In this paper, some new multidimensional partially balanced (MDPB) association schemes are defined, and the various parameters of the scheme are obtained. Using such schemes, we discuss procedures for the construction of multidimensional partially balanced designs. The theory so developed is illustrated with actual examples of construction of MDPB designs.

If $Y = g(X)$ is normal $(\mu, \sigma^2)$, where $g$ is a one-to-one real function and $X$ is a random variable whose expectation exists, we may write $EX = f(\mu, \sigma^2)$. The practical importance of this observation is that we often are concerned with testing hypotheses about, and constructing confidence intervals for, known functions of both the mean and variance of a normal distribution. This may happen when we use a statistical model, such as the lognormal distribution, that is related to the normal distribution by a transformation of variables. A slightly different case occurs when a transformation of data is made before applying a statistical method, such as analysis of variance or regression analysis, that involves the assumption of mormality for the transformed data. Some familiar examples in this context are $\mathrm{(i)} Y = X^{\frac{1}{2}}, EX = \mu^2 + \sigma^2$; $\mathrm{(ii)} Y = X^{\frac{1}{3}}, EX = \mu^3 + 3\mu\sigma^2$; $\mathrm{(iii)} Y = \arcsin (X^{\frac{1}{2}}), EX = \frac{1}{2}(1 - \cos(2\mu) \exp (-2\sigma^2))$; $\mathrm{(iv)} Y = \operatorname{arcsinh} (X^{\frac{1}{2}}), EX = \frac{1}{2}(\cosh(2\mu) \exp (2\sigma^2) -1)$; $\mathrm{(v)} Y = \log(X), EX = \exp (\mu + \frac{1}{2}\sigma^2)$. The theory of statistical inference in terms of $\mu = EY$ alone or $\sigma^2 = \operatorname{Var} Y$ alone is not easily extended to problems of inference in terms of $EX$ or $\operatorname{Var} X$, parametric functions of both $\mu$ and $\sigma^2$. Minimum variance unbiased estimators (MVUE's) for $EX$ and $\operatorname{Var} X$ were obtained by Finney (1941) for the case $Y = \log X$. Solutions for a much wider class of transformations were obtained by Neyman and Scott (1960) and Hoyle (1968). However there have been no analogous achievements with respect to hypothesis tests and confidence interval estimates for $EX$ and $\operatorname{Var} X$. The present paper, in which uniformly most accurate unbiased confidence interval procedures of level $1 - \alpha$ are derived for linear functions of $\mu$ and $\sigma^2$, is an approach to these problems. The results of this paper define an optimal solution for $EX$ when $Y = \log X$, since in this case the parametric function of interest is a monotone function of $\mu + \frac{1}{2}\sigma^2$. The results also provide a basis for approximate confidence interval solutions for other parametric functions of $\mu$ and $\sigma^2$. It is helpful to consider the problem in terms of confidence regions in the half-plane of points $(\mu, \sigma^2)$. For any transformation $Y = g(X)$ likely to be of practical significance, a confidence interval for $f(\mu, \sigma^2) = EX$ or $\operatorname{Var} X$ is a region in this half-plane, bounded by one or two contours of the form $f(\mu, \sigma^2) = m$. Kanofsky (1969) has proposed a method of simultaneous confidence estimation for all functions of $\mu$ and $\sigma^2$. He constructs a trapezoidal-shaped confidence region of level $1 - \alpha$ for $\mu$ and $\sigma^2$, and for an arbitrary function $h(\mu, \sigma^2)$, defines a confidence set for this function as the set of values $m$ such that the curve $h(\mu, \sigma^2) = m$ intersects this confidence region. If one is only interested in a single function, the procedure is conservative. However, for most such functions this is the only method based on exact distribution theory, to my knowledge, that has been proposed. The usual approach to confidence interval estimation for $EX$ or $\operatorname{Var} X$ has been to rely on approximate methods. For example, a common method of confidence interval estimation for $EX$ is to transform a level $1 - \alpha$ confidence interval for $EY = E(g(X))$, say $(\mu_1, \mu_2)$, by the inverse transform. Then $(g^{-1}(\mu_1), g^{-1}(\mu_2))$ would be an approximate level $1 - \alpha$ confidence interval for $EX$ if $g$ is monotone increasing. More sophisticated versions of this method have been proposed by Patterson (1966) and Hoyle (1968). A more direct approach is to use an estimator $T$ of $f(\mu, \sigma^2)$ and an estimator $V$ of the variance of $T. T$ is then assumed to be approximately normally distributed with mean $(f(\mu, \sigma^2)$ and variance equal to the observed value of $V$. For example, the sample mean $\bar{X}$ is an estimate of $EX$, and $S_X^2/(n(n - 1)) = \sum(X_i - \bar{X})^2/(n(n - 1))$ is an estimate of the variance of $\bar{X}$ (e.g., see Aitchison and Brown (1957) Section 5.62). Hoyle (1968) has suggested letting $T$ be the MVUE of $EX$, and $V$ the MVUE of $\operatorname{Var} T$, which he has given for a number of transformations. In this paper an optimal exact confidence interval procedure is presented for linear functions of $\mu$ and $\sigma^2$. That is, the procedure gives uniformly most accurate unbiased joint confidence regions of level $1 - \alpha$ for $\mu$ and $\sigma^2$, bounded by one or two contours of form $\mu + \lambda\sigma^2 = m$, for arbitrary $\lambda$. This provides an optimal confidence interval procedure for $EX$ when $Y = \log (X)$ is normal. Also it provides the basis for a new approximate confidence interval method for $EX$ in the general case $Y = g(X)$. That is, by a proper choice of the linear coefficient $\lambda$, it seems reasonable that a confidence region bounded by one or two contours of form $f(\mu, \sigma^2) = m$ might be approximated with some success by a confidence region bounded by contours of the form $\mu + \lambda\sigma^2 = m$. Certainly the degree of approximation possible should be better than that obtainable using only vertical bounding contours, as when a confidence interval for $\mu$ is transformed to give an approximate confidence interval for $EX$. Also, if the contours $f(\mu, \sigma^2) = m$ are fairly straight within a convex joint confidence region of level $1 - \alpha$ for $\mu$ and $\sigma^2$, it is not unreasonable to hope that an approximate confidence region should be possible that would have a true level near $1 - \alpha$, and that would be less conservative than a level $1 - \alpha$ region for $f(\mu, \sigma^2)$ determined by Kanofsky's method. The main result of the paper is the derivation in Section 2 of uniformly most powerful unbiased level $\alpha$ hypothesis tests for linear functions of $\mu$ and $\sigma^2$. The theoretical interest of this section is mainly in the analytic detail of how a well-known theorem applies to this somewhat unusual case. A numerical example follows, illustrating the use of the tables of critical values given in the Appendix. It is not obvious that the confidence procedures defined by these tests in Section 4 define confidence sets that are intervals, an extremely desirable property both for ease of calculation and for practical usefulness of the confidence sets. The proof in Section 5 that the one-sided tests define one-sided confidence intervals provided that $v$, the number of degrees of freedom available for the estimate of $\sigma^2$, is at least two, is the second major result of the paper. In Section 6 it is shown that this property does not obtain when $v = 1$. The analogous result in the two-sided case is proved only for $v = 2$ in Section 7. However it is conjectured that, as in the one-sided case, the desired property also holds for all larger values of $v$. The final section contains a brief discussion of applications of the method to confidence interval estimation for $EX$ when $Y = g(X)$ is normal. It is shown that essentially the only direct application is to the case where $Y = \log(X)$, and that there are no nontrivial direct applications where $EX$ is a function of $\mu$ or $\sigma^2$ alone. The construction of normal tolerance limits involves confidence interval estimation of functions of the form $\mu + \delta\sigma$ (Owen (1958)). However it is shown here that there are essentially no transformations to normality such that $EX$ is a function of $\mu + \delta\sigma$ for some $\delta$. A more complete discussion of approximate applications of the method is left for a subsequent paper.

Since the appearance of P. Billingsley's monograph [2] on the large sample inference in Markov processes in which the weak consistency and asymptotic normality of the maximum likelihood estimate was investigated, there has been considerable interest in the further development of the theory along other directions. Billingsley's work was mainly concerned with extending the results of H. Cramer ([4] page 500). Among more recent developments one might mention the proof of the almost sure consistency of the maximum likelihood estimator following the ideas of A. Wald by G. Roussas [7], and the results on asymptotic Bayes estimates obtained by Lorraine Schwartz [9]. In the present paper we extend to Markov processes one of the fundamental results in the asymptotic theory of inference, viz., the approach of the posterior density (in a sense to be made precise later) to the normal. When the observed chance variables are independent and identically distributed, this result was obtained by L. LeCam in [5] (page 309). The same author offers another derivation of this result in [6]. Special cases of the theorem were first given by S. Bernstein and R. von Mises (for reference see [5]). The regularity conditions satisfied by the transition probability density of the Markov process are given in Section 1. We prove in Theorem 2.4 of Section 2 those properties of the maximum likelihood estimator that are needed for the proof of the main result of the paper given in Section 3 (Theorem 3.1). Theorem 3.1 is stated in a form which is general enough to include the Bernstein-von Mises theorem as well as the somewhat sharper versions that are available when it is known that the prior probability distribution has a finite absolute moment of order $m$. Theorem 3.2 deduces these results as a consequence of Theorem 3.1. The latter result also enables us to prove a theorem on the asymptotic behavior of regular Bayes estimates. This is done in Theorem 4.1 of Section 4.

In this paper, the exact distribution of Wilks' likelihood ratio criterion, $\Lambda$, for MANOVA in the noncentral linear case i.e. when the alternative hypothesis is of unit rank, has been obtained and explicit expressions for the same for $p = 2, 3, 4$ and 5, where $p$ is the number of variables and for general $f_1$ and $f_2$ are given. A general form of the distribution of $\Lambda$ in this case, for any $p$, is also given. It has been shown that the total integral of the series obtained by taking a few terms only, rapidly approaches the theoretical value one as more terms are taken into account. Further the accuracy of the approximation, suggested by Posten and Bargmann [11], is examined numerically and it has been shown that the approximation is excellent except when $f_1$ and $f_2$ are both small and the noncentrality parameter $\lambda^2$ is large.

The well-known space $D\lbrack 0, 1\rbrack$ is generalized to $k$ time dimensions and some properties of this space $D_k$ are derived. Then, following the "classical" lines as presented in Billingsley [1], a Skorohod-metric, tightness criteria and some other results concerning weak convergence are given. The theory is applied to prove weak convergence of two generalizations of the one-dimensional empirical process and of the Kolmogorov-Smirnov test statistic of independence. Stochastic processes with multidimensional time parameter and their weak convergence have been investigated by several authors. Dudley [4] established a theory of convergence of stochastic processes with sample functions in nonseparable metric spaces. Later on, Wichura [11] (see also Wichura [12]) modified the concepts of Dudley and developed them systematically. He applied his theory to a space which is with minor changes our space $D_k$. Weak convergence in the sense of Wichura [12] and ours differ usually, but both concepts coincide if the limit process has--with probability one--continuous sample functions only. From here it follows that the results of Dudley and Wichura concerning weak convergence of multivariate empirical processes are equivalent to ours. At least two further authors proved the convergence of multivariate empirical processes, namely LeCam [8] and Bickel [1]. Our proof follows the classical approach of Parthasarathy [9] using an argument of Kuelbs [7] to carry over the proof from 1 to $k$ dimensions. Kuelbs however deals properly with the "interpolated sum" process for two-dimensional time parameter. The space $D_k$ seems to be defined for the first time in connection with multivariate processes by Winkler [13], yet his investigations are not concerned with weak convergence. Another generalization of the space $D\lbrack 0, 1\rbrack$ and the Skorohod metric to functions on more general spaces than $E_k$ is given in the paper [10] of Straf, in which there are applications to genuinely discontinuous limit processes.

We treat a model describing the continued formation and growth of mutant biological populations. At each transition time of a Poisson process a new mutant population begins its evolution with a fixed number of elements and evolves according to the laws of a continuous time positive recurrent Markov Chain $Y(t)$ with stationary transition probabilities $P_{ik}(t), i, k = 0,1,2,\cdots, t \geqq 0$. Our principal concern is the asymptotic behavior of moments and of the distribution function of the functional $S(t) = \{$number of different sizes of mutant populations at time $t\}$. When the recurrence time distribution to any state of the Markov Chain $Y(t)$ has a finite second moment, the moments of $S(t)$ and limit behavior of its distribution function are controlled by the stationary measure associated with $Y(t)$. When the second moment of the recurrence time distribution is infinite, then a local limit theorem and speed of convergence estimate for $P_{ik}(t)$ with $k = k(t) \rightarrow \infty, t \rightarrow \infty$ are required to establish asymptotic formulas for moments of $S(t)$.

Balanced weighing designs have been investigated by Bose and Cameron [2], [3]. This is a part of the work on the extensions of balanced weighing designs to a situation of "partial balance" based on association schemes [1]. Here some methods are developed to construct these new designs (called partially balanced weighing designs) with two association classes.

Maximally invariant loss functions are constructed in a decision theoretic framework, and sufficient conditions for their measurability are given.

The present investigation is a follow up of [7] to a class of multiple regression problems, and is devoted to the construction of an estimate of regression parameter vector based on suitable rank statistics. Asymptotic linearity of these rank statistics in the multiple regression set up is established and the asymptotic multi-normality of the derived estimates is deduced. There exists the choice of the score-generating function to every basic distribution so that the asymptotic distribution of the estimates is the same as that of maximal-likelihood estimates.

Let $X_1, X_2, \cdots$ be independent identically distributed random variables. You observe the $X$'s sequentially, knowing that their distribution is one of countably many different probabilities. Within an arbitrary error level, can you decide which one? This is the general problem of sequential discrimination. Freedman [4] showed that the discriminability of a family $\Theta$ is equivalent to a seemingly weaker condition. Namely, for any error level $\alpha$ and any particular $\theta \in \Theta$ there is a uniformly powerful fixed sample size test of $\{\theta\}$ versus $\Theta-\{\theta\}$ with error level uniformly as small as $\alpha$. The proof is constructive. Given the fixed sample size tests there is a recipe for manufacturing a sequential procedure to decide among all the members of $\Theta$. The fixed sample size tests are, however, still required. Hoeffding and Wolfowitz [5] considered this problem at length. LeCam and Schwartz [7] also touched upon it briefly. Both papers considered separations in various topologies and structures. Here we return to the original problem and ask whether likelihood ratios can be sensibly used. A rule is easy to specify0. For each $\theta \in \Theta$ you pick a number greater than one. Now watch $X_1, X_2, \cdots$. At each step compute the likelihood ratio for every pair of probabilities. Eventually it may happen that for some $\theta$ all the ratios with $\theta$ in the numerator are as big as the pre-assigned number. If so, stop and declare $\theta$ to be the true distribution. This rule is the extension to the countable case of the general sequential probability ratio test proposed by Barnard [2] and detailed by Armitage [1]. It does require the computation of all the likelihood ratios; but since $\Theta$ is countable, there is always at least one base for calculating densities. Any one will do. Likelihood ratio procedures have the advantage of being easy to formulate. Also, the comparison of densities seems to be a reasonably natural technique. However, it does not always work. An example will illustrate that it may fail spectacularly. When do likelihood ratio procedures work? The principal result is a characterization of families which are likelihood ratio discriminable. Check each probability $\theta$ separately. There may be a number $K(\theta)$ bigger than one which will be eventually exceeded simultaneously by all the ratios with $\theta$ in the numerator. If not, likelihood ratios will not work. If so, then the values $K(\theta)$ may be chosen so as to limit the error to any desired level. Despite their failures, likelihood ratio procedures may work when other natural conditions fail. Freedman showed that if each $\theta \in \Theta$ is isolated in the topology of setwise convergence, then $\Theta$ is disciminable. The converse is false. There is a family which is likelihood ratio discriminable, but which has one element in the setwise closure of all the others. This is a direct consequence of a recent theorem by LeCam [8]. Finally, with many familiar families likelihood ratio procedures have finite expected stopping time. Cases vary, however, and there is a discriminable family which has infinite expected stopping time for sampling under one of its elements.

Classification of a sample from a zero mean, stationary, Gaussian time series into populations distinguished by characteristics of the spectrum can be done with a decision theoretic procedure or spectral analysis. Decision theory requires that each population be characterized by a probability distribution on the space of spectral density functions. In this paper, we relate the two methods by showing that under many conditions, as the sample length increases, the expected cost of the Bayes test formed from spectral estimates by approximating their sampling distribution by a product of chi-squared distributions approaches the expected cost of the Bayes test formed from the original data. The amount of smoothing that can be used in the spectral estimates depends on the prior knowledge of the order of differentiability of the spectrum. This result is related to but weaker than the notion that spectral estimates are asymptotically sufficient statistics for the second order properties of a stationary Gaussian time series.

In this paper Chow and Robbins' (1965) sequential theory has been extended to construct a confidence region with prescribed maximum width and prescribed coverage probability for the linear regression parameters under weaker conditions than Srivastava (1967), Albert (1966), and Gleser (1965). An extension to multivariate case has also been carried out.

This is an extension of Wald's asymptotic test procedure based on unrestricted maximum-likelihood estimators. Wald showed that under certain regularity conditions the test statistic has a limiting central chi-square distribution under the hypothesis and a limiting noncentral chi-square distribution under a sequence of local alternatives. We extend this procedure, allowing it to be based on a broader class of estimators and to obey simpler and less restrictive conditions. Sufficient conditions for validity of the limiting distributions are local twice-differentiability of the left side of the hypothesis and, under a sequence of local alternatives, asymptotic normality of the estimator of the parameter defining the distribution and stochastic convergence (to the appropriate asymptotic value) of the estimator of the covariance matrix. The required asymptotic behavior is verified for the case of independent sampling from two normal distributions and formulas are presented which aid in computing the test statistic.

The literature of mathematical statistics is filled with theorems on (weakly) consistent estimators. Even though most statisticians want stronger evidence of an estimator's worth, these theorems have provided some comfort for the applied statistician. In this paper, we begin an investigation into the concept of consistency and, more specifically, investigate the extent to which a consistent sequence of estimators identifies the parameter they estimate. It will be recalled that for any sequence of random variables which converge in probability to a limit, there is a subsequence which converges almost surely to that limit. This would seem to suggest that if one is given a consistent sequence of estimators $\hat{\phi}_1, \hat{\phi}_2, \cdots$ converging to $\phi(\theta) (\theta \in \Theta)$, say, then one can find a subsequence which converges almost surely. That is, whenever there exists a weakly consistent sequence of estimators there exists a strongly consistent sequence as well. Unfortunately, the specific subsequence may depend upon the unknown parameter value $\theta$. Still, an applied statistician might be able to choose sequentially which observed estimators to include in a (random) subsequence. This is easily seen to be equivalent to postulating the existence of functions $g_n(x_1, \cdots, x_n)(n \geqq 1)$ such that $g_n(\hat{\phi}_1, \cdots, \hat{\phi}_n)$ converges to $\phi(\theta)$ with probability one $(\theta\in\Theta)$. In Section 2, we will show that such functions do not always exist. It seems appropriate, therefore, to question whether the values of the entire sequence $\hat{\phi}_1, \hat{\phi}_2, \cdots$ (always) allow one to determine the value of $\phi(\theta)$ with probability one. We prefer the following mathematical reformulation of this question: Let $(\Omega, \mathscr{A}, P)$ be a probability space and $\mathscr{E}$ denote the set of infinite dimensional random vectors $\mathscr{X} = (X_1, X_2, \cdots)$ defined on this space whose coordinates converge in probability to a random variable (which we shall continue to denote as) $p^\mathscr{X}$. The question becomes: Does there always exist a function $f$ which maps $R^\infty$ (infinite-dimensional Euclidean space) into $\mathbf{R}$ (the reals) such that for every $\mathscr{X}\in\mathscr{E}$, the set \begin{equation*}\tag{1} \lbrack f(\mathscr{X}) \neq p\mathscr{X} \rbrack \text{is contained in a null set of} \mathscr{A}?\end{equation*} We shall refer to any function $f$ which satisfies (1) for all $\mathscr{X} \in\mathscr{F} \subset \mathscr{E}$ as a probability limit identification function (PLIF) on $\mathscr{F}$. We partially justify the reformulation as follows: In Section 3, we will show that the vector of estimators $(\hat{\phi}, \hat{\phi}_2,\cdots)$ can be interpreted as defined on the same probability space for every $\theta \in \Theta$. As such, consistent estimators are equivalent to a family of vectors $\mathscr{F} = \{\mathscr{X}_\theta, \theta \in \Theta\} \subset \mathscr{E}$. Identifying $\phi(\theta)$ becomes equivalent to showing that there exists a PLIF on $\mathscr{F}$. In Section 4, we show that there exists a PLIF on $\mathscr{E}$ if there exists a PLIF on $\mathscr{E}^\ast$, the set of $\mathscr{X} \in \mathscr{E}$ whose coordinates are Bernoulli variables and whose probability limit $p\mathscr{X}$ is almost surely a constant (necessarily zero or one). With values of $\theta$ corresponding to vectors $\mathscr{X} \in \mathscr{E}^\ast$ and $\phi(\theta)$ corresponding to $p\mathscr{X}$, the reformulation becomes complete. We do not know whether a PLIF always exists on $\mathscr{E}^\ast$ except for certain elementary probability spaces. It is hoped that the current paper will stimulate further research into this question. If they do not always exist, this will cast further doubt on the importance of consistency. Breiman, LeCam and Schwartz [2] have discussed an interesting problem whose formulation is closely related to the current one. They assume that they have a family of probability measures $\{P_\theta(\cdot), \theta \in \Theta\}$ each defined on the same measurable space $(\Omega, \mathscr{A})$ (with points $\omega \in \Omega$). They assume that $\phi(\theta)$ is measurable with respect to a $\sigma$-field defined on $\Theta$ and find necessary and sufficient conditions for the existence of an $\mathscr{A}$-measurable estimator $\hat{\phi}(\omega)$ for which $P_\theta\{\hat{\phi}(\omega) = \phi(\theta)\} = 1\quad \text{for all} \theta \in \Theta$ (and also for a closely related condition). The question of the existence of a measurable PLIF on $\mathscr{E}^\ast$ translates into the question of the existence of a certain "zero-one set" in their context. Skibinski [3] has connected their work on zero-one sets with some work of Bahadur [1].

Let $X^{(n)} = (X_1, \cdots, X_n)$ be a random sample of size $n$ from the distribution of a real-valued random variable $X$ with an absolutely continuous distribution function $F$ and a density function $f$. Rosenblatt (1956) showed that in this setting there exists no unbiased estimator of $f$ based on the order statistics. His result follows from the fact that the empirical distribution function is not absolutely continuous. He also assumed that $f$ is continuous, but this condition is unnecessary. Rosenblatt's result also arises as a consequence of general results by Bickel and Lehmann (1969) on unbiased estimation in convex families, such as the family of all such $F$ (above). A number of writers (Kolmogorov (1950), Schmetterer (1960), Ghurye and Olkin (1969)) have obtained unbiased estimators of particular normal-related families as well as for other estimable functions. Washio, Morimoto and Ikeda (1956) considered related questions for the Koopman-Pitman family of densities, and Tate (1959) confined his attention to functions of scale and location parameters. A question arises as to exactly when unbiased--uniform minimum variance unbiased (UMVU)--estimators of density functions exist and when they do not. In a recent publication, Lumel'skii and Sapozhnikov (1969) considered such a question in relation to estimating the density function at a point, whereas, in this paper our definition of unbiasedness requires the estimator to be unbiased at every point. The so-called "Bayesian" methods they employ yield estimators for most of the well-known families of distributions as well as for several types of $p$-dimensional discrete distributions. In Section 2 we formulate the problem in a fairly general setting and obtain results in terms of unbiased estimators of probability measures (or distribution functions) which always exist. In Section 3 we consider examples to illustrate the theory of the preceding section and in Section 4 give a theorem which generalizes a lemma stated by Ghurye and Olkin (1969) which formalizes the approach used by Schmetterer (1960) for obtaining unbiased estimators of certain types of parametric functions.

Suppose that independent normally distributed random vectors $W^{n\times 1}$ and $T^{k\times 1}$ are observed with $E(W) = 0, E(T) = \mu, \operatorname{Cov} (W) = \sigma^2 I$ and $\operatorname{Cov} (T) = \sigma^2 I$. In this paper it is shown that each member of a certain class of estimators of $\mu + \eta\sigma$ for a given vector $\eta$ is inadmissible if loss is dimension-free quadratic loss. This class includes the best invariant estimator. The proof is carried out by exhibiting, for each member, $\hat{\theta}$, of the class, an estimator depending on $\hat{\theta}$ whose risk is uniformly smaller than that of $\hat{\theta}$.

For a resolvable balanced incomplete block design (BIBD), Bose (1942) obtained an inequality $b \geqq v + r - 1$. Stanton (1957) showed that this inequality is equivalent to an inequality $r \geqq \lambda + k$. The main purpose of this note is to improve Bose's inequality to $b \geqq 2\nu + r - 2$ for a resolvable BIBD which is not affine resolvable.

It seems to be generally known that the proof of the continuity part of Theorem 1 in Hodges' and Lehmann's paper (1963) is incorrect. The fact that the theorem is incorrect is--perhaps--not so well known. We show this by constructing independent real random variables $X_1,\cdots, X_n$, each having the same non-atomic symmetric distribution, and an odd translation invariant estimator $h(X_1,\cdots, X_n)$ such that $P(h(X_1,\cdots, X_n) = 0) > 0. h$ may be chosen symmetric provided $n \geqq 3$.

In an earlier paper the author proved the existence of a best test for testing hypotheses about a random parameter with unknown distribution. This paper gives a result which helps one find the best test approximately for several of the examples considered in the previous paper.

Let $X_{i1} < \cdots < X_{in_i}(i = 1,\cdots, c)$ be the order statistics from an absolutely continuous $\operatorname{cdf} F_i(x) = F(x - \theta_i)$ where $F(x)$ has symmetric density. The problem of testing the hypothesis $H_0:\theta_1 = \cdots = \theta_c$, which has been discussed by many authors, will be considered in this paper. We are concerned with the tests based on only the middle $n_i - 2k_i$ random variables $X_{ik_i+1} < \cdots < X_{in_i-k_i}$ where $k_i = \lbrack n_i\alpha \rbrack$ is the largest integer not exceeding $n_i\alpha$ for any $\alpha, 0 < \alpha < \frac{1}{2}$. A test of Bhapkar's type [Bhapkar, V. P. (1961). A nonparametric test for the problem of several samples. Ann. Math. Statist. 32 1108-1117] is proposed for this problem and it is shown that, for some distributions with heavy tails, the asymptotic relative efficiency of the proposed test relative to Bhapkar's test, which is based on the complete samples, is larger than one. The work presented in this paper is an attempt toward generalizing Hettmansperger's results [Hettmansperger, T. P. (1968). On the trimmed Mann-Whitney statistics. Ann. Math. Statist. 39 1610-1614] to the $c$-sample problem.