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.

In this paper an approximate confidence interval is found for the expected value of the difference between two quantities which are independently distributed proportionally to $\chi^2$ variates. Three methods are used. The first is based on the work of Welch [13], [14] and Aspin [1], [2] on the generalized "Student's" problem, and involves neglecting successively higher powers of the reciprocal of one of the degrees of freedom. This method is used to check the other two solutions, both of which involve neglecting successive increasing and decreasing powers, respectively, of a nuisance parameter. Finally a solution is formed using those resulting from the second and third methods, and is more accurate than those solutions. The order of accuracy, and the use of the final solution, are discussed. The paper does not present a method of computing confidence intervals in a form suitable for immediate practical application. Series developments of a certain hypothetical function are given; more remains to be said about the relation between the series and the function, and the problem of computing tables. A computational exploration of the solution is at present in hand.

In this paper a simple sequential test is suggested. Distribution of the sample size, its moment generating function, the power function of the test, and the ASN (average sample number) function are obtained. The determination of the set of relative optimum zones for making decisions is shown to be unique. The existence of a class of sets of absolute optimum zones is proved. The suggested test is shown to be consistent. Some possible applications are discussed and a few numerical efficiencies are calculated.

The purpose of this paper is to discuss the probability distribution that arises when the probability of success at any trial depends linearly upon the number of previous successes. Such a scheme has obvious uses in both biological and economic fields. It will be shown that by assuming a simple linear relationship between the number of previous successes and the probability of success in the next trial, we can derive a distribution that is reasonably easy to handle, provides as good a fit as more usual distributions, and has parameters which are capable of easy physical interpretation. Moreover, for appropriate values of the parameters the negative binomial and the Gram-Charlier systems can be shown to be close approximations.

This paper investigates the problem of identifiability and estimability of linear structures in $n$ dimensions. The concept of identifiability is examined to elucidate the senses in which it may be interpreted in the present problem. Particular attention is given to the question of treating linear subspaces rather than specific coordinate systems. Necessary and sufficient conditions for identifiability are obtained under the assumption that the "errors" follow a multinormal distribution.

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.

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.

The Cramer-Rao inequality provides, under certain regularity conditions, a lower bound for the variance of an estimator [7], [15]. Various generalizations, extensions and improvements in the bound have been made, by Barankin [1], [2], Bhattacharyya [3], Chapman and Robbins [5], Fraser and Guttman [11], Kiefer [12], and Wolfowitz [16], among others. Further considerations of certain inequality properties of a measure of information, discussed by Kullback and Leibler [14], yields a greater lower bound for the information measure (formula (4.11)), and leads to a result which may be considered a generalization of the Cramer-Rao inequality, the latter following as a special case. The results are used to define discrimination efficiency and estimation efficiency at a point in parameter space.

This paper is a follow-up of a previous paper [1], the full implications of some of the results there being brought out here in terms that are physically more meaningful. Two cases of simultaneous confidence bounds, I and II, are given, in each case with a confidence coefficient which is to be greater than or equal to a preassigned level. Case I relates to the characteristic roots of $\sigma$ and $\sigma_1\sigma^{-1}_2$, where $\sigma$ stands for the dispersion matrix of one $p$-variate and $\sigma_1$ and $\sigma_2$ for the dispersion matrices of two $p$-variate normal populations. Case II relates to a $(p + q)$-variate normal population $(p \leqq q)$, for which the matrix of regression of the $p$-set on the $q$-set is defined in a natural manner. This matrix is denoted by $\beta(p \times q)$ and simultaneous confidence bounds are given on all bilinear compounds of this matrix (with arbitrary coefficient vectors of unit modulus). Confidence bounds on the characteristic roots of $\sigma$ and $\sigma_1\sigma^{-1}_2$ are given respectively by (3.1.3) and (3.2.8). Confidence bounds on the bilinear compounds of the regression matrix $\beta$ are given by (4.7).

Consider a random sample of size $n$ taken from a continuous distribution $f(x)$. Let another random sample, independent of the first sample and also of size $n$, be drawn from the same population. Let $U^n_r$ be the random variable associated with the number of values in the second sample which exceed the $r$th smallest value in the first sample. Similarly let $V^n_s$ be the random variable associated with the number of values in the second sample which exceed the $s$th largest value in the first sample. Due to the fact that the $r$th smallest value in a sample of size $n$ is at the same time the $s$th largest value in the sample with $s = n - r + 1$, it follows that \begin{equation*}\tag{1} \mathrm{Pr} (U^n_r = x) \equiv \mathrm{Pr} (V^n_s = x),\end{equation*} $s = n - r + 1; \quad r = 1,2, \cdots, n;\quad x = 0,1,2, \cdots, n.$ The probability distribution of $U^n_r$ (and hence of $V^n_s)$ is given by: \begin{equation*}\tag{2} \mathrm{Pr} (U^n_r = x) = \binom{n-x+r-1}{r-1}\binom{n-r+x}{x}\binom/{2n}{n} = \frac{1}{2}P_{n-x+r-1,r-1} P_{n-r+x,x}/P_{2n,n},\end{equation*} $x = 0, 1, 2, \cdots, n.$ Formula (2) can be proved by combinatorial methods; details are omitted. An alternative formula, derived in another way [3], is \begin{equation*}\tag{2a} \mathrm{Pr}(U^n_r = x) = \frac{1}{2}\binom{n-1}{r-1}\binom{n}{x}/\binom{2n-1}{n-r_x} = \frac{1}{2} P_{n-1,r-1}P_{n,x}/P_{2n-1,n-r+x}.\end{equation*} In formulae (2) and (2a), $P_{n,x} = (\frac{1}{2})^n\binom{n}{x}$. Formulae in terms of $P_{n,x}$ are particularly convenient for hand computation, since one can use the extensive tables of the binomial probability distribution published by the National Bureau of Standards. If the values of $\mathrm{Pr} (U^n_r \leqq x), \text{for} x = 0, 1, 2, \cdots, n - 1, r = 1,2, \cdots, n$ are written (for fixed $n$) in matrix form, one notes certain useful symmetries, which can be expressed by the identities \begin{align*}\tag{3} \mathrm{Pr} (U^n_r &\leqq x) = \mathrm{Pr} (U^n_{x+1} \leqq r - 1), \\ \tag{4} \mathrm{Pr} (U^n_r &\leqq x) + \mathrm{Pr} (U^n_{n-r+1} \leqq n - x - 1) = 1. \\ \end{align*} If one takes $x = n - r$ in (4) and uses the relation (3), it is readily verified that \begin{equation*}\tag{5} \mathrm{Pr} (U^n_r \leqq n - r) = \frac{1}{2}.\end{equation*} Proofs of (3), (4), and (5) can be obtained by using the results of pages 257-258 of [3]. Because of these symmetries, the complete matrix (for any fixed $r$) can be constructed if one knows only the quantities, $\mathrm{Pr} (U^n_r \leqq x), r = 1(1)\lbrack n/2\rbrack, x = r - 1, r, r + 1, \cdots, n - r - 1.$ In Table 1 these values are given for $n = 2(1)15(5)20.$ To see how the complete matrix is obtained from Table 1, it is interesting to verify, using (3), (4), and (5), that the complete matrix, in the special case $n = 5,$ is given by Table 2. A somewhat different, but related, exceedance problem is to take two random samples of size $n$ from a continuous distribution $f(x).$ Let us for convenience attach the letter $x$ to one of the samples and the letter $y$ to the other sample. Further let $x_{r,n}$ and $y_{r,n}$ be respectively the $r$th smallest observations in each of the samples. Let us define $Z_{r,n} = \max (x_{r,n}, y_{r,n}).$ If $Z_{r,n} = x_{r,n}$, count the number of $y$'s which are $\geqq x_{r,n};$ if $z_{r,n} = y_{r,n}$ count the number of $x$'s which are $\geqq y_{r,n}.$ Denoting the number of exceedances as $W^n_r,$ it is readily seen from (1) that the probability distribution of $W^n_r$ is given by \begin{equation*}\tag{6} \mathrm{Pr}(W^n_r = x) = 2\binom{n-z+r-1}{r-1}\binom{n-r+x}{x}/\binom{2n}{n},\quad x = 0,1,2,\cdots, n - r.\end{equation*} It is evident from the definition that, \begin{equation*}\tag{7} \mathrm{Pr}(W^n_r \leqq x) = 1,\quad x \geqq n - r.\end{equation*} Clearly one can find the values of $\mathrm{Pr}(W^n_r \leqq x)$ by using Table 1. Thus, for example, in the special case $n = 5$ one obtains Table 3.

A family of probability distributions is called "factor-closed" (f.c.) if it is closed under the operation of factorization. The classical binomial family and certain generalizations of its are shown to be f.c. The multinomial family is also f.c. Most families of infinitely divisible distributions are not f.c.

A systematic approach to distributions having the reproductive property (see [1] p. 171) is attempted, and necessary and sufficient conditions are given. The case of distributions depending on $k (> 1)$ parameters is considered; it need not be a straightforward generalization of the one-parameter case.

We consider cases where we have a finite number of decisions and a finite number of possible distributions, and we confine attention to procedures which have zero probability of continuing beyond the $N$th observation, where $N$ is a given positive integer. We find a class $C$ of procedures such that given any procedure $R$, there is a member of $C$, say $R'$, such that the probabilities of coming to the various decisions under the various distributions when using $R'$ are at least as desirable as when using $R$, and such that we are at least as likely to take fewer than $n$ observations under $R'$ as under $R$, for any $n$. Various extensions are indicated.

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.

An asymptotic expression is given for the probability integral of range for samples from a symmetrical unimodal population. Its accuracy is investigated for the case of a normal parent population and for sample sizes from 20 to 100. Over this range errors are small, and by using a correction based on values given below the probability integral can be found with a maximum error of 0.0001. Percentage points of range in the normal case are tabled for $n =$ 20, 40, 60, 80 and 100.