Criteria for evaluating goodness-of-fit tests are reviewed and two additional criteria proposed. The several goodness-of-fit tests which have been proposed are studied in the light of these criteria. It is shown that it is relatively easy to evaluate the maximum and minimum power of those tests which are "partially ordered" against alternatives at a fixed "distance" from the hypothesis. A comparison is made of five tests on the basis of such minimum and maximum power functions.

Many commonly employed symmetrical designs such as Balanced Incomplete Block Designs (BIBD's), Latin Squares (LS's), Youden Squares (YS's), etc., are shown to have optimum properties among the class of non-randomized designs (Section 3). This represents an extension of a property first proved by Wald for LS's in [1]; a similar property demonstrated by Ehrenfeld for LS's in [2] (as well as a third optimum property considered here) is shown to be an immediate consequence of the Wald property, and the Wald property is shown to be the more relevant when one considers optimality rigorously (Section 2). Surprisingly, all of these optimum properties fail to hold if randomized designs are considered (Section 4); the results of Sections 2 and 3, as well as those appearing previously in the literature (as in [1], [2], [3]) must be interpreted in this sense. Generalizations of the BIBD's and YS's, for which analogous results hold, are introduced.

Several renewal processes related to the Type I and Type II counter models are defined and studied. The distribution and characteristic functions for the secondary (or output) process of the Type I counter model are obtained explicitly. Both the non-stationary and stationary probabilities of the state of the counter, (locked or unlocked), are derived. Integral equations determining the distribution and characteristic functions for the secondary process of the Type II counter model are obtained. Also it is shown that a more general model proposed by Albert and Nelson [1] may be solved explicitly in terms of a corresponding Type II counter model. An example of this general model is given. Related with each model is a discrete renewal process which is also studied.

A minimum variance unbiased estimator is provided for the parameter of a truncated Poisson distribution in the case of truncation on the left. In this connection the distribution is obtained for the sum of $n$ independent identically distributed truncated Poisson random variables, and then well-known properties of sufficient statistics are employed to obtain the estimator. For the case of truncation away from the zero value results are expressed in terms of Stirling numbers of the second kind. The estimator has a particularly simple form and tables are available for its computation. For the general case results are expressed in terms of what we call generalized Stirling numbers. As a by-product of the statistical considerations there arises an identity between generalized Stirling numbers which may be useful in the study of Difference Equations.

R. C. Bose [1] has considered the problem of balancing in symmetrical factorial experiments. In all the designs considered in that paper, the block size is a power of $S$, the number of levels of a factor. The purpose of the present paper is to consider a general class of designs, where a `complete balance' is achieved over different effects and interactions. It is proved in this paper (Theorems 4.1 and 4.2) that if a `complete balance' is achieved over each order of interaction, the design must be a partially balanced incomplete block design. Its parameters are found. The usual method of analysis (of a PBIB design [2]) which is not so simple, can be simplified a little for these designs (section 5), on account of the balancing of the interactions of various orders. The simplified method of analysis is illustrated by a worked out example 5.1. Finally, the problem of balancing is dealt with for asymmetrical factorial experiments also. Incidentally, it may be observed that the generalised quasifactorial designs discussed by C. R. Rao [4] are the same as found by the author, from considerations of balancing.

The decision problem considered here is that of deciding which element of a finite parametric family of probability distributions $p(x, \mu)$ represents the true distribution of the statistic $X$. It is assumed that $p(x, \mu)$ satisfies certain regularity conditions which essentially require that the parameter $\mu$ be integer-valued with known bounds and that $p(x, \mu_1)/p(x, \mu_0)$ be an increasing function of $x$ whenever $\mu_0 < \mu_1$. Complete classes are characterized for various loss functions $W(\mu, \alpha)$ which are convex functions of the decision $\alpha$ for each fixed value of $\mu$. Minimax procedures are considered for the case $W(\mu, \alpha) = |\alpha - \mu|^k$.

The estimation of a parameter lying in a subset of a set of possible parameters is considered. This subset is the null space of a well-behaved function and the estimator considered lies in the subset and is a solution of likelihood equations containing a Lagrangian multiplier. It is proved that, under certain conditions analogous to those of Cramer, these equations have a solution which gives a local maximum of the likelihood function. The asymptotic distribution of this `restricted maximum likelihood estimator' and an iterative method of solving the equations are discussed. Finally a test is introduced of the hypothesis that the true parameter does lie in the subset; this test, which is of wide applicability, makes use of the distribution of the random Lagrangian multiplier appearing in the likelihood equations.

In this paper confidence bounds are obtained (i) on the ratio of variances of a (possibly) correlated bivariate normal population, and then, by generalization, (ii) on a set of parametric functions of a (possibly) correlated $p + p$ variate normal population, which plays the same role for a $2p$-variate population as the ratio of variances does for the bivariate case, (iii) on the ratio of means of the population indicated in (i), and, by generalization, (iv) on a set of parametric functions of the population indicated in (ii), which plays the same role for this problem as the ratio of means does for the bivariate case. For (i) and (iii) the confidence coefficient is any preassigned $1 - \alpha$ and the distribution involved is the central $t$-distribution, while for (ii) and (iv), the confidence statement in each case is a simultaneous one with a joint confidence coefficient greater than or equal to a preassigned $1 - \alpha$. For (ii) the distribution involved is that of the central largest canonical correlation coefficient (squared), and for (iv) the distribution involved is that of the central Hotelling's $T^2$. As far as the authors are aware the results on (ii) and (iv) are new and so perhaps that on (i). But the result on (iii) has been in the field for a long time in various superficially different forms. An important point to keep in mind on these problems is that, for such confidence bounds and the associated tests of hypotheses to be physically meaningful, the two variates for the bivariate distribution should be comparable. For example, they might refer to the same characteristic of a set of individuals before and after a feed. Likewise, for a $(p + p)$-variate distribution, the $p$ variates of the first set should be comparable to $p$ variates of the second set. For example, they might refer to several characteristics of a set of individuals before and after a treatment. In each case the confidence bounds are obtained by inverting the test of a certain hypothesis, which is indicated at its proper place. Thus, for the $(p + p)$-variate problem, we assume that there are $p$ pairs of comparable variates and it is the pairwise comparison for these $p$ pairs that seems, in this situation, to be physically more meaningful than anything else. Any general bounds that will be obtained in this paper are to be regarded, in a large measure, as a means to this end, although there could conceivably be physical questions, some of which will be illustrated in a later applied paper to be published elsewhere, to which these more general bounds would be pertinent.

If $y_1, \cdots, y_N$ are observations on a stationary time series at equal intervals of time and it is known that $Ey_t = 0$ for $t = 1, \cdots, N$, the most natural definition of a serial correlation coefficient with lag unity would be $$r^{\ast} = \Bigg(\sum^{N - 1}_{i = 1} y_iy_{i + 1}\Bigg) \Bigg\lbrack\Bigg(\sum^{N - 1}_{i = 1} y^2_i\Bigg)\Bigg(\sum^{N - 1}_{i = 1} y^2_{i + 1}\Bigg)\Bigg\rbrack^{-1/2}$$ if the denominator $\neq 0$. This is the ordinary correlation coefficient between $(y_1, \cdots, y_{N - 1}$ and $(y_2, \cdots, y_N)$, except that instead of taking deviations from the sample mean, we have taken deviations from the population means. Due to the seemingly unsurmountable mathematical difficulties involved in obtaining the distribution of $r^{\ast}$ even on the hypothesis of independence and normality of the observations, several alternative definitions have been proposed as approximations to $r^{\ast}$. However, it is desirable to consider some relevant properties of the distribution of $r^{\ast}$. In this paper the distribution of $r^{\ast}$ near the extremities of its range will be considered. The observations will be assumed to be distributed as independent $N(0, 1)$ variates. There is no loss of generality in assuming the variance to be unity as $r^{\ast}$ is independent of the scale parameter. A geometrical approach suggested by Hotelling seemed to be particularly suitable in obtaining the order of contact of the distribution curve at $r^{\ast} = \pm 1$. Hotelling [1] shows how to determine the order of contact of frequency curves of some statistics with the variate axis at the ends of the range even though the actual distributions are unknown. It will be shown here that if for a number $r_0$ in $\lbrack 0, 1\rbrack$ and close to $1, P(r^{\ast} \geqq r_0)$ is expanded in a series of powers of $(1 - r_0)$, the first non-zero coefficient is that of the power $(N - 2)/2$. Upper and lower bounds for the coefficient of this power will be calculated. The lower bound is positive and the upper bound gives an approximation for an upper bound on $P(r^{\ast} \geqq r_0)$.

The charts presented in this paper are designed to facilitate the estimation of the number of observations per treatment required for analysis of variance tests of specified power. They are intended for use by experimenters dealing with fixed treatments effects. With these charts the experimenter may answer the following question: How many observations must I use per treatment to obtain a power of $P_1$ against alternative hypothesis $H_1$? Charts are presented for use with tests of treatments effects involving two to five levels of the treatment variable. The charts are strictly valid only for the completely randomized design; however they may be applied with relatively little error to tests of treatments effects in the randomized block and factorial designs, the latter employing a within-cells estimate of error variance.

The mixed model in a 2-way analysis of variance is characterized by a fixed classification, e.g., treatments, and a random classification, e.g., plots or individuals. If we consider $k$ different treatments each applied to everyone of $n$ individuals, and assume the usual analysis of variance assumptions of uncorrelated errors, equal variances and normality, an appropriate analysis for the set of $nk$ observations $x_{ij}, i = 1, 2, \cdots n, j = 1, 2, \cdots k$, is ???? where the $F$ ratio under the null hypothesis has the $F$ distribution with $(k - 1)$ and $(k - 1)(n - 1)$ degrees of freedom. As is well known, if we extend the situation so that the errors have equal correlations instead of being uncorrelated, the $F$ ratio has the same distribution. Under the null hypothesis, the numerator estimates the same quantity as the denominator, namely, $(1 - \rho)\sigma^2$, where $\rho$ is the constant correlation coefficient among the treatments. This case can also be considered as a sampling of $n$ vectors (individuals) from a $k$-variate normal population with variance-covariance matrix $$V = \sigma^2 \begin{pmatrix} 1 & \rho & \cdots & \rho \\ \rho & & & \vdots \\ \vdots & & & \rho \\ \rho & \cdots & \rho & 1\end{pmatrix}.$$ If we consider this type of formulation and suppose the $k$ treatment errors to have a multivariate normal distribution with unknown variance-covariance matrix (the same for each individual), then the usual test described above is valid for $k = 2$. For $k > 2$, and $n \geqq k$, Hotelling's $T^2$ is the appropriate test for the homogeneity of the treatment means. However, the working statistician is sometimes confronted with the case where $k > n$, or he does not have the adequate means for computing large order inverse matrices and would therefore like to use the original test ratio which in general does not have the requisite $F$ distribution. Box [1] and [2] has given an approximate distribution of the test ratio to be $F\lbrack(k - 1)\epsilon, (k - 1)(n - 1)\epsilon\rbrack$ where $\epsilon$ is a function of the population variances and covariances and may further be approximated by the sample variances and covariances. We show in Section 3 that $\epsilon \geqq (k - 1)^{-1}$, and therefore a conservative test would be $F(1, n - 1)$. Box referred only to one group of $n$ individuals. We shall extend his results to a frequently occurring case, namely, the analysis of $g$ groups where the $\alpha$th group has $n_\alpha$ individuals, $\alpha = 1, 2, \cdots g$, and $\Sigma^g_{\alpha = 1} n_\alpha = N$. We will show that the treatment mean square and the treatment $\times$ group interaction can be tested in the same approximate fashion by using the Box procedure.

This note is concerned with possible definitions of relative efficiency for two sequences of tests of the same hypothesis. For two examples of one kind of definition, relative efficiencies of the Student test and sign test against normal alternatives are calculated for fixed sample size and asymptotically.

It is shown that for a given set of parameters ($b$ blocks, $k$ plots per block and $v$ treatments), among the class of connected incomplete block designs, a balanced incomplete block design (if it exists) is the design which maximizes the minimum efficiency, efficiency being defined as $$\frac{\text {Variance of an estimated treatment contrast in a randomized block}{Variance of the estimated treatment contrast in the incomplete block}}.$$ The proof will be preceded by a lemma.

The notation P.B.I.B. $(m)$ will mean partially balanced incomplete block design with $m$ associative classes. It is found that the $C$ matrix of a P.B.I.B $(m)$ may be expressed as a linear function of $m + 1$ commutative and linearly independent matrices. The author feels that this decomposition may be of interest to those studying the properties of P.B.I.B. designs.

Cohen [1] and Des Raj [2] have shown that in estimating the parameters of truncated type III populations, it is necessary to calculate for several values of $x$ the Mills ratio of the ordinate of the standardized type III curve at $x$ to the area under the curve from $x$ to $\infty$. Des Raj [3] has also noted that for large values of $x$ the existing tables of Salvosa [4] are inadequate for this purpose and he has found lower and upper bounds for the ratio. The object of this note is to improve these bounds, by obtaining monotonic sequences of lower and upper bounds through the use of continued fractions.