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Showing papers in "Annals of Mathematical Statistics in 1950"


Journal ArticleDOI
TL;DR: In this paper, an empirical study of a number of approximations for variance stabilization is presented, some intended for significance and confidence work and others for variance stabilisation for Poisson data.
Abstract: The use of transformations to stabilize the variance of binomial or Poisson data is familiar(Anscombe [1], Bartlett [2, 3], Curtiss [4], Eisenhart [5]). The comparison of transformed binomial or Poisson data with percentage points of the normal distribution to make approximate significance tests or to set approximate confidence intervals is less familiar. Mosteller and Tukey [6] have recently made a graphical application of a transformation related to the square-root transformation for such purposes, where the use of "binomial probability paper" avoids all computation. We report here on an empirical study of a number of approximations, some intended for significance and confidence work and others for variance stabilization. For significance testing and the setting of confidence limits, we should like to use the normal deviate $K$ exceeded with the same probability as the number of successes $x$ from $n$ in a binomial distribution with expectation $np$, which is defined by $\frac{1}{2\pi} \int^K_{-\infty} e^{-\frac{1}{2}t^2} dt = \operatorname{Prob} \{x \leq k |mid \operatorname{binomial}, n, p\}.$ The most useful approximations to $K$ that we can propose here are $N$ (very simple), $N^+$ (accurate near the usual percentage points), and $N^{\ast\ast}$ (quite accurate generally), where $N = 2 (\sqrt{(k + 1)q} - \sqrt{(n - k)p)}.$ (This is the approximation used with binomial probability paper.) $N^+ = N + \frac{N + 2p - 1}{12\sqrt{E}},\quad E = \text{lesser of} np \text{and} nq, N^\ast = N + \frac{(N - 2)(N + 2)}{12} \big(\frac{1}{\sqrt{np + 1}} - \frac{1}{\sqrt{nq + 1}}\big), N^{\ast\ast} = N^\ast + \frac{N^\ast + 2p - 1}{12 \sqrt{E}}\cdot\quad E = \text{lesser of} np \text{and} nq.$ For variance stabilization, the averaged angular transformation $\sin^{-1}\sqrt{\frac{x}{n + 1}} + \sin^{-1} \sqrt{\frac{x + 1}{n+1}}$ has variance within $\pm 6%$ of $\frac{1}{n + \frac{1}{2}} \text{(angles in radians)}, \frac{821}{n + \frac{1}{2}} \text{(angles in degrees)},$ for almost all cases where $np \geq 1$. In the Poisson case, this simplifies to using $\sqrt{x} + \sqrt{x + 1}$ as having variance 1.

2,196 citations


Journal ArticleDOI
TL;DR: In this article, the significance of the largest observation in a sample of size $n$ from a normal population was investigated and the authors proposed a new statistic, S^2_n/S^2, to test whether the two largest observations are too large.
Abstract: The problem of testing outlying observations, although an old one, is of considerable importance in applied statistics. Many and various types of significance tests have been proposed by statisticians interested in this field of application. In this connection, we bring out in the Histrical Comments notable advances toward a clear formulation of the problem and important points which should be considered in attempting a complete solution. In Section 4 we state some of the situations the experimental statistician will very likely encounter in practice, these considerations being based on experience. For testing the significance of the largest observation in a sample of size $n$ from a normal population, we propose the statistic $\frac{S^2_n}{S^2} = \frac{\sum^{n-1}_{i=1} (x_i - \bar x_n)^2}{\sum^n_{i=1} (x_i - \bar x)^2}$ where $x_1 \leq x_2 \leq \cdots \leq x_n, \bar x_n = \frac{1}{n - 1} \sum^{n-1}_{i=1} x_i$ and $\bar x = \frac{1}{n}\sum^{n}_{i=1} x_i.$ A similar statistic, $S^2_1/S^2$, can be used for testing whether the smallest observation is too low. It turns out that $\frac{S^2_n}{S^2} = 1 - \frac{1}{n - 1} \big(\frac{x_n - \bar x}{s}\big)^2 = 1 - \frac{1}{n - 1} T^2_n,$ where $s^2 = \frac{1}{n}\sigma(x_i - \bar x)^2,$ and $T_n$ is the studentized extreme deviation already suggested by E. Pearson and C. Chandra Sekar [1] for testing the significance of the largest observation. Based on previous work by W. R. Thompson [12], Pearson and Chandra Sekar were able to obtain certain percentage points of $T_n$ without deriving the exact distribution of $T_n$. The exact distribution of $S^2_n/S^2$ (or $T_n$) is apparently derived for the first time by the present author. For testing whether the two largest observations are too large we propose the statistic $\frac{S^2_{n-1,n}}{S^2} = \frac{\sum^{n-2}_{i=1} (x_i - \bar x_{n-1,n})^2}{\sum^n_{i=1} (x_i - \bar x)^2},\quad\bar x_{n-1,n} = \frac{1}{n - 2} \sum^{n-2}_{i=1} x_i$ and a similar statistic, $S^2_{1,2}/S^2$, can be used to test the significance of the two smallest observations. The probability distributions of the above sample statistics $S^2 = \sum^n_{i=1} (x_i - \bar x)^2 \text{where} \bar x = \frac{1}{n} \sum^n_{i=1} x_i$ $S^2_n = \sum^{n-1}_{i=1} (x_i - \bar x_n)^2 \text{where} \bar x_n = \frac{1}{n-1} \sum^{n-1}_{i=1} x_i$ $S^2_1 = \sum^n_{i=2} (x_i - \bar x_1)^2 \text{where} \bar x_1 = \frac{1}{n-1} \sum^n_{i=2} x_i$ are derived for a normal parent and tables of appropriate percentage points are given in this paper (Table I and Table V). Although the efficiencies of the above tests have not been completely investigated under various models for outlying observations, it is apparent that the proposed sample criteria have considerable intuitive appeal. In deriving the distributions of the sample statistics for testing the largest (or smallest) or the two largest (or two smallest) observations, it was first necessary to derive the distribution of the difference between the extreme observation and the sample mean in terms of the population $\sigma$. This probability$X_1 \leq x_2 \leq x_3 \cdots \leq x_n$ $s^2 = \frac{1}{n} \sum^n_{i=1} (x_i - \bar x)^2 \quad \bar x = \frac{1}{n} \sum^n_{i=1} x_i$ distribution was apparently derived first by A. T. McKay [11] who employed the method of characteristic functions. The author was not aware of the work of McKay when the simplified derivation for the distribution of $\frac{x_n - \bar x}{\sigma}$ outlined in Section 5 below was worked out by him in the spring of 1945, McKay's result being called to his attention by C. C. Craig. It has been noted also that K. R. Nair [20] worked out independently and published the same derivation of the distribution of the extreme minus the mean arrived at by the present author--see Biometrika, Vol. 35, May, 1948. We nevertheless include part of this derivation in Section 5 below as it was basic to the work in connection with the derivations given in Sections 8 and 9. Our table is considerably more extensive than Nair's table of the probability integral of the extreme deviation from the sample mean in normal samples, since Nair's table runs from $n = 2$ to $n = 9,$ whereas our Table II is for $n = 2$ to $n = 25$. The present work is concluded with some examples.

1,401 citations




Journal ArticleDOI
TL;DR: In this paper, the properties of a measure of dependence $q'$ between two random variables are studied and it is shown (Sections 3-5) that under fairly general conditions, the measure has an asymptotically normal distribution and provides approximate confidence limits for the population analogue of $q'.
Abstract: The properties of a measure of dependence $q'$ between two random variables are studied. It is shown (Sections 3-5) that $q'$ under fairly general conditions has an asymptotically normal distribution and provides approximate confidence limits for the population analogue of $q'$. A test of independence based on $q'$ is non-parametric (Section 6), and its asymptotic efficiency in the normal case is about 41% (Section 7). The $q'$-distribution in the case of independence is tabulated for sample sizes up to 50.

306 citations



Journal ArticleDOI
TL;DR: In this article, the consistency of the estimates and the asymptotic distributions of the estimate and the test criteria are studied under conditions more general than those used in the derivation of these estimates and criteria.
Abstract: In a previous paper [2] the authors have given a method for estimating the coefficients of a single equation in a complete system of linear stochastic equations. In the present paper the consistency of the estimates and the asymptotic distributions of the estimates and the test criteria are studied under conditions more general than those used in the derivation of these estimates and criteria. The point estimates, which can be obtained as maximum likelihood estimates under certain assumptions including that of normality of disturbances, are consistent even if the disturbances are not normally distributed and (a) some predetermined variables are neglected (Theorem 1) or (b) the single equation is in a non-linear system with certain properties (Theorem 2). Under certain general conditions (normality of the disturbances not being required) the estimates are asymptotically normally distributed (Theorems 3 and 4). The asymptotic covariance matrix is given for several cases. The criteria derived in [2] for testing the hypothesis of over-identification have, asymptotically, $\chi^2$-distributions (Theorem 5). The exact confidence regions developed in [2] for the case that all predetermined variables are exogenous (that is, that the difference equations are of zero order) are shown to be consistent and to hold asymptotically even when this assumption is not true (Theorem 6).

263 citations


Journal ArticleDOI
TL;DR: In this paper, the mean and variance of normal populations from singly and doubly truncated samples having known truncation points are estimated with the aid of standard tables of areas and ordinates of the normal frequency function.
Abstract: This paper is concerned with the problem of estimating the mean and variance of normal populations from singly and doubly truncated samples having known truncation points. Maximum likelihood estimating equations are derived which, with the aid of standard tables of areas and ordinates of the normal frequency function, can be readily solved by simple iterative processes. Asymptotic variances and covariances of these estimates are obtained from the information matrices. Numerical examples are given which illustrate the practical application of these results. In Sections 3 to 8 inclusive, the following cases of doubly truncated samples are considered: I, number of unmeasured observations unknown; II, number of unmeasured observations in each `tail' known; and III, total number of unmeasured observations known, but not the number in each `tail'. In Section 9, singly truncated samples are treated as special cases of I and II above.

220 citations




Journal ArticleDOI
TL;DR: In this paper, the authors investigated the asymptotic form of the distribution of distance in a hypersphere and showed that the distance is almost always nearly equal to the distance between the extremities of two orthogonal radii.
Abstract: Deltheil ([1], pp. 114-120) has considered the distribution of distance in an $n$-dimensional hypersphere. In this paper I put his results (17) in a more compact form (16); and I investigate in greater detail the asymptotic form of the distribution for large $n$, for which the rather surprising result emerges that this distance is almost always nearly equal to the distance between the extremities of two orthogonal radii. I came to study this distribution by the need to compute a doubly-threefold integral, which measures the damage caused to plants by the presence of radioactive tracers in their fertilizers; for the distribution affords a method of evaluating numerically certain multiple integrals. I hope to describe elsewhere this application of the theory.

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


Book ChapterDOI
TL;DR: In this paper, the authors proposed the likelihood ratio principle as a general applicable criterion for hypothesis testing, which has proved extremely successful; nearly all tests now in use for testing parametric hypotheses are likelihood ratio tests, and many of them have been shown to possess various optimum properties.
Abstract: The likelihood ratio principle. The development of a theory of hypothesis testing (as contrasted with the consideration of particular cases), may be said to have begun with the 1928 paper of Neyman and Pearson [16]. For in this paper the fundamental fact is pointed out that in selecting a suitable test one must take into account not only the hypothesis but also the alternatives against which the hypothesis is to be tested, and on this basis the likelihood ratio principle is proposed as a generally applicable criterion. This principle has proved extremely successful; nearly all tests now in use for testing parametric hypotheses are likelihood ratio tests, (for an extension to the non-parametric case see [33]), and many of them have been shown to possess various optimum properties.

Journal ArticleDOI
TL;DR: In this article, a two-sample procedure was used to test the ratio of means of two normal populations with power independent of the unknown variances, and the results showed that no non-trivial single sample test exists whose power is independent of ε-sigma_1 and ε -sigma-2.
Abstract: Stein [4] has exhibited a double sampling procedure to test hypotheses concerning the mean of normal variables with power independent of the unknown variances. This procedure is here adapted to test hypotheses concerning the ratio of means of two normal populations, also with power independent of the unknown variances. The use of a two sample procedure in a regression problem is also considered. Let $\{X_{ij}\} (i = 1, 2) (j = 1, 2, 3, \cdots)$ be independent random variables distributed according to $N(m_i, \sigma_i):$ all parameters are assumed to be unknown. Defining $k$ by the equation \begin{equation*}\tag{(1)} m_1 = km_2\end{equation*} we wish to test the hypothesis $H$ that $k$ has a specified value $k_0$. If $k_0 = 1$ the hypothesis $H$ reduces to a classical problem, often referred to in the literature as the Behrens-Fisher-problem (cf. Scheffe [3] for a bibliography). At the present time it is still an open question whether it is possible (or desirable) to find a non-trivial single sample test for $H$ with the size of the critical region independent of $\sigma_1$ and $\sigma_2$. In any case it is a simple extension of the result of Dantzig [1] (cf. also Stein [4]) to show that no non-trivial single sample test exists whose power is independent of $\sigma_1$ and $\sigma_2$. On the other hand the case $k_0 eq 1$ may be expected to occur frequently in fields of application where a choice must be made between different products, methods of experimentation etc. which involve different costs. The statistician must make a choice on the basis of results relative to the ratio of costs involved. Nevertheless this problem appears to have received little attention in the literature. In general tests based on a two-sample procedure may not be as "efficient" in the sense of Wald [5] as a strict sequential procedure. On the other hand the two sample procedure reduces the number of decisions to be made by the experimenter and it will, in certain fields, simplify the experimental procedure.

Journal ArticleDOI
TL;DR: In this article, the consequences of performing a preliminary $F$-test in the analysis of variance were described and a more stable procedure was recommended for performing the preliminary test in which the two mean squares are pooled only if their ratio is less than twice the 50% point.
Abstract: The paper describes the consequences of performing a preliminary $F$-test in the analysis of variance. The use of the 5% or 25% significance level for the preliminary test results in disturbances that are frequently large enough to lead to incorrect inferences in the final test. A more stable procedure is recommended for performing the preliminary test in which the two mean squares are pooled only if their ratio is less than twice the 50% point.


Journal ArticleDOI
TL;DR: In this article, the authors continue the study of sequential decision functions, as follows: a) the proof of the optimum character of the sequential probability ratio test was based on a certain property of Bayes solutions for sequential decisions between two alternatives, the cost function being linear.
Abstract: The study of sequential decision functions was initiated by one of the authors in [1]. Making use of the ideas of this theory the authors succeeded in [4] in proving the optimum character of the sequential probability ratio test. In the present paper the authors continue the study of sequential decision functions, as follows: a) The proof of the optimum character of the sequential probability ratio test was based on a certain property of Bayes solutions for sequential decisions between two alternatives, the cost function being linear. This fundamental property, the convexity of certain important sets of a priori distributions, is proved in Theorem 3.9 in considerable generality. The number of possible decisions may be infinite. b) Theorem 3.10 and section 4 discuss tangents and boundary points of these sets of a priori distributions. (These results for finitely many alternatives were announced by one of us in an invited address at the Berkeley meeting of the Institute of Mathematical Statistics in June, 1948) c) Theorem 3.6 is an existence theorem for Bayes solutions. Theorem 3.7 gives a necessary and sufficient condition for a Bayes solution. These theorems generalize and follow the ideas of Lemma 1 of [4] d) Theorems 3.8 and 3.8.1 are continuity theorems for the average risk function. They generalize Lemma 3 in [4] e) Other theorems give recursion formulas and inequalities which govern Bayes solutions.



Journal ArticleDOI
TL;DR: In this article, it was shown that under the restriction of impartial decision, the rule $d_k =$ "Always select only the population corresponding to the greatest X_i$" cannot be improved, no matter what $x$ or the true parameter values may be.
Abstract: In two recent papers, Paulson [1] and Mosteller [2] have called attention to several unsolved problems in $k$-sample theory. A problem which is typical of the ones considered in this paper is as follows. Let $\pi_1, \pi_2, \cdots, \pi_k$ be a set of normal populations, $\pi_i$ having an unknown mean $m_i$ and variance $\sigma^2, G(x, \theta_i)$ being the distribution function which characterizes $\pi_i$. Samples of equal size are drawn from each population, $\bar X_i$ being the sample means, and $S^2$ the estimate of $\sigma^2$ obtained. The problem is to construct a suitable decision rule $d = d(\{\bar X_i\}; S^2)$ to select one or more populations, the object being to minimize the expected value of the random distribution function $G(x \mid s(d)) = \sum^k_{i=1} Z_i(d) \cdot G(x, \theta_i) / \sum^k_{i=1} Z_i(d),$ where $Z_i(d) = 1$ if $\pi_i$ is selected by $d$, and $= 0$ otherwise. It is shown that under the restriction of impartial decision, the rule $d_k =$ "Always select only the population corresponding to the greatest $\bar X_i$" cannot be improved, no matter what $x$ or the true parameter values may be. It follows (i) that $d_k$ is the uniformly best decision rule in the class of impartial decision rules for all weight functions of type $W = \max_i \{m_i\} - \big(\sum^k_{i=1} z_im_i / \sum^k_{i=1} z_i\big)$, and (ii) that the customary $F$ and $t$ tests of analysis of variance are not relevant to the problem. This result is an application of Theorem 1 which applies to a number of similar problems concerning $k$ populations, especially when the populations admit sufficient statistics for their parameters. Two examples of statistical applications are given in Section 6.

Journal ArticleDOI
TL;DR: In this article, the authors considered the observations are considered to be normally distributed with constant variance and means consisting of linear combinations of certain trigonometric functions, and the likelihood ratio criterion for testing the independence of the observations against the alternatives of circular serial correlation of a given lag is found to be a function of the CSC coefficient for residuals from the fitted Fourier series.
Abstract: In this paper the observations are considered to be normally distributed with constant variance and means consisting of linear combinations of certain trigonometric functions. The likelihood ratio criterion for testing the independence of the observations against the alternatives of circular serial correlation of a given lag is found to be a function of the circular serial correlation coefficient for residuals from the fitted Fourier series (Section 4). The exact distribution (Section 5), the moments (Section 6), and approximate distributions (Section 7) are given for the cases of greatest interest. From these results significance levels have been found (Section 3). The use of these levels is indicated (Section 2), and an example of their use is given (Section 3).


Journal ArticleDOI
TL;DR: In this article, a method is described by which it is possible to derive the distribution of the sum of roots of a certain determinantal equation under the condition that $m = 0.
Abstract: This paper is in continuation of the author's first two papers [1] and [2]. In this paper a method is described by which it is possible to derive the distribution of the sum of roots of a certain determinantal equation under the condition that $m = 0$. This condition implies, when the results are applied to canonical correlations, that the numbers of variates in the two sets differ by unity. The distributions for the sum of roots under this condition have been obtained for $l$ = 2, 3 and 4 and are given in this paper. This paper also derives the moments of these distributions.

Journal ArticleDOI
TL;DR: In this article, it was shown that the existence of a symmetrical balanced incomplete block design with parameters $b = v, b, r, k$ and λ = 2 is not guaranteed.
Abstract: An arrangement of $v$ varieties or treatments in $b$ blocks of size $k, (k < v),$ is known as a balanced incomplete block design if every variety occurs in $r$ blocks and any two varieties occur together in $\lambda$ blocks. These parameters obviously satisfy the equations \begin{equation*}\tag{1} bk = vr\end{equation*}\begin{equation*}\tag{2} \lambda(v - 1) = r(k - 1)\end{equation*} Fisher [1] has also proved that the inequality \begin{equation*}\tag{3} b \geq v, \quad r \geq k\end{equation*} must hold. If $v, b, r, k$ and $\lambda$ are positive integers satisfying (1), (2) and (3), then a balanced incomplete block design with these parameters possibly exists, but the actual existence of a combinatorial solution is not ensured. These conditions are thus necessary but not sufficient for the existence of a design. Fisher and Yates in their tables [2] have listed all designs with $r \leq 10$ and given combinatorial solutions, where known. A balanced incomplete block design in which $b = v$, and hence $r = k$ is called a symmetrical balanced incomplete block design. The impossibility of the symmetrical designs with parameters $v = b = 22, r = k = 7, \lambda = 2$ and $v = b = 29, r = k = 8, \lambda = 2$ was first demonstrated by Hussain [3], [4] essentially by the method of enumeration. The object of the present note is to give an alternative simple proof of the impossibility of these designs and to show that the only unknown remaining symmetrical design in Fisher and Yates' tables, viz. $v = b = 46, r = k = 10, \lambda = 2,$ is definitely impossible. Symmetrical designs with $\lambda \leq 5, r, k \leq 20$, which are impossible combinatorially, are also listed.

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

Journal ArticleDOI
TL;DR: In this paper, a characterization of unbiased estimates with minimum variance is given for two fairly broad classes of problems, and solutions are given which are more readily applicable to other problems in statistics.
Abstract: Subject to certain restrictions, a characterization of unbiased estimates with minimum variance is obtained. For two fairly broad classes of problems, solutions are given which are more readily applicable. These are used to obtain such estimates in some particular cases. The applicability of the results to problems of sequential estimation is pointed out. The problem of unbiased estimation is not at present of much practical importance, but is of some theoretical interest and has been treated by many statisticians. Also, the method used in this paper may be applicable to other problems in statistics.

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

Book ChapterDOI
TL;DR: In this paper, Girshick, Mosteller, Savage and Wolfowitz have considered the uniqueness of unbiased estimates depending only on an appropriate sufficient statistic for sequential sampling schemes of binomial variables.
Abstract: Recently, in a series of papers, Girshick, Mosteller, Savage and Wolfowitz have considered the uniqueness of unbiased estimates depending only on an appropriate sufficient statistic for sequential sampling schemes of binomial variables. A complete solution was obtained under the restriction to bounded estimates. This work, which has immediate consequences with respect to the existence of unbiased estimates with uniformly minimum variance, is extended here in two directions. A general necessary condition for uniqueness is found, and this is applied to obtain a complete solution of the uniqueness problem when the random variables have a Poisson or rectangular distribution. Necessary and sufficient conditions are also found in the binomial case without the restriction to bounded estimates. This permits the statement of a somewhat stronger optimum property for the estimates, and is applicable to the estimation of unbounded functions of the unknown probability.

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