The estimation of variance by use of successive differences of higher order is discussed in this paper. Heretofore, attention has been focused, in published works, on estimates of variance obtained by employing the sum of squares of deviations from the mean and also by using mean square successive differences of the first order [1], [2], [3], [9]. A concise description of the method employing differences of any order with appropriate formulae for the precision of estimates so obtained and also a practical example on the use of the technique are given in section 11. Fundamental contributions to the estimation of variance from higher order differences, a study of the efficiency of the technique and proper orientation of the subject matter in the field of mathematical statistics are given in sections 2-10 of the paper.

Let $n$ successive independent observations be made on the same chance variable whose distribution function $f(x, \theta)$ depends on a single parameter $\theta$. The number $n$ is a chance variable which depends upon the outcomes of successive observations; it is precisely defined in the text below. Let $\theta^\ast(x_1, \cdots, x_n)$ be an estimate of $\theta$ whose bias is $b(\theta)$. Subject to certain regularity conditions stated below, it is proved that $\sigma^2(\theta^\ast) \geq \big(1 + \frac{db}{d\theta}\big)^2\big\lbrack EnE\big(\frac{\partial\log f}{\partial\theta}\big)^2\big\rbrack^{-1}.$ When $f(x, \theta)$ is the binomial distribution and $\theta^\ast$ is unbiased the lower bound given here specializes to one first announced by Girshick [3], obtained under no doubt different conditions of regularity. When the chance variable $n$ is a constant the lower bound given above is the same as that obtained in [2], page 480, under different conditions of regularity. Let the parameter $\theta$ consist of $l$ components $\theta_1, \cdots, \theta_l$ for which there are given the respective unbiased estimates $\theta^\ast_1(x_1, \cdots, x_n), \cdots, \theta^\ast_1(x_1, \cdots, x_n)$. Let $\|\lambda_{ij}\|$ be the non-singular covariance matrix of the latter, and $\|\lambda^{ij}\|$ its inverse. The concentration ellipsoid in the space of $(k_1, \cdots, k_l)$ is defined as $\sum_{i,j} \lambda^{ij}(k_i - \theta_i)(k_j - \theta_i) = l + 2.$ (This valuable concept is due to Cramer). If a unit mass be uniformly distributed over the concentration ellipsoid, the matrix of its products of inertia will coincide with the covariance matrix $\|\lambda)_{ij}\|$. In [4] Cramer proves that no matter what the unbiased estimates $\theta^\ast_1, \cdots, \theta^\ast_l$, (provided that certain regularity conditions are fulfilled), when $n$ is constant their concentration ellipsoid always contains within itself the ellipsoid $\sum_{i,j} \mu_{ij}(k_i - \theta_i)(k_j - \theta_j) = l + 2$ where $\mu_{ij} = nE\big(\frac{\partial\log f}{\partial\theta_i}\frac{\partial\log f}{\partial\theta_i}\big).$ Consider now the sequential procedure of this paper. Let $\theta^\ast_1, \cdots, \theta^\ast_l$ be, as before, unbiased estimates of $\theta_1, \cdots, \theta_l$, respectively, recalling, however, that the number of $n$ of observations is a chance variable. It is proved that the concentration ellipsoid of $\theta^\ast_1, \cdots, \theta^\ast_l$ always contains within itself the ellipsoid $\sum_{i,j} \mu'_{ij}(k_i - \theta_i)(k_j - \theta_j) = l + 2$ where $\mu'_{ij} = EnE\big(\frac{\partial\log f}{\partial\theta_i}\frac{\partial\log f}{\partial\theta_j}\big).$ When $n$ is a constant this becomes Cramer's result (under different conditions of regularity). In section 7 is presented a number of results related to the equation $EZ_n = EnEX$, which is due to Wald [6] and is fundamental for sequential analysis.

In this article certain contributions are made to the theory of estimating linear functions of cell proportions in connection with the methods of (1) least squares, (2) minimum chi-square, and (3) maximum likelihood. Distinctions among these three methods made by previous writers arise out of (1) confusion concerning theoretical vs. practical weights, (2) neglect of effects of correlation between sampling errors, and (3) disagreement concerning methods of minimization. Throughout the paper the equivalence of these three methods from a practical point of view has been emphasized in order to facilitate the integration and adaptation of existing statistical techniques. To this end: 1. The method of least squares as derived by Gauss in 1821-23 [6, pp. 224-228] in which weights in theory are chosen so as to minimize sampling variances is herein called the ideal method of least squares and the theoretical estimates are called ideal linear estimates. This approach avoids confusion between practical approximations and theoretical exact weights. The ideal method of least squares is applied to uncorrelated linear functions of correlated sample frequencies to determine the appropriate quantity to minimize in order to derive ideal linear estimates in sample-frequency problems. This approach leads to a sum of squares of standardized uncorrelated linear functions of sampling errors in which statistics are to be substituted in numerators. 3. A new elementary method is used to reduce the sum of squares in (2)--before substitution of statistics--to Pearson's expression for chi-square. In this result, obtained without approximation, appropriate substitution of statistics shows that the denominators of chi-square should be treated as constant parameters in the differentiation process in order to minimize chi-square in conformity with the ideal method of least squares. 4. The ideal method of minimum chi-square, derived in (3) as the sample-frequency form of the ideal method of least squares, yields ideal linear estimates in terms of the unknown parameters in the denominators of chi-square. When these parameters are estimated by successive approximations in such a way as to be consistent with statistics based on them, it is shown that the method of minimum chi-square leads to maximum likelihood statistics. 5. An iterative method which converges to maximum likelihood estimates is developed for the case in which observations are cross-classified and first order totals are known. In comparison with Deming's asymptotically efficient statistics, it is shown that, in a certain sense, maximum likelihood statistics are superior for any given value of $n$--especially in small samples. 6. The method of proportional distribution of marginal adjustments is developed. This method yields estimates of expected cell frequencies whose efficiency is 100 per cent when universe cell frequencies are proportional--a condition closely approximated in most practical surveys for which first order totals are available from complete censuses. Whether this favorable condition is satisfied or not, the method yields results which are easy to interpret and it has many computational advantages from the point of view of economy of time and effort. Throughout the article discussion is confined to the estimation of parameters whose relationships to cell proportions are linear. However, most of the results can be extended to the case of non-linear relationships, the necessary qualifications being similar to those in curve-fitting problems when the functions to be fitted is not linear in its parameters. In this case, of course, least squares estimates are not linear estimates. In particular, obvious extensions of the general proofs in sections 5 and 6 make them applicable to the non-linear case. Thus even when relationships are non-linear, it can be shown that the method of minimum chi-square is the sample-frequency form of the method of least squares which leads (by means of appropriate successive approximations) to maximum likelihood statistics in sample-frequency problems. This principle which establishes the equivalence of the methods of least squares, minimum chi-square, and maximum likelihood greatly facilitates the integration and adaptation of existing techniques developed in connection with these important methods of estimation.

Let $x$ and $y$ follow a normal bivariate probability function with means $\bar X, \bar Y$, standard deviations $\sigma_1, \sigma_2$, respectively, $r$ the coefficient of correlation, and $\rho_1 = \bar X/\sigma_1, \rho_2 = \bar Y/\sigma_2$. Professor C. C. Craig [1] has found the probability function of $z = xy/\sigma_1\sigma_2$ in closed form as the difference of two integrals. For purposes of numerical computation he has expanded this result in an infinite series involving powers of $z, \rho_1, \rho_2$, and Bessel functions of a certain type; in addition, he has determined the moments, semin-variants, and the moment generating function of $z$. However, for $\rho_1$ and $\rho_2$ large, as Craig points out, the series expansion converges very slowly. Even for $\rho_1$ and $\rho_2$ as small as 2, the expansion is unwieldy. We shall show that as $\rho_1$ and $\rho_2 \rightarrow \infty$, the probability function of $z$ approaches a normal curve and in case $r = 0$ the Type III function and the Gram-Charlier Type A series are excellent approximations to the $z$ distribution in the proper region. Numerical integration provides a substitute for the infinite series wherever the exact values of the probability function of $z$ are needed. Some extensions of the main theorem are given in section 5 and a practical problem involving the probability function of $z$ is solved.

The modified $t$-test considered by Daly$^1$ (see [1]) is used to develop one-sided significance tests to decide whether the mean of a new normal population exceeds the mean of an old normal population having the same variance. Significance tests are also developed to decide whether the mean of the new population is less than the mean of the old population. These tests require very little computation for their application and are approximately as powerful as the most powerful tests of these hypotheses.