the chi square test of goodness of fit, intended for the student and user of statistical theory rather than for the expert. Part I describes the historical development of the distribution theory on which the test rests. Research bearing on the practical application of the test--in particular on the minimum expected number per class and the construction of classes--is discussed in Part II. Some varied opinions about the extent to which the test actually is useful to the scientist are presented in Part III. Part IV outlines a number of tests that have been proposed as substitutes for the chi square test (the $\omega^2$ test, the smooth test, the likelihood ratio test) and Part V a number of supplementary tests (the run test, tests based on low moments, subdivision of chi square into components).
834 citations
TL;DR: For the estimation of a single quantity of form, the optimum allocation comprises two or three sources as discussed by the authors, and the corresponding number is 2 or 3 for estimation of both parameters, the best proportions are indicated in Sections 2 and 4 below.
Abstract: If for the estimation of $\beta_1, \beta_2$ different observations (``sources'') of form (1.1) are potentially available, each of them being repeatable as many times as we please, the question arises which of them the experimenter should utilize, and in what proportions. With appropriate optimality conventions the answer is the following. For the estimation of a single quantity of form $\theta = \alpha_1\beta_1 + \alpha_2\beta_2$ the optimum allocation comprises two sources only; for the estimation of both parameters, the corresponding number is two or three; the best proportions are indicated in Sections 2 and 4 below. Generalizations to more than two parameters and to observations at different costs are briefly discussed. The problem is related to Hotelling's weighing problem [2] and to the topics treated by David and Neyman in [1].
435 citations
TL;DR: In this article, a test of the null hypothesis against alternatives of the form $F_i(x) = F(x - \theta_i)-quad (\text{all} x, i = 1, \cdots, C) was discussed.
Abstract: Suppose that $C$ independent random samples of sizes $n_1, \cdots, n_c$ are to be drawn from $C$ univariate populations with unknown cumulative distribution functions $F_1, \cdots, F_c$. This paper discusses a test of the null hypothesis $F_1 = F_2 = \cdots = F_c$ against alternatives of the form $F_i(x) = F(x - \theta_i)\quad (\text{all} x, i = 1, \cdots, C)$ with the $\theta_i$'s not all equal, or against alternatives of a much more general sort to be specified in Section 5. The test to be discussed has as its critical region large values of the ordinary $F$-ratio for one-way analysis of variance, computed after the observations have been replaced by their ranks in the $\sum n_i$-fold over-all sample. This use of ranks simplifies the distribution theory, and permits application of the test to cases where the ranks are available but the numerical values of the observations are difficult to obtain. Briefly, then, we shall consider a non-parametric analogue, based on ranks, of one-way analysis of variance. It is shown in Section 4 that, under quite general conditions, the proposed test statistic, $H$, is asymptotically chi-square with $C - 1$ degrees of freedom when the null hypothesis holds. Section 5 derives a necessary and sufficient condition that the natural family of sequences of tests based on large values of $H$ all be consistent against a given alternative. Section 6 derives the variance of $H$ under the null hypothesis, Section 7 derives the maximum value of $H$, and Section 8 gives a difference equation which may be used to obtain exact small-sample distributions under the null hypothesis. These derivations are made on the assumption of continuity for the cumulative distribution functions; Section 9 considers extensions to the possibly discontinuous case.
434 citations
TL;DR: In this paper, a class of statistical quality control procedures and continuous inspection procedures which are optimum for a specified income function and a production model which can only be in one of four states, two of which are states of repair, with known transition probabilities are presented.
Abstract: This paper deals with a class of statistical quality control procedures and continuous inspection procedures which are optimum for a specified income function and a production model which can only be in one of four states, two of which are states of repair, with known transition probabilities. The Markov process, generated by the model and the class of decision procedures, approaches a limiting distribution and the integral equations from which the optimum procedures can be derived are given.
366 citations
TL;DR: In this article, the power of a family of nonparametric tests, known as tests based on permutations of observations, was investigated and shown to be asymptotically (as the sample size tends to ∞) as powerful as certain related standard parametric tests.
Abstract: The paper investigates the power of a family of nonparametric tests which includes those known as tests based on permutations of observations. Under general conditions the tests are found to be asymptotically (as the sample size tends to ∞) as powerful as certain related standard parametric tests. The results are based on a study of the convergence in probability of certain random distribution functions. A more detailed summary will be found at the end of the Introduction.
349 citations
TL;DR: In this article, the authors generalize the notion of a set of orthogonal Latin squares, and give a method of constructing these arrays by considering a class of polynomials with coefficients in the finite Galois field.
Abstract: In this paper we shall proceed to generalize the notion of a set of orthogonal Latin squares, and we term this extension an orthogonal array of index unity. In Section 2 we secure bounds for the number of constraints which are the counterpart of the familiar theorem which states that the number of mutually orthogonal Latin squares of side $s$ is bounded above by $s - 1$. Curiously, our bound depends upon whether $s$ is odd or even. In Section 3 we give a method of constructing these arrays by considering a class of polynomials with coefficients in the finite Galois field $GF(s)$, where $s$ is a prime or a power of a prime. In the concluding section we give a brief discussion of designs based on these arrays.
336 citations
TL;DR: In this article, the known upper bounds for the maximum possible number of constraints for arrays of strength 2 and 3 have been improved, and certain methods for constructing these arrays have been given.
Abstract: Orthogonal arrays can be regarded as natural generalizations of orthogonal Latin squares, and are useful in various problems of experimental design. In this paper the known upper bounds for the maximum possible number of constraints for arrays of strength 2 and 3 have been improved, and certain methods for constructing these arrays have been given.
TL;DR: Doob's approach can be extended to a wide class of asymptotic theorems as discussed by the authors, such as Kolmogorov's and Smirnov's.
Abstract: Doob [1] has given heuristically an appealing methodology for deriving asymptotic theorems on the difference between the empirical distribution function calculated from a sample and the actual distribution function of the population being sampled. In particular he has applied these methods to deriving the well known theorems of Kolmogorov [2] and Smirnov [3]. In this paper we give a justification of Doob's approach to these theorems and show that the method can be extended to a wide class of such asymptotic theorems.
TL;DR: In this article, the authors propose to re-strict attention to decision procedures whose maximum risk does not exceed the minimax risk by more than a given amount, where the average risk is minimized with respect to some guessed a priori distribution suggested by previous experience.
Abstract: Instead of minimizing the maximum risk it is proposed to re-strict attention to decision procedures whose maximum risk does not exceed the minimax risk by more than a given amount. Subject to this restriction one may wish to minimize the average risk with respect to some guessed a priori distribution suggested by previous experience. It is shown how Wald’s minimax theory can be modified to yield analogous results concerning such restricted Bayes solutions. A number of examples are discussed, and some extensions of the above criterion are briefly considered.
TL;DR: Group divisible incomplete block designs (GD) as discussed by the authors are a subclass of partially balanced block designs with two associate classes, and they may also be regarded as a special case of intra-and inter-group balanced incomplete blocks designs.
Abstract: Group divisible incomplete block designs are an important subclass of partially balanced designs [1], [2] with two associate classes, and they may also be regarded as a special case of intra- and inter-group balanced incomplete block designs [3], [4]. They may be defined as follows. An incomplete block design with $v$ treatments each replicated $r$ times in $b$ blocks of size $k$ is said to be group divisible (GD) if the treatments can be divided into $m$ groups, each with $n$ treatments, so that the treatments belonging to the same group occur together in $\lambda_1$ blocks and treatments belonging to different groups occur together in $\lambda_2$ blocks. If $\lambda_1 = \lambda_2 = \lambda$ (say), then every pair of treatments occurs in $\lambda$ blocks, and the design becomes a balanced incomplete block design, which has been extensively studied [5], [6], [7], [8]. We shall therefore confine ourselves to the case $\lambda_1
eq \lambda_2$. The object of this paper is to study the combinatorial properties of these designs. It is shown that the GD designs can be divided into three exhaustive and mutually exclusive classes: (a) Singular GD designs characterized by $r - \lambda_1 = 0$; (b) Semi-regular GD designs characterized by $r - \lambda_1 > 0, rk - v\lambda_2 = 0$; (c) Regular GD designs characterized by $r - \lambda_1 > 0, rk - v\lambda_2 > 0$. Certain inequality relations between the parameters necessary for the existence of the design have been derived in each case. Some other interesting theorems about the structure of these designs have also been obtained. Methods of constructing GD designs will be given in a separate paper.
TL;DR: In this article, the authors consider the problem of estimation of parameters from a sample in which only the first order observations are known and show that the maximum likelihood estimate of a given parameter is consistent, asymptotically normally distributed, and of minimum variance for large samples.
Abstract: In this paper we consider the problem of estimation of parameters from a sample in which only the first $r$ (of $n$) ordered observations are known. If $r = \lbrack qn \rbrack, 0 < q < 1$, it is shown under mild regularity conditions, for the case of one parameter, that estimation of $\theta$ by maximum likelihood is best in the sense that $\hat{\theta}$, the maximum likelihood estimate of $\theta$, is (a) consistent, (b) asymptotically normally distributed, (c) of minimum variance for large samples. A general expression for the variance of the asymptotic distribution of $\hat{\theta}$ is obtained and small sample estimation is considered for some special choices of frequency function. Results for two or more parameters and their proofs are indicated and a possible extension of these results to more general truncation is suggested.
TL;DR: In this article, the authors derived the most powerful rank order test against specific parametric alternatives and showed that the test criterion was $c_1(R) = \sum d_iEZN, r_i}, where EZN is the expected value of the standard normal order statistic and R = (r_1, \cdots, d_N)$ is the permutation of the ranks.
Abstract: The most powerful rank order tests against specific parametric alternatives are derived. Following the methods of Hoeffding [4], we derive the most powerful rank order test of whether $N$ observations come from the same but unknown population against the alternative that the observations $Z_1, \cdots, Z_N$ come from populations which have the joint density $\Pi^N_{i = 1} \frac{1}{\sigma\sqrt{2\pi}} \exp \big\lbrack - \frac{1}{2\sigma^2} (z_i - d_i\xi - \eta)^2 \big\rbrack,$ where $d_1, \cdots, d_N$ are given constants, not all equal, and $\xi/\sigma$ is sufficiently small. The test criterion was found to be $c_1(R) = \sum d_iEZ_{N, r_i}$, where $EZ_{Ni}$ is the expected value of the $i$th standard normal order statistic and $R = (r_1, \cdots, r_N)$ is the permutation of the ranks. The distribution of this statistic was shown to be asymptotically normal providing the known constants $d_1, \cdots, d_N$ satisfied Noether's condition [9]. The two-sample distribution is a special case, and the resultant statistic $c_1(R)$ is shown to be asymptotically normal. The approximation of the distribution of the $c_1(R)$ statistic to the distribution $C(1 - x^2)^{\frac{1}{2}N-2}, - 1 \leqq x \leqq 1$, is investigated. This statistic is then compared to the existing Mann and Whitney $U$ statistic. No method having been found for analytical evaluation of the power of this test, the power was examined experimentally. Tables are appended giving the exact distribution of the $c_1(R)$ statistic for all possible subsample sizes whose total size is less than or equal to 10 together with the corresponding Mann and Whitney $U$ value. Table 2 gives critical values of $c_1(R)$ for $N \leqq 10, p \leqq.10$.
TL;DR: In this paper, it is shown under certain regularity assumptions that unbiased tests of H do not exist and other types of minimax tests are derived under suitable monotonicity conditions.
Abstract: Let the distribution of some random variables depend on real parameters θ1, • • •, θs and consider the hypothesis H: θi ≦ θi*, i = 1, • • •,s. It is shown under certain regularity assumptions that unbiased tests of H do not exist. Tests of minimum bias and other types of minimax tests are derived under suitable monotonicity conditions. Certain related multidecision problems are discussed and two-sided hypotheses are considered very briefly.
TL;DR: In this paper, a multidimensional analogue of the von Mises statistic is considered for the case of sampling from a multi-dimensional uniform distribution and the limiting distribution of the statistic is shown to be that of a weighted sum of independent chi-square random variables with one degree of freedom.
Abstract: A multidimensional analogue of the von Mises statistic is considered for the case of sampling from a multidimensional uniform distribution The limiting distribution of the statistic is shown to be that of a weighted sum of independent chi-square random variables with one degree of freedom The weights are the eigenvalues of a positive definite symmetric function A modified statistic of the von Mises type useful in setting up a two sample test is shown to have the same limiting distribution under the null hypothesis (both samples come from the same population with a continuous distribution function) as that of the one-dimensional von Mises statistic We call the statistics mentioned above von Mises statistics because they are modifications of the $\omega^2$ criterion considered by von Mises [5] The paper makes use of elements of the theory of stochastic processes
TL;DR: In this article, the authors investigated certain statistical problems arising in the determination of the ''best'' of $k$ categories when comparing $k - 1$ experimental categories with a standard or control.
Abstract: This paper investigates certain statistical problems arising in the determination of the ``best'' of $k$ categories when comparing $k - 1$ experimental categories with a standard or control. The discussion is limited to the case of a single stage sampling procedure with an equal number of observations on each of the $k$ categories. Results both of an exact and of an approximate nature are obtained when (a) the observations with each category are normally distributed, and (b) the observations with each category have a binomial distribution.
TL;DR: In this paper, a slippage problem for normal distributions is formulated as a multiple decision problem, and a solution is obtained which has certain optimum properties for the fixed sample case with the same number of observations from each distribution.
Abstract: A slippage problem for normal distributions is formulated as a multiple decision problem, and a solution is obtained which has certain optimum properties. The discussion is confined to the fixed sample case with the same number of observations from each distribution, and the normal distributions involved are assumed to have a common but unknown variance.
TL;DR: In this paper, the first analytical methods for the analysis of unsymmetrical balanced incomplete block designs were developed for the first time, and for such designs in general, inequalities were found for the number of treatments common to two blocks.
Abstract: In this paper there are developed for the first time analytical methods for the investigation of the structure of unsymmetrical balanced incomplete block designs. Two unsymmetrical balanced incomplete block designs are proved to be impossible, and for such designs in general, inequalities are found for the number of treatments common to two blocks.
TL;DR: In this paper, it was shown that it is possible to weaken conditions (25, (26), (27), (28) and (27) by replacing them by condition (3) (abc) below.
Abstract: In their interesting and pioneering paper Robbins and Monro [1] give a method for "solving stochastically" the equation in $x: M(x) = \alpha$, where $M(x)$ is the (unknown) expected value at level $x$ of the response to a certain experiment. They raise the question whether their results, which are contained in their Theorems 1 and 2, are valid under a condition (their condition (4'), our condition (1) below) which is statistically plausible and is weaker than the condition which they require to prove their results. In the present paper this question is answered in the affirmative. They also ask whether their conditions (33), (34), and (35) (our conditions (25), (26) and (27) below) can be replaced by their condition (5") (our condition (28) below). A counterexample shows that this is impossible. However, it is possible to weaken conditions (25), (26) and (27) by replacing them by condition (3) (abc) below. Thus our results generalize those of [1]. The statistical significance of these results is described in [1].
TL;DR: In this paper, the problem of finding the "best" linear function for discriminating between two multivariate normal populations, without limitation to the case of equal covariance matrices, is considered.
Abstract: The problem considered is that of finding the "best" linear function for discriminating between two multivariate normal populations, $\pi_1$ and $\pi_2$, without limitation to the case of equal covariance matrices. The "best" linear function is found by maximizing the divergence, $J'(1, 2)$, between the distributions of the linear function. Comparison with the divergence, $J(1, 2)$, between $\pi_1$ and $\pi_2$ offers a measure of the discriminating efficiency of the linear function, since $J(1, 2) \geq J'(1, 2)$. The divergence, a special case of which is Mahalanobis's Generalized Distance, is defined in terms of a measure of information which is essentially that of Shannon and Wiener. Appropriate assumptions about $\pi_1$ and $\pi_2$ lead to discriminant analysis (Sections 4, 7), principal components (Section 5), and canonical correlations (Section 6).
TL;DR: In this article, the classification statistics discussed in the literature can be written as functions of the elements of a symmetric random matrix, and an analytic derivation of the distribution of such a matrix is given.
Abstract: Two classification statistics discussed in the literature can be written as functions of the elements of a $2 \cdot 2$ symmetric random matrix $M$. An analytic derivation is given of the distribution of $M$, and of a related matrix $M^\ast$, extending earlier work on distribution theory by Wald [1] and Anderson [2].
TL;DR: Bose as discussed by the authors gave an interesting generalization of MacNeish's result, which is stated as a theorem in the next section and is published at his suggestion at the Institute of Mathematical Statistics at Chapel Hill, North Carolina.
Abstract: In 1922 MacNeish [1] considered the problem of orthogonal Latin squares and showed that if the number $s$ is written in standard form: $s = p^{n_0}_0p^{n_1}_1 \cdots p^{n_k}_k,$ where $p_0, p_1, \cdots, p_k$ are primes, and if $r = \min(p^{n_0}_0, p^{n_1}_1, \cdots, p^{n_k}_k),$ then we can construct $r - 1$ orthogonal Latin squares of side $s$. An alternative proof was also given by Mann [2]. At the April, 1950 meeting of the Institute of Mathematical Statistics at Chapel Hill, North Carolina, R. C. Bose announced an interesting generalization of this result [3] which is stated as a theorem in the next section. The proof given here is simpler than Bose's original proof and is published at his suggestion.
TL;DR: In this paper, it was shown that a particular decision rule is the uniformly best decision rule for many problems of this class and that the conditional expectation of an unbiased decision rule given the sufficient statistics is also an impartial decision rule.
Abstract: A class of decision problems concerning $k$ populations was considered in [1] and it was shown that a particular decision rule is the uniformly best `impartial' decision rule for many problems of this class. The present paper provides certain improvements of this result. The authors define impartiality in terms of permutations of the $k$ samples rather than in terms of the $k$ ordered values of an arbitrarily chosen real-valued statistic as in the earlier paper. They point out that (under conditions which are satisfied in the standard cases of $k$ independent samples of equal size) if the same function is a sufficient statistic for each of the $k$ samples then the conditional expectation of an impartial decision rule given the $k$ sufficient statistics is also an impartial decision rule. A characterization of impartial decision rules is given which relates the present definition of impartiality with the one adopted in [1]. These results, together with Theorem 1 of [1], yield the desired improvements. The argument indicated here is illustrated by application to a special case.
TL;DR: In this paper, it was shown that various statistics based on the number of runs up and down have an asymptotic multivariate normal distribution under a number of different alternatives to randomness.
Abstract: It is shown that various statistics based on the number of runs up and down have an asymptotic multivariate normal distribution under a number of different alternatives to randomness. The concept of likelihood ratio statistics is extended to give a method for deciding what function of these runs should be used, and it is shown that the asymptotic power of these tests depends only on the covariance matrix, calculated under the hypothesis of randomness, and the expected values, calculated under the alternative hypothesis. A general method is given for calculating these expected values when the observations are independent, and these calculations are carried through for a constant shift in location from one observation to the next and for normal and rectangular populations.