The Bayes sequential design is obtained for an optimization problem involving the choice of experiments. Given are experiments $A, B$, densities $p_1, p_2$, a positive integer $N$ and a number $\xi \varepsilon \lbrack 0, 1\rbrack$. A sequence of $N$ observations is to be made such that at each stage either $A$ or $B$ is observed, the loss being 1 if the experiment with density $p_2$ is chosen, 0 otherwise. $\xi$ is the prior probability that $A$ has density $p_1$. If the mean of $p_1$ is bigger than the mean of $p_2$ one obtains a more common version of the "two-armed bandit" (see e.g. [1]). The principal result of this paper is a proof of optimality for the procedure which at each stage chooses the experiment with higher posterior probability of being correct. Some attention is also given to the problem of computing risk functions.

If $\{Y_n\}$ is a stationary ergodic Markov process taking on values in a finite set $\{1, 2, \cdots, A\}$, then its entropy can be calculated directly. If $\phi$ is a function defined on $1, 2, \cdots, A$, with values $1, 2, \cdots, D$, no comparable formula is available for the entropy of the process $\{Y_n = \phi(Y_n)\}$. However, the entropy of this functional process can be approximated by the monotonic functions $\bar{G}_n = h(X_n \mid X_{n-1}, \cdots, X_1)$ and $\underline{G}_n = h(X_n \mid X_{n-1}, \cdots, X_1, Y_0)$, the conditional entropies. Furthermore, if the underlying Markov process $\{Y_n\}$ has strictly positive transition probabilities, these two approximations converge exponentially to the entropy $H$, where the convergence is given by $0 \leqq \bar{G}_n - H \leqq B_\rho^{n-1}$ and $0 \leqq H - \underline{G}_n \leqq B\rho^{n-1}$ with $0 < \rho < 1, \rho$ being independent of the function $\phi$.

This paper considers the problem of approximating a discrete time, discrete states, non-Markov process by a continuous diffusion process. The problem is set in the context of population genetics but may have more general application. In the genetic situation, most authors have treated the stochastic behavior of a gene frequency subject to evolutionary factors by assuming its probability distribution may be approximated by the solution of the Fokker-Planck diffusion equation. It is here shown that under certain sufficient conditions such an approximation is valid, even for genetic models in which the gene frequency does not necessarily form a Markov process. A summary of some old and new results concerning the asymptotic behavior of gene frequency is given, with special emphasis on the case when mutation is absent so that an absorbing state will ultimately be reached.

Location and scale parameters, on the one hand, and distributions admitting sufficient statistics for the parameters, on the other, have played a large role in the development of modern statistics. This paper deals with the problem of finding those distributions involved in the intersection of these two domains. In Sections 2 through 4 the preliminary definitions and lemmas are given. The main results found in Theorems 1 through 4 may be considered as a strengthening of the results of Dynkin [3] and Lindley [8]. Theorem 1 discovers the only possible forms assumed by the density of an exponential family of distributions having a location parameter. These forms were discovered by Dynkin under the superfluous assumptions that a density with respect to Lebesgue measure exist and have piecewise continuous derivatives of order one. Theorem 2 consists of the specialization of Theorem 1 to one-parameter exponential families of distributions. The resulting distributions, as found by Lindley, are either (1), the distributions of $(1/\gamma) \log X$, where $X$ has a gamma distribution and $\gamma \neq 0$, or (2), corresponding to the case $\gamma = 0$, normal distributions. In Theorem 3, the result analogous to Theorem 2 for scale parameters is stated. In Theorem 4, those $k$-parameter exponential families of distributions which contain both location and scale parameters are found. If the parameters of a two-parameter exponential family of distributions may be taken to be location and scale parameters, then the distributions must be normal. The final section contains a discussion of the family of distributions obtained from the distributions of Theorem 2 and their limits as $\gamma \rightarrow \pm \infty$. These limits are "non-regular" location parameter distributions admitting a complete sufficient statistic. This family of distributions is a main class of distributions to which Basu's theorem (on statistics independent of a complete sufficient statistic) applies. Furthermore, this family is seen to provide a natural setting in which to prove certain characterization theorems which have been proved separately for the normal and gamma distributions. Concluding the section is a theorem which, essentially, characterizes the gamma distribution by the maximum likelihood estimate of its scale parameter.

It is well known that the family of Pareto distributions with densities \begin{equation*}\begin{align*}f(x; a, p) = pa^p/x^{p+1},\qquad x > a > 0, \\ = (1.1) \\ 0,\qquad x \leqq a, p > 0,\end{align*}\end{equation*} provides reasonably good fits to many empirical distributions, e.g., to distributions of income and of property values. In most of these cases, ancillary information is present, which could be utilized if an appropriate multivariate Pareto distribution were available. The objects of this note are (i) to suggest two families of bivariate Pareto distributions with the property that both marginal distributions are of univariate Pareto form; (ii) to extend these to multivariate forms; and (iii) to discuss estimation of the parameters in the bivariate distributions.

The operational method of Mellin transforms is employed here to derive some interesting distribution functions. The distribution of the product of two non-central chi-square variates is obtained and some special cases exhibited. Finally, an application of the derived distributions to a problem in products of complex numbers is discussed.

The purpose of this paper is to point out (and to use) the relations of certain statistical concepts with "statistical" thermodynamics. (A) It is observed that Gibbs's "canonical distribution" of energy is precisely what statisticians have later labeled a "distribution of the exponential type". It follows that a rigorous treatment of the canonical law can be based upon the concept of "sufficiency", which is thereby related to the physical idea of "thermal equilibrium" and to the "zero-th principle of thermodynamics". In other words, the theory of physical fluctuations can be based upon "principles" very similar to those of the "phenomenological", or "classical, non-statistical" thermodynamics. Naturally, our results will be less detailed than those of statistical mechanics. However, the foundations of the latter theory still raise a host of unanswered problems, and it seems good in the meantime to show that the less powerful phenomenological theory has a wider scope than is commonly thought (see also [15]). The possibility of a purely phenomenological approach to statistical thermodynamics is not in itself a new idea. A procedure somewhat similar to ours has indeed been long ago suggested in Szilard's admirable, but very difficult and neglected, paper [18]--not to be confused with his [19]. Of course, Szilard used a quite different vocabulary; but, with hindsight, one may now say that he has co-invented the concept of sufficiency with R. A. Fisher; by showing that, under certain regularity conditions, Gibbs's canonical law is the only probability distribution with a single scalar sufficient statistic, Szilard also anticipated the results of G. Darmois [2], B. O. Koopman [10] and E. J. G. Pitman [16], but was partly anticipated by Poincare [17]. (B) The second thesis of the paper is independent of Szilard, and concerns the concept of temperature. For systems with a canonical energy, the temperature is the parameter of the Gibbs distribution; as such it is undefined for isolated systems with a determined energy. However, it is necessary to generalize the concept of temperature to isolated systems. Several definitions have been proposed and, although they all safely converge mutually for the usual very large systems, the temperature remains mathematically ambiguous for small isolated systems; it also becomes physically meaningless. We shall show that the temperature for systems-in-isolation should be viewed as a statistical estimate of the parameter of a conjectural canonical distribution, from which the presently isolated system may be presumed to have once been drawn. This interpretation explains the nature of the ambiguity of the concept of temperature; it also meets the actual practice of physicists; finally, some of the a priori conditions, which the physicists impose upon their "estimators", turn out to correspond to the statistical conditions of consistency, unbiasedness, and efficiency. Physicists also use two very interesting variants of consistency and unbiasedness, which we shall study under the names of "self-consistency" and "self-unbiasedness". The most commonly used temperature, due to Ludwig Boltzmann, turns out to be the maximum likelihood estimator. In summary, we hope to show that it is a great pity that mathematical and physical statistics should have developed largely independently of each other, while using the same concepts. By combining the rigor of modern statistics with the intuitive vigor of thermodynamics, both should be served well. However, as things stand, the mathematical statistician should not hope to unearth in the literature of physics any result as yet unknown to him. An important open problem suggested by this paper is the following. When sufficiency and estimation are defined in the most general terms, it seems that one should also be able to generalize the scope of thermodynamics. However, an approach such as that of P. R. Halmos and L. J. Savage [5] could not be applied to thermodynamics without substantial restrictions, as we shall show in Section 7. It remains to study these restrictions in greater detail, before one can assert that a non-void generalization of thermodynamics is possible. The problem is addressed to both mathematicians and physicists. We shall strive to reduce to the minimum the detailed knowledge of physics required to read this paper. If the reader's appetite for information about thermodynamics has been awakened, he could do no better than to make use of references [11] and [20].

In this paper the case of multivariate regression with stochastic predictors is considered, the joint distribution of the predictors being unknown, and the conditional distribution of the predict and given the predictors being normal with an unknown standard deviation. Sequential procedures of Stein's [13] type, terminating with probability one, are developed to obtain tests, confidence regions, and point estimates for the regression parameters. For the tests, the power function does not depend on the unknown distribution of the predictors or any nuisance parameters; for the confidence regions, the "span" is fixed and known; and for the point estimates, the expected loss, for a particular type of loss function, is a known constant. The procedure is subsequently modified to get more useful and "efficient" tests and estimates. Some study of the distribution and expectation of the sample size is also made for the sequential procedures developed.

Various approaches are available for the formulation of linear models for the analysis of variance. The mixed model in which one factor is a fixed-effects factor, and one factor is a random-effects factor can be obtained for instance as a limiting case of the general models of Cornfield and Tukey [3], and of Wilk and Kempthorne [8]. An entirely different approach is that given by Scheffe [6], and is the one considered in the present paper. Whatever the approach used, a hypothesis of interest will usually be the hypothesis of no fixed-effects. Consider a two-way layout in which $A$ denotes a fixed-effects factor and $B$ a random-effects factor. Let $I$ and $J$ be the numbers of levels of factors $A$ and $B$ respectively at which measurements are taken $(I > 1, J > 1)$. Let $K$ be the number of replications performed in each cell $(K > 1)$. In the light of $K$ be the number of replications performed in each cell $(K > 1)$. In the light of Table 1 of [3], Table 3 of [8], and formulas (46) and (54) of [6], an adequate $F$-type statistic to use for testing the hypothesis $H_A$ that all main-effects corresponding to the levels of factor $A$ are zero appears to be \begin{equation*}\tag{1.1}\mathscr{F} = (\mathrm{MS})_A/(\mathrm{MS})_{AB}.\end{equation*} The usual mean squares $(\mathrm{MS})_A$ and $(\mathrm{MS})_{AB}$ corresponding to factor $A$ and to $A \times B$ interactions are explicitly defined below. With the normal theory models which are commonly used in the case of two fixed-effects factors and in the case of two random-effects factors (e.g., in [7]), the criterion (1.1) has under the hypothesis $H_A$ the $F$-distribution with $I - 1$ and $(I - 1)(J - 1)$ d.f. In Scheffe's mixed model [6], this is no longer the case. When $J \geqq I$, a Hotelling $T^2$ statistic can then be constructed for the test of $H_A$. When multiplied by a constant factor, this statistic has the $F$-distribution with $I - 1$ and $J - I + 1$ d.f. While requiring a larger amount of computational work, the $T^2$ test will have little power when $J - I + 1$ is small. It is therefore tempting to construct a test of $H_A$, based on the ratio (1.1), by assuming that the law of $\mathscr{F}$ is not much different under $H_A$ from that of $F$ with $I - 1$ and $(I - 1)(J - 1)$ d.f. In Sub-section 4.1, we investigate the possible ill-effects of this assumption. They can be considerable, and remedies are suggested in Subsections 4.2 and 4.3.

In this paper we shall propose a test of linearity of a median regression curve against an alternative of convexity. To be specific, we shall test $H_0: Y_i = \alpha + \beta X_i + \epsilon_i,\qquad i = 0, 1, \cdots, n,$ against $H_1: Y_i = \phi(X_i) + \epsilon_i,\qquad i 0, 1, \cdots, n,$ where $\alpha, \beta$ and $\phi$ are unspecified, and $\phi(x)$ is a nonlinear convex function. The basic assumption underlying the test is that the $\epsilon_i$ are independent identically distributed random variables with median zero and with a continuous density function $f(\epsilon)$ such that $f(0) > 0$. The $X_i$ are fixed and known. The test consists in estimating a line by the Mood-Brown procedure (using medians) from a central subset of the observations, making a weighted count of the number of remaining observations lying above the line, and rejecting $H_0$ if this number, $R_n$, is large. The test can easily be adapted to a one-sided alternative of concavity or to a two-sided alternative of either convexity or concavity. Section 2 is devoted to a discussion of the line estimation procedure, and in particular, the asymptotic distribution of the estimator is obtained under the null and alternative hypotheses. In Section 3 the $R$ test of convexity is introduced, the asymptotic null and alternative hypothesis distributions of the test statistic $R_n$ are obtained, and a formula for the asymptotic power is given. The test is shown to be consistent against twice differentiable convex alternatives. In Section 4 we obtain the relative asymptotic efficiency of the $R$ test as compared to the least squares test for parabolic alternatives with errors normally distributed, and make recommendations for the use of the test. Finally, in the Appendix, results of some Monte Carlo experiments used to investigate the small sample behavior of $R_n$ under $H_0$ are presented. The author is not aware of the existence of other tests of linearity against general convex alternatives. In the case where the alternatives in mind can be expressed in a linear regression scheme, both the least squares test and the median test suggested by Mood in [5] are possible competitors of the convexity test presented in this paper. The least squares test is to be preferred when errors are known to be normally distributed with common variance, but for more general types of errors there is at present no obvious way of determining a "best" test, and consequently the choice of test to be used in a given situation must be based largely upon subjective considerations.

Having observed $X_i = \alpha + \beta c_i + \sigma Y_i$, we test the hypothesis $\beta = 0$ against the alternative $\beta > 0$. We suppose that the square root of the probability density $f(x)$ of the residuals $Y_i$ possesses a quadratically integrable derivative and define a class of rank order tests, which are asymptotically most powerful for given $f$. The main result is exposed in the following succession: theorem, corollaries and examples, comments, preliminaries and proof. The proof is based on results by Hajek [6] and LeCam [8], [9]. Section 6 deals with asymptotic efficiency of rank-order tests, which is shown, on the basis of Mikulski's results [10], to be presumably never less than the asymptotic efficiency of corresponding parametric tests of Neyman's type [11]. This would extend the well-known result obtained by Chernoff and Savage [2] for the Student $t$-test. Furthermore, it is shown that the efficiency may be negative, i.e., asymptotic power may be less than the asymptotic size. In Section 7 we consider parallel rank-order tests of symmetry for judging paired comparisons. Section 8 is devoted to rank-order tests for densities such that $(f(x))^{\frac{1}{2}}$ does not possess a quadratically integrable derivative. In Section 9, we construct a test which is asymptotically most powerful simultaneously for all densities $f(x)$ such that $(f(x))^{\frac{1}{2}}$ possesses a quadratically integrable derivative.

The Cramer-von Mises $\omega^2$ criterion for testing that a sample, $x_1, \cdots, x_N$, has been drawn from a specified continuous distribution $F(x)$ is \begin{equation*}\tag{1}\omega^2 = \int^\infty_{-\infty} \lbrack F_N(x) - F(x)\rbrack^2 dF(x),\end{equation*} where $F_N(x)$ is the empirical distribution function of the sample; that is, $F_N(x) = k/N$ if exactly $k$ observations are less than or equal to $x(k = 0, 1, \cdots, N)$. If there is a second sample, $y_1, \cdots, y_M$, a test of the hypothesis that the two samples come from the same (unspecified) continuous distribution can be based on the analogue of $N\omega^2$, namely \begin{equation*}\tag{2} T = \lbrack NM/(N + M)\rbrack \int^\infty_{-\infty} \lbrack F_N(x) - G_M(x)\rbrack^2 dH_{N+M}(x),\end{equation*} where $G_M(x)$ is the empirical distribution function of the second sample and $H_{N+M}(x)$ is the empirical distribution function of the two samples together [that is, $(N + M)H_{N+M}(x) = NF_N(x) + MG_M(x)\rbrack$. The limiting distribution of $N\omega^2$ as $N \rightarrow \infty$ has been tabulated [2], and it has been shown ([3], [4a], and [7]) that $T$ has the same limiting distribution as $N \rightarrow \infty, M \rightarrow \infty$, and $N/M \rightarrow \lambda$, where $\lambda$ is any finite positive constant. In this note we consider the distribution of $T$ for small values of $N$ and $M$ and present tables to permit use of the criterion at some conventional significance levels for small values of $N$ and $M$. The limiting distribution seems a surprisingly good approximation to the exact distribution for moderate sample sizes (corresponding to the same feature for $N\omega^2$ [6]). The accuracy of approximation is better than in the case of the two-sample Kolmogorov-Smirnov statistic studied by Hodges [4].

The first part of this paper (Section 2) is devoted to the derivation of the asymptotic distribution of the sample proportion above a normal sample mean. The second part of the paper, leaning on the first, treats the asymptotic joint distribution of runs of various lengths above the sample mean (Section 3). The approach here proves general enough to cover asymptotic run distributions arising when the dichotomy criterion is magnitude relative to a sample function other than the sample mean and the population is other than normal.

For continuous symmetrical distributions and specified tolerance interval characteristics, distribution-free tolerance intervals are presented that require substantially smaller sample sizes than those required for arbitrary continuous populations. In the case of one-sided tolerance intervals, about twice as many sample values are required for the general situation as for the symmetrical situation. For two-sided intervals, the additional sample values required for the general situation vary from about 35 percent to about 85 percent in the cases considered, depending on the specified tolerance interval characteristics. These intervals furnish at least rough protection against all possible violations of the symmetry assumption. In addition, some specialized one-sided and two-sided tolerance intervals are developed for life-test situations involving continuous symmetrical populations. Results for continuous symmetrical populations with known central median values are also presented; the life-testing versions of these intervals are especially useful.

The properties of linear flats in finite projective geometry are greatly used by many authors to construct incomplete block designs. In this paper the incidence properties of linear flats contained in quadrics in a finite projective geometry are exploited to construct two associate class partially balanced incomplete block designs.

This paper concerns a generalization of the gamma distribution, the specific form being suggested by Liouville's extension to Dirichlet's integral formula [3]. In this form it also may be regarded as a special case of a function introduced by L. Amoroso [1] and R. d'Addario [2] in analyzing the distribution of economic income. (Also listed in [4] and [5].) In essence, the generalization (1) herein is accomplished by supplying a positive parameter, $p$, as an exponent in the exponential factor of the gamma distribution. The moment generating function is shown, and cumulative probabilities are related directly to the incomplete gamma function (tabulated in [6]). Distributions are given for various functions of independent "generalized gamma variates" thus defined, special attention being given to the sum of such variates. Convolution results occur in alternating series form, with coefficients whose evaluation may be tedious and lengthy. An upper bound is provided for the modulus of each term, and simplified computation methods are developed for some special cases. A corollary is derived showing that the researches of Robbins in [7] apply to a larger class of problems than was treated in [7]. Extensions of his methods lead to iterative formulae for the coefficients in series obtained for an even larger class of problems.

If $r = x/y$ is the ratio of two independent continuous positive random variables, its distribution can be estimated by generating random samples from the distribution of $x$ and $y$, and then proceeding in various ways. It is shown, using well-known results in the theory of Wilcoxon's test that the uniformly minimum variance unbiased estimate of $H(A) = P(r \leqq A)$ is obtained by computing Wilcoxon's statistic for the random variables $u_i = x_i, v_i = Ay_i(i = 1, \cdots, N)$. The variance of the estimate of $H(A)$ is readily estimated. The computations required by this approach are more arduous than those needed to estimate $H(A)$ from the quantities $r_i = x_i/y_i$, but may be worthwhile where the major part of the computations lies in generating the $x_i$ and $y_i$. Extension of the reasoning leads to choosing different numbers of $x$'s and $y$'s if they are of different complexity to generate. Further, if the distribution of one of the quantities $x$ or $y$ is known then an effectivity infinite sample from that population is already available and the distribution of $r$ can be estimated by sampling only the variable with unknown distribution, which may (or may not) result in economy of effort.

It is known that the moments of several test criteria in multivariate analysis can be expressed in terms of Gamma functions. Various attempts have been made to determine the exact distributions from the known expressions for the moments. Wilks [11] identified such a distribution with that of a product of independent Beta variables, and he gave explicit results in some particular cases. Later, Nair [5], using the theory of Mellin transforms, expressed the distributions as solutions of certain differential equations, which he solved in some special cases. Kabe [3] expressed them in terms of Meijer $G$-functions. Asymptotic expansions of the distribution functions have been given by several authors; notably by Box [2], Tukey and Wilks [8], and Banerjee [1]. In this paper we give exact results in some important cases, in which it is found that the distribution is identical with that of a linear function of Gamma variates.

In this investigation we derive a few properties of the intersection of blocks in a balanced incomplete block (b.i.b. for conciseness) design with parameters $v, b, r, k, \lambda$ and discuss the divisibility of $v$ by $k$. The last section deals with certain characteristic properties of symmetric and affine resolvable b.i.b. designs.