Showing papers in "Annals of Mathematical Statistics in 1954"

Some Theorems on Quadratic Forms Applied in the Study of Analysis of Variance Problems, I. Effect of Inequality of Variance in the One-Way Classification

Abstract: This is the first of two papers describing a study of the effect of departures from assumptions, other than normality, on the null-distribution of the $F$-statistic in the analysis of variance. In this paper, certain theorems required in the study and concerning the distribution of quadratic forms in multi-normally distributed variables are first enunciated and simple approximations tested numerically. The results are then applied to determine the effect of group-to-group inequality of variance in the one-way classification. It appears that if the groups are equal, moderate inequality of variance does not seriously affect the test. However, with unequal groups, much larger discrepancies appear. In a second paper, similar methods are used to determine the effect of inequality of variance and serial correlation between errors in the two-way classification.

Saddlepoint Approximations in Statistics

Abstract: It is often required to approximate to the distribution of some statistic whose exact distribution cannot be conveniently obtained. When the first few moments are known, a common procedure is to fit a law of the Pearson or Edgeworth type having the same moments as far as they are given. Both these methods are often satisfactory in practice, but have the drawback that errors in the "tail" regions of the distribution are sometimes comparable with the frequencies themselves. The Edgeworth approximation in particular notoriously can assume negative values in such regions. The characteristic function of the statistic may be known, and the difficulty is then the analytical one of inverting a Fourier transform explicitly. In this paper we show that for a statistic such as the mean of a sample of size $n$, or the ratio of two such means, a satisfactory approximation to its probability density, when it exists, can be obtained nearly always by the method of steepest descents. This gives an asymptotic expansion in powers of $n^{-1}$ whose dominant term, called the saddlepoint approximation, has a number of desirable features. The error incurred by its use is $O(n^{-1})$ as against the more usual $O(n^{-1/2})$ associated with the normal approximation. Moreover it is shown that in an important class of cases the relative error of the approximation is uniformly $O(n^{-1})$ over the whole admissible range of the variable. The method of steepest descents was first used systematically by Debye for Bessel functions of large order (Watson [17]) and was introduced by Darwin and Fowler (Fowler [9]) into statistical mechanics, where it has remained an indispensable tool. Apart from the work of Jeffreys [12] and occasional isolated applications by other writers (e.g. Cox [2]), the technique has been largely ignored by writers on statistical theory. In the present paper, distributions having probability densities are discussed first, the saddlepoint approximation and its associated asymptotic expansion being obtained for the probability density of the mean $\bar{x}$ of a sample of $n$. It is shown how the steepest descents technique is related to an alternative method used by Khinchin [14] and, in a slightly different context, by Cramer [5]. General conditions are established under which the relative error of the saddlepoint approximation is $O(n^{-1})$ uniformly for all admissible $\bar{x}$, with a corresponding result for the asymptotic expansion. The case of discrete variables is briefly discussed, and finally the method is used for approximating to the distribution of ratios.

A Single-Sample Multiple Decision Procedure for Ranking Means of Normal Populations with known Variances

Abstract: This paper is concerned with a single-sample multiple decision procedure for ranking means of normal populations with known variances. Problems which conventionally are handled by the analysis of variance (Model I) which tests the hypothesis that $k$ means are equal are reformulated as multiple decision procedures involving rankings. It is shown how to design experiments so that useful statements can be made concerning these rankings on the basis of a predetermined number of independent observations taken from each population. The number of observations required is determined by the desired probability of a correct ranking when certain differences between population means are specified.

On the Distribution of the Likelihood Ratio

Abstract: A classical result due to Wilks [1] on the distribution of the likelihood ratio $\lambda$ is the following. Under suitable regularity conditions, if the hypothesis that a parameter $\theta$ lies on an $r$-dimensional hyperplane of $k$-dimensional space is true, the distribution of $-2 \log \lambda$ is asymptotically that of $\chi^2$ with $k - r$ degrees of freedom. In many important problems it is desired to test hypotheses which are not quite of the above type. For example, one may wish to test whether $\theta$ is on one side of a hyperplane, or to test whether $\theta$ is in the positive quadrant of a two-dimensional space. The asymptotic distribution of $-2 \log \lambda$ is examined when the value of the parameter is a boundary point of both the set of $\theta$ corresponding to the hypothesis and the set of $\theta$ corresponding to the alternative. First the case of a single observation from a multivariate normal distribution, with mean $\theta$ and known covariance matrix, is treated. The general case is then shown to reduce to this special case where the covariance matrix is replaced by the inverse of the information matrix. In particular, if one tests whether $\theta$ is on one side or the other of a smooth $(k - 1)$-dimensional surface in $k$-dimensional space and $\theta$ lies on the surface, the asymptotic distribution of $\lambda$ is that of a chance variable which is zero half the time and which behaves like $\chi^2$ with one degree of freedom the other half of the time.

The Use of Maximum Likelihood Estimates in {\chi^2} Tests for Goodness of Fit

Abstract: The usual test that a sample comes from a distribution of given form is performed by counting the number of observations falling into specified cells and applying the χ2 test to these frequencies. In estimating the parameters for this test, one may use the maximum likelihood (or equivalent) estimate based (1) on the cell frequencies, or (2) on the original observations. This paper shows that in (2), unlike the well known result for (1), the test statistic does not have a limiting χ2-distribution, but that it is stochastically larger than would be expected under the χ2 theory. The limiting distribution is obtained and some examples are computed. These indicate that the error is not serious in the case of fitting a Poisson distribution, but may be so for the fitting of a normal.

Truncated Life Tests in the Exponential Case

Abstract: It is frequently desirable on practical grounds to terminate a life test by a preassigned time $T_0$. In this paper we consider life tests which are truncated as follows. With $n$ items placed on test, it is decided in advance that the experiment will be terminated at $\min (X_{r0,n}, T_0)$, where $X_{r0,n}$ is a random variable equal to the time at which the $r_0$th failure occurs and $T_0$ is a truncation time, beyond which the experiment will not be run. Both $r_0$ and $T_0$ are assigned before experimentation starts. If the experiment is terminated at $X_{r0,n}$ (that is, if $r_0$ failures occur before time $T_0$), then the action in terms of hypothesis testing is the rejection of some specified null-hypothesis. If the experiment is terminated at time $T_0$ (that is, if the $r_0$th failure does not occur before time $T_0$), then the action in terms of hypothesis testing is the acceptance of some specified null-hypothesis. While truncated procedures can be considered for any life distribution, we limit ourselves here to the case where the underlying life distribution is specified by a p.d.f. of the exponential form, $f(x; \theta) = \theta^{-1}e^{-x/\theta}, x > 0, \theta > 0$. The practical justification for using this kind of distribution as a first approximation to a number of test situations is discussed in a recent paper by Davis [1]. It is a common assumption for electron tube life. Two situations are considered. The first is the nonreplacement case in which a failure occurring during the test is not replaced by a new item. The second is the replacement case where failed items are replaced at once by new items drawn at random from the same p.d.f. as the original $n$ items. Formulae are given for $E_\theta(r)$, the expected number of observations to reach a decision; for $E_\theta(T)$, the expected waiting time to reach a decision; and for $L(\theta)$, the probability of accepting the hypothesis that $\theta = \theta_0$, the value associated with the null-hypothesis, when $\theta$ is the true value. Some procedures are worked out for finding truncated tests meeting specified conditions, and practical illustrations are given. It is an intrinsic feature of all life test decision procedures that they are in some sense truncated, although not necessarily by a fixed time $T_0$. In Section 3 we give exact formulae for $E_\theta(r)$ and $E_\theta(T)$ for a decision procedure given in [2]. There is a close relation between these results and those in Section 2.

Some Theorems on Quadratic Forms Applied in the Study of Analysis of Variance Problems, II. Effects of Inequality of Variance and of Correlation Between Errors in the Two-Way Classification

Abstract: Theorems already enunciated in a previous paper on quadratic forms are used to determine the effects of inequality of variance and first order serial correlation of errors in the two-way classification on the analysis of variance. It is found that when the appropriate null hypothesis is true, inequality of variance from column to column results in an increased chance of exceeding the significance point for the test on homogeneity of column means, and a decreased chance for the corresponding test on row means. For moderate differences in variance neither effect is large. First order serial correlation within rows produces a large effect on the "between rows" comparisons, but little effect on the "between columns" comparisons.

Multidimensional Stochastic Approximation Methods

Abstract: Multidimensional stochastic approximation schemes are presented, and conditions are given for these schemes to converge a.s. (almost surely) to the solutions of $k$ stochastic equations in $k$ unknowns and to the point where a regression function in $k$ variables achieves its maximum.

On the Asymptotic Efficiency of Certain Nonparametric Two-Sample Tests

Abstract: In this paper the following asymptotic efficiencies are computed for the given two-sample tests against normal alternatives to the null hypothesis: $\text{rank test for location}\ldots\ldots\ldots\ldots 3/\pi \cong 95{\tt\%}$ $\text{median test for location}\ldots\ldots\ldots\ldots 2/\pi \cong 64{\tt\%}$ $\text{run test for location}\ldots\ldots\ldots \ldots 0$ $\text{run test for dispersion}\ldots\ldots\ldots\ldots 0$ $\text{square rank test for dispersion}\ldots\ldots\ldots\ldots 15/2\pi^2 \cong 76{\tt\%}$ Also, general expressions for means and variances of some of these test criteria are found for distributions alternative to the null hypothesis.

On a Stochastic Approximation Method

Abstract: Asymptotic properties are established for the Robbins-Monro [1] procedure of stochastically solving the equation $M(x) = \alpha$. Two disjoint cases are treated in detail. The first may be called the "bounded" case, in which the assumptions we make are similar to those in the second case of Robbins and Monro. The second may be called the "quasi-linear" case which restricts $M(x)$ to lie between two straight lines with finite and nonvanishing slopes but postulates only the boundedness of the moments of $Y(x) - M(x)$ (see Sec. 2 for notations). In both cases it is shown how to choose the sequence $\{a_n\}$ in order to establish the correct order of magnitude of the moments of $x_n - \theta$. Asymptotic normality of $a^{1/2}_n(x_n - \theta)$ is proved in both cases under a further assumption. The case of a linear $M(x)$ is discussed to point up other possibilities. The statistical significance of our results is sketched.

Normal Multivariate Analysis and the Orthogonal Group

Abstract: New methods are introduced for deriving the sampling distributions of statistics obtained from a normal multivariate population. Exterior differential forms are used to represent the invariant measures on the orthogonal group and the Grassmann and Stiefel manifolds. The first part is devoted to a mathematical exposition of these. In the second part, the theory is applied; first, to the derivation of the distribution of the canonical correlation coefficients when the corresponding population parameters are zero; and secondly, to split the distribution of a normal multivariate sample into three independent distributions, (a) essentially the Wishart distribution, (b) the invariant distribution of a random plane which is given by the invariant measure on the Grassmann manifold, (c) the invariant distribution of a random orthogonal matrix. This decomposition provides derivations of the Wishart distribution and of the distribution of the latent roots of the sample variance covariance matrix when the population roots are equal.

Some Theorems Relevant to Life Testing from an Exponential Distribution

Abstract: A life test on $N$ items is considered in which the common underlying distribution of the length of life of a single item is given by the density \begin{equation*}\tag{1} p(x; \theta, A) = \begin{cases}\frac{1}{\theta} e^{-(x-A)/\theta},\quad\text{for} x \geqq A \\ 0,\quad\text{otherwise}\end{cases}\end{equation*} where $\theta > 0$ is unknown but is the same for all items and $A \geqq 0.$ Several lemmas are given concerning the first $r$ out of $n$ observations when the underlying p.d.f. is given by (1). These results are then used to estimate $\theta$ when the $N$ items are divided into $k$ sets $S_j$ (each containing $n_j > 0,$ items, $\sum^k_{j=1} n_j = N)$ and each set $S_j$ is observed only until the first $r_j$ failures occur $(0 < r_j \leqq n_j).$ The constants $r_j$ and $n_j$ are fixed and preassigned. Three different cases are considered: 1. The $n_j$ items in each set $S_j$ have a common known $A_j (j = 1, 2, \cdots, k).$ 2. All $N$ items have a common unknown $A.$ 3. The $n_j$ items in each set $S_j$ have a common unknown $A_j (j = 1, 2, \cdots, k).$ The results for these three cases are such that the results for any intermediate situation (i.e. some $A_j$ values known, the others unknown) can be written down at will. The particular case $k = 1$ and $A = 0$ is treated in [2]. The constant $A$ in (1) can be interpreted in two different ways: (i) $A$ is the minimum life, that is life is measured from the beginning of time, which is taken as zero. (ii) $A$ is the "time of birth", that is life is measured from time $A$. Under interpretation (ii) the parameter $\theta,$ which we are trying to estimate, represents the expected length of life.

Approximation Methods which Converge with Probability one

Abstract: Let $H(y\mid x)$ be a family of distribution functions depending upon a real parameter $x,$ and let $M(x) = \int^\infty_{-\infty} y dH(y \mid x)$ be the corresponding regression function. It is assumed $M(x)$ is unknown to the experimenter, who is, however, allowed to take observations on $H(y\mid x)$ for any value $x.$ Robbins and Monro [1] give a method for defining successively a sequence $\{x_n\}$ such that $x_n$ converges to $\theta$ in probability, where $\theta$ is a root of the equation $M(x) = \alpha$ and $\alpha$ is a given number. Wolfowitz [2] generalizes these results, and Kiefer and Wolfowitz [3], solve a similar problem in the case when $M(x)$ has a maximum at $x = \theta.$ Using a lemma due to Loeve [4], we show that in both cases $x_n$ converges to $\theta$ with probability one, under weaker conditions than those imposed in [2] and [3]. Further we solve a similar problem in the case when $M(x)$ is the median of $H(y \mid x).$

Sufficiency and Statistical Decision Functions

Abstract: This paper contains an account, in abstract terms, of sufficiency and of its role in statistical decision problems. The study of sufficiency in abstract terms was initiated by Halmos and Savage [1], and the present paper, although self-contained, is to be regarded as a continuation of their work. The main objects of the paper are to show that the justification for the use of sufficient statistics in statistical methodology which is sketched in the final section of [1] is valid under certain quite general conditions, and to extend this justification to the case of sequential experiments. The paper falls into two parts of which the first (Sections 2-7) is mainly expository and provides an account of the theory of sufficiency in the nonsequential case. The second part (Sections 8-11) then extends the theory to sequential experiments.

Extreme Values in Samples from $m$-Dependent Stationary Stochastic Processes

Abstract: The limiting distributions for the order statistics of $n$ successive observations in a sequence of independent and identically distributed random variables are shown to hold also when the sequence is generated by a stationary stochastic process of a certain moving average type. A sequence of random variables $\{x_i\}$ has been called $m$-dependent [3] if $| i - j | > m$ implies that $x_i$ and $x_j$ are independent. If the variables in a strictly stationary sequence are $m$-dependent and have a finite upper bound to their range of variation, the largest in a sample of $n$ successive members tends with probability one to this upper bound. This is a simple extension of Dodd's results [1] for the case of independence. The following theorem shows that when this upper bound is infinite, the asymptotic distribution of the largest in such a sample is the same as in the case of independence.

Universal Bounds for Mean Range and Extreme Observation

Abstract: Consider any distribution $f(x)$ with standard deviation $\sigma$ and let $x_1, x_2 \cdots x_n$ denote the order statistics in a sample of size $n$ from $f(x).$ Further let $w_n = x_n - x_1$ denote the sample range. Universal upper and lower bounds are derived for the ratio $E(w_n)/\sigma$ for any $f(x)$ for which $a\sigma \leqq x \leqq b\sigma,$ where $a$ and $b$ are given constants. Universal upper bounds are given for $E(x_n)/\sigma$ for the case $- \infty < x < \infty.$ The upper bounds are obtained by adopting procedures of the calculus of variation on lines similar to those used by Plackett [3] and Moriguti [4]. The lower bounds are attained by singular distributions and require the use of special arguments.

The Maxima of the Mean Largest Value and of the Range

Abstract: R. L. Plackett derived the maximum of the ratio of mean range to the standard deviation as function of the sample size, and gave the initial (symmetrical) distribution for which this maximum is actually reached. On the other hand, Moriguti derived the maximum for the mean largest value under the assumption that the distribution from which the maximum is taken is symmetrical. His mean value turned out to be one half of the value given by Plackett. In the following, these results will be generalized for an arbitrary (not necessarily symmetrical) continuous variate. The mean and the standard deviation of the largest value and the mean range will be given for two distributions: one where the mean largest value is a maximum, and another one where the mean range is a maximum. Obviously, a mean largest value can exist if and only if the initial mean exists. In addition we postulate in both cases the existence of the second moment.

A Property of the Normal Distribution

Abstract: The following theorem is proved. Let $X_1, X_2, \cdots, X_n$ be $n$ independently (but not necessarily identically) distributed random variables, and assume that the $n$th moment of each $X_i(i = 1, 2, \cdots, n)$ exists. The necessary and sufficient conditions for the existence of two statistically independent linear forms $Y_1 = \sum^n_{s=1} a_sX_s$ and $Y_2 = \sum^n_{s=1}b_sX_s$ are: (A) Each random variable which has a nonzero coefficient in both forms is normally distributed. $(B) \sum^n_{s=1}a_sb_s\sigma^2_s = 0$. Here $\sigma^2_s$ denotes the variance of $X_s (s = 1, 2, \cdots, n)$. For $n = 2$ and $a_1 = b_1 = a_2 = 1, b_2 = -1$ this reduces to a theorem of S. Bernstein [1]. Bernstein's paper was not accessible to the authors, whose knowledge of his result was derived from a statement of S. Bernstein's theorem contained in a paper by M. Frechet [3]. A more general result, not assuming the existence of moments was obtained earlier by M. Kac [4]. A related theorem, assuming equidistribution of the $X_i (i = 1, 2, \cdots n)$ is stated without proof in a recent paper by Yu. V. Linnik [5].

The estimation of biological populations

Abstract: : A number of statistical models, underlying the methods used in the estimation of the sizes and other parameters of animal population, are set up. The relevant estimation equations are given, with their variances and covariances. For the most part, the theory is designed for large populations. An effort was made to have the models conform as closely as possible to the practices of animal sampling. The complexities of estimating the birth, death, emigration, and immigration rates indicate that it will be necessary to set up special experiments to determine these factors adequately.

A Single-Sample Multiple Decision Procedure for Ranking Variances of Normal Populations

Abstract: A single-sample multiple decision procedure for ranking variances of normal populations is described. Exact small-sample methods and a large-sample method are given for computing the sample sizes necessary to guarantee a preassigned probability of a correct ranking under specified conditions on certain variance ratios. Some tables computed by these methods are provided.

Some Properties of Beta and Gamma Distributions

Abstract: The object of this paper is to present certain important properties of the Gamma distribution and the two kinds of Beta distributions, and to indicate certain useful applications of these two to sampling problems The distribution of the Studentised $D^2$-statistic under the null hypothesis is obtained in two different ways

Estimation of the mean and standard deviation by order statistics

Abstract: The aim of this paper is: (i) to find the best linear estimates of the means and standard deviations of the rectangular, triangular, exponential and double exponential populations; (ii) to compare the efficiencies of these estimates with some other estimates for small samples; (iii) to discuss the variation of coefficients in the best linear estimates as the population varies.

Asymptotic Behavior of Some Rank Tests for Analysis of Variance

Abstract: The $H$ test and the median test have been proposed for testing the hypothesis of the equality of $c$ probability distributions. Assuming translation-type alternatives, we find the limiting distributions of the $H$ and median test statistics. These results are used to derive general formulas for the asymptotic relative efficiencies of these tests with respect to one another and to the classical $F$ test. A short discussion of the computation of approximate power functions of these tests is also included.

Power Under Normality of Several Nonparametric Tests

Abstract: Presented are tabulations of the power and power efficiency of four nonparametric tests (rank-sum, maximum deviation, median, and total number of runs) for the difference in means of two samples drawn from normal populations with equal variance. The cases considered are for equal sample sizes of three, four and five observations and alternatives $\delta = | \mu_1 - \mu_2 |/\sigma$.

Spacing of Information in Polynomial Regression

Abstract: The purpose of this paper is to investigate a problem in the spacing of information in certain applications of polynomial regression. It is shown that for a polynomial of degree $m,$ the variance-covariance matrix of the estimated polynomial coefficients given by a spacing of information at more than $m + 1$ values of the sure variate can always be attained by spacing the same information at only $m + 1$ values of the sure variate, these spacing values being bounded by the first spacing values. The presented results are of use in experimental design involving polynomial regression when a choice of sure variate values is possible but restricted to a specified range. Let the polynomial under consideration be \begin{equation*}\tag{1.1} P(x) = \alpha_1 + \alpha_2x + \cdots + \alpha_{m+1}x^m, m \geqq 1,\end{equation*} and let $P(x_\epsilon) = y(x_\epsilon) + \delta_\epsilon, \epsilon = 1(1)N, N \geqq (m + 1).$ The $y(x_\epsilon)$ are observed uncorrelated variates with random error $\delta_\epsilon$ having mean zero and finite variance $\sigma^2_\epsilon > 0.$ The $x_\epsilon$ are observed variates without error, there being at least $(m + 1)$ distinct $x_\epsilon.$ The following notation is introduced. Let $\overset{\rightarrow}{x} = (1, x, x^2, \cdots, x^m), X = (\overset{\rightarrow}{x}_\epsilon), \epsilon = 1(1)N,$ and let $W$ be the $N \times N$ diagonal matrix with entry $w_\epsilon = 1/\sigma^2_\epsilon$ in the $(\epsilon, \epsilon)$ position. $w_\epsilon$ will henceforth be referred to as the "information" of $y(x_\epsilon),$ and $Q = \sum w_\epsilon, \epsilon = 1(1)N,$ will be referred to as the "total information." The matrix $X'WX$ will be called the "information matrix." The problem is to show that given a spacing of total information $Q$ at locations $x_\epsilon, \epsilon = 1(1)N, N \geqq (m + 1),$ there being at least $(m + 1)$ distinct $x_\epsilon,$ it is always possible to re-space $Q$ at $(m + 1)$ distinct locations $r_j, j = 1(1)(m + 1),$ in such a manner that $\min x_\epsilon \leqq r_j \leqq \max x_\epsilon, \epsilon = 1(1)N, j = 1(1)(m + 1),$ and $X'WX = R'UR$, with $R'UR$ being the information matrix of the re-spacing. The problem is solved by prescribing a method for finding the required $U$ and $R$ which determine the spacing of the total information. The motivation for the problem is as follows. In experimentation in the chemical engineering industry, we most often have control over our sure variates. The sure variate $x$ could be the pressure level of our process equipment, and we would be permitted to choose any operating pressure $x$ in the pressure range $\min x$ to $\max x,$ tolerated by our equipment. Quite often, and in particular with isotopic measurements, laboratory analytical determinations are required for our $y$-variates with the laboratory being the major source of error. With each laboratory determination having variance $\sigma^2,$ we can request $n_x$ determinations on the material sample taken at sure variate $x$. Using the average of the laboratory determinations, the corresponding $y$ variate has variance $\sigma^2/n_x.$ Specifying $Q$ then amounts to specifying total laboratory effort expended on the experiment. It might be set by such usual factors as the dollar allowance on the experiment; if the material is highly radioactive, it might be set by such unusual factors as exposure time allowed the laboratory analysts. Furthermore, in experimentation with fairly large equipment, it is important to minimize the distinct levels of operation, that is, the distinct number of sure $x$'s. The time required to make the change and to reach sufficient equilibrium representing steady-state operation of the process is often long. In any case we lose time, and with production line equipment, we also lose production. These are the reasons for minimizing the distinct number of sure $x$'s in the experiment. The equivalence $X'WX = R'UR$ gives the required minimization. If the functional relationship between $y$ and $x$ is adequately represented by a polynomial of degree $m$, the equivalence assures that only $(m + 1)$ distinct sure $x$'s are required to maintain the same efficiency of statistical evaluation of the experimental results, since most statistical evaluation will require $(X'WX)^{-1},$ which can now be replaced by $(R'UR)^{-1}.$ It may be seen that such experiments, common in physico-chemical industry, present a formulation and require a mathematical model not found in ordinary regression theory, where usually it is not possible to assign various values to the corresponding $y$ variances. With the indicated background in mind, the results of this paper find application in experimental design. The determination of a spacing which optimizes some criteria involving the information matrix is made simpler. A familiar example arising in point estimation is minimizing $\overset{\rightarrow}{p}(X'WX)^{-1}\overset\rightarrow{p}'$ for a specified row vector $\overset\rightarrow{p}.$ An example from interpolation is minimizing the maximum of $\overset{\xi}(X'WX)^{-1}\overset{\Xi}'$ with $\overset{\Xi} = (1, \Xi, \Xi^2 \cdots \xi^m)$ and $\min x_\epsilon \leqq \xi \leqq \max x_\epsilon;$ the extrapolation problem is similar. The advantage of applying the above result to such problems is that the spacing of information is at once reduced to $(m + 1)$ distinct locations, any larger number being unnecessary. The matrix $X$ then is the matrix of a Vandermonde determinant. The properties of these matrices are well known and attractive. These uses will be illustrated by an example given in Section 4.

On the Problem of Construction of Orthogonal Arrays

Abstract: A method of constructing orthogonal arrays of an arbitrary strength $t$ is formulated. This method is a modification of the method based on differences, formulated by R. C. Bose [1] for the purpose of constructing orthogonal arrays of strength 2. It is shown further that each of the multifactorial designs of R. L. Plackett and J. P. Burman [2], in which each factor takes on two levels, provide a scheme for constructing orthogonal arrays of strength 3, consisting of the maximum possible number of rows. An orthogonal array (36, 13, 3, 2) is constructed. The method used for its construction cannot lead to a number of constraints greater than 13. It is known however [3] that 16 is an upper bound for the number of constraints in this case; the problem as to whether this bound can actually be attained remains unsolved.