It is well known that the future life distribution of a device remains the same regardless of the time it was previously in use, if and only if the life distribution of that device is exponential. For this reason exponential life distributions are accepted as characterizing the phenomenon of no wear. The problem of finding a class of life distributions which would similarly reflect the phenomenon of wear-out has been under investigation for some time. In answer to this problem we introduce in this paper the class of IHRA (Increasing Hazard Rate Average) distributions and show that it has, among others, the following optimal properties: (i) it contains the limiting case of no wear, i.e., all exponential distributions, (ii) whenever components which have IHRA life distributions are put together into a coherent system, this system again has an IHRA life distribution, i.e., a system wears out when its components wear out, and (iii) the IHRA class is the smallest class with properties (i) and (ii).

Networks with independent components are considered. Assuming all components to have a probability $p$ of functioning we study the properties of the reliability function $R(p)$ that the network functions. In particular, we investigate networks of high order. It is shown that an arbitrary (randomized) network can be approximated by a pure one with an approximation error of the order $n^{-1}$. Bounds are obtained for the maximum difference quotient and derivative of the first and second order of $R(p)$. As a corollary we obtain an asymptotic result concerning the best possible approximation of a function of Lipschitz type by reliability functions.

For the problem of optimum prediction by means of $k$th degree polynomial regression, it is shown in [3] how to find the observation points and respective proportions of observations in the interval $\lbrack-1, 1\rbrack$ in order to obtain the minimax variance over the interval $\lbrack -1, t\brack$ of the predicted regression value for all $t \geqq t_1 > 1; t_1$ is the point outside the interval of observations at which the Chebyschev polynomial of degree $k$ is equal to the maximum value of the variance of the least squares estimate in $\lbrack -1, 1\rbrack$. It is shown herein that if the observation points and proportions are chosen as specified in [3], then the maximum of the "least squares" variance in the interval $\lbrack -1, 1\rbrack$ is at -1. As a consequence, an equation is developed which permits the evaluation of $t_1$ as a function of $k$. Moreover, it is shown that $t_1 \rightarrow 1$ as $k \rightarrow \infty$, so that, for large $k$, the solution given in [3] yields an approximation to the minimax variance over the interval $\lbrack -1, t \rbrack$, all $t > 1$.

This paper is concerned with tests based on sample spacings for the two sample problem with nuisance location and scale parameters--Part A--Sections 1 through 4, and for the goodness of fit problem with nuisance scale and location parameters, Part B, Sections 5 through 7. Estimation of the nuisance scale parameter is also considered. The notation and methods of proof, as well as the basic results of the preceding paper, Blumenthal (1966) are used without restatement. That paper is referred to as Spacings I, throughout the sequel.

Lower bounds for the Pitman efficiency of the one-sided Kolmogorov test for all normal alternatives and for the Smirnov test for normal shift alternatives are obtained. In the one-sample case this does not fall below 36 per cent at the usual levels of significance and power. In the two-sample case the lower bound depends on the ratio of the two sample sizes and does not fall below 36 per cent when the ratio is larger than 40. We assume that the results of [1] are known.

A closed form expression for the exact distribution of the Mann-Whitney-Wilcoxon $U$ test has been derived. From this, expressions for the exact power of the $U$ test for exponential and rectangular alternatives have been derived. Several determinations of the power of combined sample sizes of 11, 15 and 21 have been compared with the corresponding determinations of the power of Mood's median test procedure. Also, the asymptotic efficiency of the $U$ test relative to Mood's median test for exponential and rectangular translation alternatives is considered.

Consideration of a sequence of statistical decision problems having identical generic structure constitutes a sequential compound decision problem. The risk of a sequential compound decision problem is defined as the average risk of the component problems. In the case where the component decisions are between two fully specified distributions $P_1$ and $P_2, P_1 \neq P_2$, Samuel (Theorem 2 of [9]) gives a sequential decision function whose risk is bounded from above by the risk of a best "simple" procedure based on knowing the proportion of component problems in which $P_2$ is the governing distribution plus a sequence of positive numbers converging to zero uniformly in the space of parameter-valued sequences as the number of problems increases. Related results are abstracted by Hannan in [2] for the sequential compound decision problem where the parameter space in the component problem is finite. The decision procedures in both instances rely on the technique of "artificial randomization," which was introduced and effectively used by Hannan in [1] for sequential games in which player I's space is finite. In the game situation such randomization is necessary. However, in the compound decision problem such "artificial randomization" is not necessary as is shown in this paper. Specifically, we consider the case where each component problem consists of making one of $n$ decisions based on an observation from one of $m$ distributions. Theorems 4.1, 4.2, and 4.3 give upper bounds for the difference in the risks (the regret function) of certain sequential compound decision procedures and a best "simple" procedure which is Bayes against the empirical distribution on the component problem parameter space. None of the sequential procedures presented depend on "artificial randomization." The upper bounds in these three theorems are all of order $N^{-\frac{1}{2}}$ and are uniform in the parameter-valued sequences. All procedures depend at stage $k$ on substitution of estimates of the $k - 1$st (or $k$th) stage empirical distribution $p_{k-1}$ (or $p_k$) on the component parameter space into a Bayes solution of the component problem with respect to $p_{k-1}$ (or $p_k$). Theorem 4.1 (except in the case where the estimates are degenerate) and Theorem 4.3 when specialized to the compound testing case between $P_1$ and $P_2$ (Theorems 5.1 and 5.2) yield a threefold improvement of Samuel's results mentioned above by simultaneously eliminating the "artificial randomization," by improving the convergence rate of the upper bound of the regret function to $N^{-\frac{1}{2}}$, and by widening the class of estimates. Higher order uniform bounds on the regret function in the sequential compound testing problem are also given. The bounds in Theorems 5.3 and 5.4 (or Theorems 5.5 and 5.6) are respectively of $O((\log N)N^{-1})$ and $o(N^{-\frac{1}{2}})$ and are attained by imposing suitable continuity assumptions on the induced distribution of a certain function of the likelihood ratio of $P_1$ and $P_2$. Theorem 6.1 extends Theorems 4.1, 4.2, and 4.3 to the related "empirical Bayes" problem. Also lower bounds of equivalent or better order are given for all theorems. The next section introduces notation and preliminaries to be used in this paper and in the following paper [15].

This paper is concerned with repetitive sequential play in finite statistical games (decision problems) from the statistician's point of view. We shall assume that the statistician's move at stage $k$ may depend on the previous $k - 1$ moves of Nature as well as the random variable $\mathbf{X}_k = (X_1, \cdots, X_k)$, where the $X_i$ are independent observations (r.v.'s) (possibly vector-valued) from the sequence of statistical games, $k = 1, 2, \cdots$. The play is repetitive in the sense that each component game is identical in structure, with only the moves of the statistician and Nature changing. Furthermore, we impose no assumptions regarding the behavior of the parameter sequence of Nature's moves. The statistician does have the added disadvantage that the finite class of distributions in the component game is not fully specified. However, he does know that class in question has: either (i) all members with discrete distributions or (ii) all members with $q$-dimensional a.e. continuous Lebesgue densities. This same problem when the distributions are fully known has been treated in [6] for statistical as well as more general games in which Nature's space is finite. In the case where the distributions are completely specified but the history of the past moves is unknown to the statistician, see [20], [22], [27], and [28]. The development in this paper is closely connected to and motivated by these results, particularly those of the preceding paper [27]. If for fixed $N$, the empirical distribution $p_N$ of Nature's moves is known, then the statistician could use as a rule for each of the $N$ component games a strategy Bayes against $p_N$ having risk $\phi(p_N)$. In all the papers cited in the previous paragraph, the aim was to construct for the statistician, when $p_N$ is unknown and $N$ not specified, a sequence of randomized decision functions whose $N$th average loss minus $\phi(p_N)$ approaches zero (or has an upper bound approaching zero) in a suitable sense as the number of repetitions of play, $N$, increases. However, in the case of statistical games, all of the above results require that the finite class of distributions be fully specified. In this paper we remove that assumption by estimating the distributions sequentially based on past moves and observations. Then in the present play of the component game the statistician substitutes these estimators into a procedure which is Bayes against the empirical distribution of Nature's previous moves. The resulting sequence of procedures is shown to be "asymptotically good" in the sense that the average loss over the $N$ games $W_N$ minus the Bayes risk $\phi(p_N)$ approaches zero (in an appropriate sense) as $N$, the number of games played, increases. In Section 2 we introduce notation and preliminaries. Section 3 discusses play in repetitive games and defines the proposed sequential procedures $\mathbf{t} = \{\mathbf{t}_k\}$. In Section 4 we prove preliminary results upon which all proofs are founded. Section 5 considers the discrete case giving uniform (in sequences of Nature's moves) convergence theorems (as $N \rightarrow \infty$) for the quantity $W_N - \phi(p_N)$. Theorem 5.1 is a uniform convergence theorem of $O(N^{-\frac{1}{2}})$ of the expected value of $W_N - \phi(p_N)$ for finite discrete classes, each member of which is non-degenerate and satisfies a certain tail probability condition. Under the same conditions, Theorem 5.2 gives uniform convergence to zero in probability for the quantity $N^{\frac{1}{2}} (\log N)^{-1} \{W_N - \phi(p_N)\} \text{as} N \rightarrow \infty$. Uniform convergence of $W_N - \phi(p_N) \rightarrow 0$ in probability for general non-degenerate finite discrete class is presented in Theorem 5.3. Section 6 treats the estimation problem for densities needed to form the randomized strategy sequences $\mathbf{t}$ in the continuous case. The results stated are based on a paper by Cacoullos [3] generalizing the univariate results of Parzen [15]. In Section 7, we present results for the continuous case. Theorem 7.1 and its corollary give uniform convergence of $W_N - \phi(p_N)$ to zero in probability and of its expectation to zero, respectively. The finite continuous classes of Theorem 7.1 are very general in the sense that each member is a continuous a.e. density. Finally, in Section 8 we draw certain conclusions and relate our results to similar results obtained elsewhere. The novelty of the paper rests in the fact that through the past history of Nature's moves and the observations connected with past play, one can construct a sequential strategy, $\mathbf{t} = \{\mathbf{t}_k\}$, with very little knowledge about the finite class of distributions, which approaches asymptotic "optimal" play. The lack of knowledge on the finite class of distributions distinguishes this work from the related "repetitive type" problems in games and/or decision theory treated in [1], [2], [4], [6], [7], [8], [9], [10], [12], [17], [18], [19], [20], [21], [22], [24], [25], [26], [27], [28], and [29]. For possible applications of this work see Neyman [14], especially his Example 3 and his discussion relating to the work of Blackwell [2].

The distribution of the latent vectors of a sample covariance matrix was found by T. W. Anderson [1] in 1951 when the population covariance matrix is a scalar matrix, $\Sigma = \sigma^2I$. The asymptotic distribution for arbitrary $\Sigma$, also, was obtained by T. W. Anderson [3] in 1963. The exact distribution of the latent vectors of a sample covariance matrix has been described by the author [10] in 1965 when the observations are obtained from a bi-variate normal distribution. The elements of each latent vector are the coefficients of a principal component (with sum of squares of coefficients being unity), and the corresponding latent root is the variance of the principal component. In this paper, the exact distribution of the latent vector corresponding to the largest latent root of a sample covariance matrix is given when the observations are from a multivariate normal distribution whose population covariance matrix is arbitrary $\Sigma$, and the distribution of the largest latent root is given when the population covariance matrix is a scalar matrix, $\Sigma = \sigma^2I$.

Many contributions have been made to the problem of characterizing the normal distribution using the property of independence of sample mean and sample variance, maximum likelihood, etc. In this paper, using certain identities among the product (linear) moments of order statistics in a random sample, the generalized truncated (both from below and above) normal distributions, the negative normal and the positive normal distributions are characterized in the class of arbitrary distributions having finite second moments. In Theorem 3.3, the normal distribution is characterized in the class of arbitrary distributions having mean zero and finite second moments. Bennett's [1] characterization of the normal distribution without the assumption of absolute continuity is a special case of Theorem 3.3, namely Corollary 3.3.2.

Results of $N$ weighing operations to determine the individual weights of $p$ objects, as envisaged in Hotelling's weighing designs [4], fit into the linear model $Y = X\beta + e$, where $Y$ is an $N \times 1$ random observed vector of the recorded results of weighings; $X = (x_{ij}), i = 1, 2, \cdots, N, j = 1, 2, \cdots, p$, is an $N \times p$ matrix of known quantities with $x_{ij} = +1, -1$ or 0, if, in the $i$th weighing operation, the $j$th object is placed respectively in the left pan, right pan, or in none; $\beta$ is a $p \times 1$ vector $(p \leqq N)$ representing the weights of the objects; $e$ is an $N \times 1$ unobserved random vector such that $E(e) = 0$ and $E(ee') = \sigma^2I_N. X$ represents the weighing design matrix. When $X$ is of full rank, that is, when $X'X$ is non-singular, the weights of the objects are given by the least squares estimates, $\hat\beta = \lbrack X'X\rbrack^{-1}X'Y$. The covariance matrix is given by $\operatorname{Cov}(\hat\beta) = \sigma^2\lbrack X'X\rbrack^{-1} = \sigma^2C. c_{ii}$, which is the $i$th diagonal element of $C$, represents the variance factor for the $i$th object. In weighing designs, we search for the elements $x_{ij}$ such that $c_{ii}$ is the least for each $i$. When, however, $X$ is singular, it is well known that, while it is not possible to have a unique (unbiased) estimate for each of the $p$ objects, it is possible to have a unique (unbiased) estimate for a linear function, $\lambda'\beta$, of the parameters, if and only if there exists a solution for $r$ in the equations $Sr = \lambda$, where $S = X'X$. Raghavarao [7] visualized that bad designing, repetitions or accidents might lead to singular weighing designs, and considered the question of taking additional weighings to make the resultant design matrix $X$ of full rank maximizing the resultant $\det. |X'X|$, if possible, as required in Mood's [5] efficiency definition. He appeared to take only the chemical balance (and not the spring balance) into consideration, and was eventually led to the question of dealing with the situation when the rank of $X$ was less than the full by only one. In this paper, the problem is, in the first place, cast into the framework of a generalized inverse (referred to as a $g$-inverse). Two harmonizing results in the area of $g$-inverses are then indicated by way of an aid to algebraic simplifications in the context of tackling the general problem. Singular weighing designs may not perhaps all be altogether useless in themselves, although one may not adopt singular weighing designs in a scheme of weighing operations to begin with. In certain given situations, singular weighing designs may even be preferred to the best available weighing designs adopted for the estimation of the weight of each individual object, and, in such situations, there may arise a necessity for comparing two singular weighing designs. Linking the problem of singular weighing designs to a $g$-inverse, as has been done in this paper, would also help institute such a comparison. Consideration is further given to the situation when the rank of $X$ may be less than the full by more than one. Finally, a special class of singular spring balance designs is also discussed.

It was shown by the authors [1], [2] how to adjust the treatment design matrix $X$ to furnish estimates of effects as orthogonal linear functions of observations for any irregular fractional replicate from an $N$ treatment factorial. The fractional replicate considered earlier was such that the design matrix $X$ was of dimensions $p \times p$ implying that $p$ effect parameters be estimated from $p$ observations. The method consisted in finding a matrix $\lambda$ such that the design matrix $X$ and the observation vector $Y$ were augmented to become $X_1 = \lbrack X'\vdots X'\lambda\rbrack'$ of dimensions $(p + m) \times p$ and $Y_1 = \lbrack Y'\vdots Y'\lambda\rbrack'$ of dimensions $(p + m) \times 1$ with $p + m = N$ in such a way that $\lbrack X_1'X_1\rbrack$ reduced to a diagonal matrix. In the present note, the earlier results have been generalized in the sense that the design matrix $X$ need not be square, that is, of dimensions $p \times p$, but is of dimensions $(p + m_1) \times p, p + m_1 < N$, implying that $p$ effect parameters be estimated from $(p + m_1)$ observations. Besides this generalization the following additional results were obtained: (i) the structural relationship between the effect parameters retained and the observations omitted was derived, (ii) a working rule was developed for constituting the irregular fractional replicate with observations that are internally consistent making it possible to estimate the effect parameters, and (iii) a desirable procedure of designing the fractional replicate to obtain maximum efficiency was set forth.

In this note, we give simpler proofs of results obtained, regarding the necessary conditions for existence of certain unsymmetrical PBIB designs having group divisible, triangular and $L_i$ association schemes, in [5]. We also give simpler proofs of the necessary conditions for the existence of affine $\alpha$-resolvable $GD(m, n), T(n), L_i(s)$ and BIB designs obtained in [6]. We follow the notation of these two papers throughout this note. We finally obtain necessary conditions for the dual of a design $D$, to be a specified design $E$.