Consider a stationary normal process $\xi(t)$ with mean zero and the covariance function $r(t)$. Properties of the sample functions in the neighborhood of zeros, upcrossings of very high levels, etc. have been studied by, among others, Kac and Slepian, 1959 [4] and Slepian, 1962 [11]. In this paper we shall study the sample functions near local maxima of height $u$, especially as $u \rightarrow -\infty$, and mainly use similar methods as [4] and [11]. Then it is necessary to analyse carefully what is meant by "near a maximum of height $u$." In Section 2 we derive the "ergodic" definition, i.e. the definition which is possible to interpret by the aid of relative frequencies in a single realisation. This definition has been treated previously by Leadbetter, 1966 [5], and it turns out to be related to Kac and Slepian's horizontal window definition. In Section 3 we give a representation of $\xi(t)$ near a maximum as the difference between a non-stationary normal process and a deterministic process, and in Section 4 we examine these processes as $u \rightarrow -\infty$. We have then to distinguish between two cases. A: Regular case. $r(t) = 1 -\lambda_2t^2/2 + \lambda_4 t^4/4! - \lambda_6 t^6/6! + o(t^6)$ as $t \rightarrow 0$, where the positive $\lambda_{2k}$ are the spectral moments. Then it is proved that if $\xi(t)$ has a maximum of height $u$ at $t = 0$ then, as $u \rightarrow -\infty$, \begin{align*} (\lambda_2\lambda_6 - \lambda_4^2)(\lambda_4 - \lambda_2^2)^{-1}\{\xi((\lambda_2\lambda_6 - \lambda_4^2)^{-\frac{1}{2}}(\lambda_4 - \lambda_2^2)^{\frac{1}{2}}t|u|^{-1}) - u\} \\ \sim |u|^{-3}\{t^4/4! + \omega(\lambda_4 - \lambda_2^2)^{\frac{1}{2}}\lambda_2^ {-\frac{1}{2}}t^3/3! - \zeta(\lambda_4 - \lambda_2^2)\lambda_2 ^{-1}t^2/2\}\end{align*} where $\omega$ and $\zeta$ are independent random variables (rv), $\omega$ has a standard normal distribution and $\zeta$ has the density $z \exp (-z), z > 0$. Thus, in the neighborhood of a very low maximum the sample functions are fourth degree polynomials with positive $t^4$-term, symmetrically distributed $t^3$-term, and a negatively distributed $t^2$-term but without $t$-term. B: Irregular case. $r(t) = 1 - \lambda_2t^2/2 + \lambda_4t^4/4! - \lambda_5|t|^5/5! + o(t^5)$ as $t \rightarrow 0$, where $\lambda_5 > 0$. Now $\xi(tu^{-2}) - u \sim |u|^{-5}\{\lambda_2\lambda_5(\lambda_4 - \lambda_2^2)^{-1} |t|^3/3! + (2\lambda_5)^{\frac{1}{2}} \omega(t) - \zeta(\lambda_4 - \lambda_2^2)\lambda_2 ^{-1}t^2/2\}$ where $\omega(t)$ is a non-stationary normal process whose second derivative is a Wiener process, independent of $\zeta$ which has the density $z \exp (-z), z > 0$. The term $\lambda_5|t|^5/5!$ "disturbs" the process in such a way that the order of the distance which can be surveyed is reduced from $1/|u|$ (in Case A) to $1/|u|^2$. The results are used in Section 5 to examine the distribution of the wave-length and the crest-to-trough wave-height, i.e., the amplitude, discussed by, among others, Cartwright and Longuet-Higgins, 1956 [1]. One hypothesis, sometimes found in the literature, [10], states that the amplitude has a Rayleigh distribution and is independent of the mean level. According to this hypothesis the amplitude is of the order $1/|u|$ as $u \rightarrow -\infty$ while the results of this paper show that it is of the order $1/|u|^3$.

In this paper we consider the following situation: An experimenter has to perform a total of $N$ trials on two Bernoulli-type experiments $E_1$ and $E_2$ with success probabilities $\alpha$ and $\beta$ respectively, where both $\alpha$ and $\beta$ are unknown to him. The trials are to be carried out sequentially and independently, except that for each trial the experimenter may choose between $E_1$ and $E_2$, using the information obtained in all previous trials. The decisions on the part of the experimenter to use $E_1$ or $E_2$ in the successive trials may be randomized, i.e. for any trial he may use a chance mechanism in order to choose $E_1$ or $E_2$ with probabilities $\delta$ and $1 - \delta$ respectively, where $\delta$ may depend on the decisions taken and the results obtained in the previous trials. A strategy $\Delta$ will be a set of such $\delta$'s, completely describing the experimenters behavior in every conceivable situation. We assume the experimenter wants to maximize the number of successes. More precisely, we assume that he incurs a loss \begin{equation*}\tag{1.1} L(\alpha, \beta, s) = N \max(\alpha, \beta)- s\end{equation*} if he scores a total of $s$ successes. If he uses a strategy $\Delta$, his expected loss is then given by the risk function \begin{equation*}\tag{1.2} R(\alpha, \beta, \Delta) = N \max(\alpha, \beta) - E(S\mid\alpha, \beta, \Delta),\end{equation*} where $S$ denotes the random number of successes obtained. Thus the risk of a strategy $\Delta$ equals the expected amount by which the number of successes the experimenter will obtain using $\Delta$ falls short of the number of successes he would score if he were clairvoyant and would use the more favorable experiment throughout the $N$ trials. It is easy to see that $R(\alpha, \beta, \Delta)$ also equals $|\alpha - \beta|$ times the expected number of trials in which the less favorable experiment is performed under $\Delta$. We say that state $(m, k; n, l)$ is reached during the series of trials if in the first $m + n$ trials $E_1$ is performed $m$ times, yielding $k$ successes, and $E_2$ is performed $n$ times, yielding $l$ successes. Clearly, under a strategy $\Delta$, the probability that this will happen is of the form \begin{equation*}\tag{1.3} \pi_{\alpha,\beta,\Delta}(m, k; n, l) = p_\Delta(m, k; n, l)\alpha^k(1 - \alpha)^{m-k} \beta^iota(1 - \beta)^{n-iota},\end{equation*} where $p_\Delta(m, k; n, l)$ depends on the state $(m, k; n, l)$ and the strategy $\Delta$, but not on $\alpha$ and $\beta$. It is easy to show (e.g. by induction on $N$) that the class of all strategies is convex in the sense that there exists, for every pair of strategies $\Delta_1$ and $\Delta_2$ and for every $\lambda \in \lbrack 0, 1 \rbrack$, a strategy $\Delta$ such that \begin{equation*}\tag{1.4} p_\Delta(m, k; n, l) = \lambda p_{\Delta_1}(m, k; n, l) + (1 - \lambda)p_{\Delta_2}(m, k; n, l)\end{equation*} for every state $(m, k; n, l)$. Moreover, this strategy $\Delta$ can always be taken to be such, that according to it the experimenter should base all his decisions exclusively on the numbers of successes and failures observed with $E_1$ and $E_2$, irrespective of the order in which these data became available. Denoting the class of all such strategies by $\mathscr{D}$ and remarking that $R(\alpha, \beta, \Delta)$ can be expressed in terms of the $\pi_{\alpha,\beta,\Delta}(m, k; n, l)$, we may conclude that $\mathscr{D}$ is an essentially complete class of strategies. We denote the probabilities $\delta$ constituting any strategy in $\mathscr{D}$ by $\delta(m, k; n, l)$: the probability with which the experimenter, having completed the first $m + n$ trials and thereby having reached state $(m, k; n, l)$, chooses $E_1$ for the next trial. We note that if $p_\Delta(m, k; n, l) = 0$ for a state $(m, k; n, l)$, then $\delta(m, k; n, l)$ does not play any role in the description of $\Delta$ and may be assigned an arbitrary value without affecting the strategy. We shall say that any strategy $\Delta'$ such that $p_\Delta'(m, k; n, l) = p_\Delta(m, k; n, l)$ for all states $(m, k; n, l)$ constitutes a version of $\Delta$. Since we are considering a symmetric problem in the sense that it remains invariant when $\alpha$ and $\beta$ are interchanged, it seems reasonable to consider strategies with a similar symmetry. Thus we are led to define the class $\mathscr{L}$ of all symmetric strategies: $\Delta\in\mathscr{L} \operatorname{iff} \Delta\in\mathscr{D}$ and $\delta(m, k; n, l) = 1 - \delta(n, l; m, k)$ for all states $(m, k; n, l)$ with $p_\Delta(m, k; n, l) \neq 0$. Clearly, for $\Delta\in\mathscr{L}$, \begin{equation*}\tag{1.5} \delta(m, k; m, k) = \frac{1}{2} \text{if} p_\Delta(m, k; m, k) \geqq 0,\quad \text{and}\end{equation*} \begin{equation*}\tag{1.6} P_\delta(m, k; n, l) = p_\delta(n, l; m, k) \text{for all states} (m, k; n, l).\end{equation*} It follows that, for $\Delta \in \mathscr{L}$ and all $(\alpha, \beta)$, \begin{equation*}\tag{1.7} R(\alpha, \beta, \Delta) = R(\beta, \alpha, \Delta)\end{equation*}. Among the contributions to the two-armed bandit problem the work of W. Vogel deserves special mention. Considering the same set-up we do, he discussed a certain subclass of the class $\mathscr{L}$ in [4], and obtained asymptotic bounds for the minimax risk for $N \rightarrow \infty$ in [5]. Since we shall not be concerned with asymptotics in this paper, we state the following result without a formal proof: The lower bound for the asymptotic minimax risk for $N \rightarrow \infty$ obtained by Vogel in [5] may be raised by a factor $2^\frac{1}{2}$. This is proved by applying the same method that was used in [5] to the optimal symmetric strategy for $\alpha + \beta = 1$ that was discussed in [4]. Combining this lower bound with the upper bound given in [5] we find that the asymptotic minimax risk must be between 0.265 $N^{\frac{1}{2}}$ and 0.376 $N^{\frac{1}{2}}$. In Section 2 we study the Bayes strategies in $\mathscr{D}$. By means of a certain recurrence relation we arrive at a complete characterization of these strategies, thus generalizing D. Feldman's well-known result in [3] for the case where the experimenter knows the values of $\alpha$ and $\beta$ except for their order. In addition we obtain expressions for the Bayes risk of any prior distribution. Using these results we proceed to derive in Section 3 certain monotonicity properties of $\delta(m, k; n, l)$ for any admissible strategy $\Delta$ in $\mathscr{D}$. Though these relations may seem intuitively evident, one does well to remember that the two-armed bandit problem has been shown to defy intuition in many aspects (cf. [2]). In Section 4 we prove the existence of an admissible symmetric minimax-risk strategy having the monotonicity properties just mentioned. This fact to some degree facilitates the search for minimax-risk strategies, but even so, the algebra involved becomes progressively more complicated with increasing $N$ and seems to remain prohibitive already for $N$ as small as 5.

This paper is concerned with a two-factor analysis of variance situation, the factors being conveniently referred to as blocks and treatments. One of the treatments has the role of a control. Attention is focused on inference about the treatment parameters, the block parameters being regarded as nuisance parameters. With a general multivariate normal form for the distribution of errors and for the prior distribution on the block and treatment parameters, the posterior distribution of the treatment parameters is derived. With a quadratic loss function an algorithm is derived for the optimum allocation of treatments over a given sample with known blocking. In special cases the optimum allocation can be written down immediately and the algorithm need not be resorted to.

We shall introduce a set of random variables and give interpretations of them in terms of coupon collection. A person collects coupons with different colors. Let there be in all $N$ different colors, which we label $1,2, \cdots, N$. The different colors may occur with different frequencies. The colors of successive coupons are independent. Let $J_\nu$ be the color of the $\nu$th coupon. Our formal assumptions are: $J_1, J_2, \cdots$ are independent random variables, all with the following distribution \begin{equation*}\tag{1.1} P(J = s) = p_s,\quad s = 1,2, \cdots, N\end{equation*} where \begin{equation*}\tag{1.2} p_s \geqq 0,\quad p_1 + p_2 + \cdots + p_N = 1.\end{equation*} Thus, $p_s$ is the probability that a coupon has color $s$. Let \begin{equation*}\tag{1.3} M_n = {\tt\#} \text{different elements among} (J_1, J_2, \cdots, J_n),\quad n = 1,2, \cdots.\end{equation*} Thus, $M_n$ is the number of different colors in the collection after $n$ coupons. Let \begin{equation*}\tag{1.4} T_n = \min \{\nu: M_\nu = n\},\quad n = 1,2, \cdots, N.\end{equation*} $T_n$ is the number of coupons needed in order to get a collection with $n$ different colors in. Define \begin{align*} \tag{1.5} D_\nu &= 1\quad\text{if} J_\nu \notin (J_1, J_2, \cdots, J_{\nu-1}),\quad \nu = 1,2, \cdots \\ &= 0\quad \text{otherwise}.\end{align*} Thus, $D_\nu$ tells if the $\nu$th coupon adds a new color to the collection or not. We shall assume that the coupons also carry a bonus value, which is a real number. All coupons with the same color have the same bonus value, while the bonus value may differ from color to color. Let $a_s$ be the bonus value of coupons with color $s, s = 1,2, \cdots, N$. The bonus sum of a collection of coupons is obtained by adding the bonus values of the different colors which are represented in the collection. Thus, duplicates do not count. Formally we define the bonus sum as follows. \begin{equation*}\tag{1.6} Q_n = a_{J_1}D_1 + a_{J_2}D_2 + \cdots + a_{J_n}D_n,\quad n = 1,2, \cdots.\end{equation*} The random variable $Q_n$ will be referred to as the bonus sum after $n$ coupons for a collector in the situation $\Omega = ((p_1, a_1), (p_2, a_2), \cdots, (p_N, a_N))$. We define for $B > 0$ \begin{equation*}\tag{1.7} W(B) = \min \{n: Q_n \geqq B\}.\end{equation*} $W(B)$ will be referred to as the waiting time to obtain bonus sum $B$ for a coupon collector in the situation $\Omega$. The following lemma, which is obvious, states that we have introduced a slight abundance of terminology and notation. LEMMA 1.1. The random variables $M_n$ and $T_n$ in (1.3) and (1.4) are respectively the bonus sum after $n$ coupons and the waiting time to obtain bonus sum $n$ for a coupon collector in the situation $((p_1), (p_2, 1), \cdots, (p_n, 1))$. Our main concern will be to study the random variable $W(B)$ and its particular case $T_n$. We confine ourselves to the case when all bonus values, $a_s$, are positive. The main result is that $W(B)$, under general conditions, is asymptotically (as $n$ and $N$ increase simultaneously) normally distributed. We give a brief sketch of the idea of proof, which is well known. When all $a$'s are positive, the distributions of the random variables $W(B)$ and $Q_n$ are related according to the formula \begin{equation*}\tag{1.8} P(W(B) > x) = P(Q_{\lbrack x \rbrack} < B),\quad x, B > 0.\end{equation*} With the aid of formula (1.8) one can "invert" results concerning either of the random variables $Q$ or $W$ to yield results concerning the other variable. In [5] we showed that $Q_n$, under general conditions, is asymptotically normally distributed. The asymptotic normality of $W$ will be derived by inversion of the results in [5]. The asymptotic behavior of the collector's waiting time has, to the best of our knowledge, earlier only been considered in the classical case, i.e. $p_s = 1/N$ and $a_s = 1, s = 1,2, \cdots, N$. In [4] Section 3, Renyi derives results about $M$ by first deriving results about $T$ and then "inverting." His basic tool is the representation \begin{equation*}\tag{1.9} T_n = U_1 + U_2 + \cdots + U_n\end{equation*} where $U_v$ is the waiting time from bonus sum $\nu - 1$ to bonus sum $\nu$. In the classical case $U_1, U_2, \cdots$ are independent random variables and $P(U_\nu = k) = ((\nu - 1)/N)^{k-1}(N - \nu + 1)/N, k = 1,2, \cdots$. Thus, results concerning the asymptotic behavior of sums of independent random variables can be applied. A complete investigation along these lines is given by Baum and Billingsley in [1]. A generalized version of the problem is considered by Ivchenko and Medvedev in [2]. In their problem, as in our problem here, a representation of the type (1.9) no longer holds. They proceed along the path we shall follow here, i.e. to obtain results about the waiting time by "inverting" results concerning the bonus sum. The following notation will be used throughout the paper. $E$ and $\sigma^2$ stand for expectation and variance. $^c$ denotes centering at expectation, i.e. $X^c = X - EX. X = _\mathscr{L} Y$ means that the random variables $X$ and $Y$ have the same distribution. $\Rightarrow$ denotes convergence in law. The normal distribution with mean $\mu$ and variance $\sigma^2$ is denoted by $N(\mu, \sigma^2)$. The integral part of a real number is denoted by \lbrack \rbrack.

Subsamples are used to generate confidence intervals for a parameter of a linear regression model, under the assumption that the error variables are independent, continuous and symmetric about 0 in distribution.

Multivariate statistical analysis has been centered around normal theory thus far. Most papers dealing with this topic are based either directly on the assumption of normality of underlying distributions or indirectly on asymptotic normality of statistics induced by some limiting theorem such as the central limit theorem when sample sizes are sufficiently large. In many if not all fields of application, however, the assumption of normality is not guaranteed even approximately and also the sample sizes are not so large compared with the dimension of the variates in question as to provide a good approximation by normal distributions. Thus, it is required to develop a theory applicable to general multivariate distributions and there are some papers in literature approaching this problem from the standpoint of a distance: [1], [4], [10] and [11] among others. The purpose of this paper is to formulate statistical inference, point and interval estimation as well as testing hypotheses, in terms of a distance or a pseudo distance defined in the family of all probability distributions over a multidimensional Euclidean space. Three specific cases, one of which does not actually give a pseudo distance but which may be useful in some situations, are discussed. The paper consists of four sections; in Section 1 we treat inference with a random sample from one distribution; in Section 2 and Section 3 we treat inference with independent random samples from two and several distributions, respectively, and in the Appendix we present some mathematical results which are used in the preceding sections as tools more practical than [3], [5] and [9] and which may be interesting in themselves.

The problem of estimating the larger of two translation parameters, when it is not known which population has the larger parameter, was studied by Blumenthal and Cohen (1968a). Blumenthal and Cohen (1968b) have also investigated various estimators for the larger of two means of two independent normal populations. Problems of estimating the largest of a set of ordered parameters, when it is known which populations correspond to each ordered parameter, have been studied in the discrete case by Sackrowitz (1969). Some questions for the continuous case where one does not know which populations correspond to each ordered parameter have been studied by Dudewicz (1969). In this paper we study various problems of estimating the largest of a set of ordered parameters, when it is known which populations correspond to each ordered parameter. The observed random variables are either normally distributed or are continuous and characterized by a translation parameter. The main portion of the study is devoted to estimating the larger of two normal means when we know which population has the larger mean. Note that in one result below, an example of an estimator which is admissible with respect to a convex loss function, but which is not generalized Bayes is given. We proceed to state the models and list the results. Let $X_i, i = 1, 2$, be independent normal random variables with means $\theta_i$, and known variances. Without loss of generality we let the variance of $X_1$ be $\tau$ and the variance of $X_2$ be 1. Assume $\theta_2 \geqq \theta_1$, and consider the problem of estimating $\theta_2$ with respect to a squared error loss function. Let $\delta(X_2)$ be any estimator based on $X_2$ alone. Consider only those $\delta(X_2)$ which are admissible for estimating $\theta_2$ when $X_1$ is not observed. The following results are obtained. (1) If the risk of $\delta(X_2)$ is bounded, then $\delta(X_2)$ is inadmissible. This result can be generalized in a few directions. In fact if $\theta_i$ are translation parameters of identical symmetric densities, then for any nonnegative strictly convex loss function $W(\cdot)$, with a minimum at $0, X_2$ is an inadmissible estimator. Suitable generalizations for arbitrary sample sizes are given. Another generalization is that if $C$ is any positive constant, then $X_2 \pm C$ is inadmissible as a confidence interval of $\theta_2$. (2) Let $U_\tau$ be the positive solution to the equation $a^2 + (\tau + 1)a - \tau = 0$. The quantity $U_\tau$ will be such that, $0 \leqq U_\tau < 1$. Then the estimators $aX_2$, for $0 \leqq a < U_\tau$, are admissible. It will be shown that no $\delta(X_2)$, such that $\delta(X_2)$ is unbounded below, can be generalized Bayes. Thus this result provides an example of an estimator which is not generalized Bayes, but which is admissible for the squared error loss function. The results above are also true for estimating the largest of $k$ ordered means with known order, in the case of equal variances. It is interesting that for some $a > 0, aX_k$ is admissible, regardless of the size of $k$. The proof of admissibility of $aX_2$ uses the methods of Blyth (1951) and Farrel (1968). (3) Consider the analogue of the Pitman estimator. That is, the estimator which is generalized Bayes with respect to the uniform prior on the space $\theta_2 \geqq \theta_1$. We prove that this estimator is admissible and minimax. In the next section the inadmissibility of $\delta(X_2)$, given that the risk of $\delta(X_2)$ is bounded, is proved. In Section 3 we prove $aX_2$ admissible for $0 \leqq a < U_\tau$. We also extend the result here to the case of estimating the largest of $k$ ordered means with known order, in the case of equal variances. In Section 4 we show that if $\delta(X_2)$ is unbounded below then it cannot be generalized Bayes. Generalizations to the symmetric translation case and arbitrary but equal sample sizes are given in Section 5. The confidence interval result is in Section 6 and the final results on the admissible and minimax property of the analogue to the Pitman estimator are given in Section 7. Throughout, the letters $C, K$, and $M$ with or without subscripts are used to denote positive constants, not necessarily the same in all cases. Also the symbols $\varphi$ and $\Phi$ are used to denote the probability density function and cumulative distribution function respectively, of the standard normal.

In this paper we are concerned with a generalization of the concept of Latin squares and orthogonality of Latin squares. The condition that every element appears once in each row and each column is replaced by the condition that it appears the same number of times in each row and each column. We call such squares $F$-squares. The usefulness of introducing and investigating the properties of $F$-squares could be justified in two directions. $F$-squares have meaningful applications in laying out experimental designs as exhibited previously by some authors. Their properties prove to be a useful tool in the studies of existence of orthogonal Latin squares and other combinatorial problems.

This paper generalizes some results of Wijsman concerning the calculation of the density of a maximal invariant. The idea of the technique is to represent the sample space as a product space, one factor $Z$ being a global cross section, i.e., essentially a set that intersects each orbit in a unique point, and the other factor being a coset space of the invariance group. Integration over the invariance group then gives the distribution of the identity function on $Z$ which is a maximal invariant. Part I of the paper gives sufficient conditions for the technique to be applicable, while Part II exhibits the technique along with an example. Part II is on a more elementary level than Part I and may be understood without a reading of Part I.

The probability is determined that the sample distribution function (df) of a random sample of any size $n$, drawn from the uniform distribution, lies below a given line segment of any slope over some $(\alpha, \beta) \subset \lbrack 0, 1\rbrack$ (Section 1, Theorem 1.1 ff.). Probabilities of related events, also under conditioning, are derived. It is well known that results of this type are equivalent to similar ones for random samples from any continuous df. A catalogue of equivalent formulae is given, the various versions being advantageous on certain ranges of the parameters. These results rest upon and generalize a formula (Theorem 2.1 below) of Dempster (1959), Dwass (1959), and Pyke (1959) (his Lemma 1), which gives an explicit expression for any $n$ that the sample df lies below some (straight) line extended over the entire unit interval. Otherwise the proof uses familiar properties of order statistics, the whole argument being essentially a combinatorial one. The present result also generalizes, in particular, results by Wald and Wolfowitz (1939), by Birnbaum and Tingey (1951) (in both papers sample df below a line segment of slope 1 over [0,1]; especially the latter paper, which improves the first, is at the root of the approach of the present article as well as of the papers by Dempster, Dwass, Pyke), by Smirnov (1944 and 1961) (sample df below a line segment with slope 1 over (0,1) or $(\alpha, 1)$), by Chang (1955) (line segment of any nonnegative slope joining the origin with some point in the open unit square $I_2^0$), Csorgo (1965) (line segment of any slope may also end in the point (1, 1)), Birnbaum and Lientz (1969) (line segment of any nonnegative slope through the origin over any subinterval of the unit interval). Apparently the first author who determined explicitly the probability that the sample df lies below a line segment over an arbitrary subinterval $(\alpha, \beta)\subset\lbrack 0, 1\rbrack$ was Takacs (1964) (Theorem 3; see also Takacs (1967), pages 176-178). (The author is indebted to Professor J. Kiefer for reminding him of this reference.) However, he had the (not very crucial) restriction that the slope $\gamma$ of the line be $\geqq 1$. Moreover, his formula contains a double sum, whereas some of ours contain a single one. This fact proves to be of great advantage in the applications we have made so far. Theorem 1.2 below gives certain conditional probabilities that the sample df lies below a line segment over some subinterval of (0,1). These probability expressions (as any of those described above), for a suitable sequence of line segments depending on $n$ tend to the probability that the Brownian bridge (or conditioned Wiener) process lies below a line segment, as $n \rightarrow \infty$ (see [18], in particular pages 182-183). This follows from the Doob-Donsker theorem on weak convergence of probability measures (see also [13]). These asymptotic results provide a starting point different from and possibly simpler than that of Section 1 to compute approximate probabilities that the sample df lies below some curved line (for an application and some of the asymptotic formulae compare [7] and [17]). It is the objective of a later paper to determine the error of the approximation. Theorem 1.3 gives similar conditional probability expressions that the sample probability (a generalization of the sample df, defined below) for intervals of variable lengths lies below a line segment. It is these formulae that are needed in a new version of the proof of the Bahadur-Kiefer representation theorem for sample quantiles [1]. The formulae derived here are explicit (though involved) in contrast to recursive ones first considered in [12] and extended, more recently, e.g., in [9] and [25]. They have been used, e.g., to compute explicitly the probability that the sample df lies below a polygon (compare a forthcoming paper of the author). Another application is the derivation of asymptotic formulae paralleling and generalizing, e.g., those of Renyi and Csorgo ([4] and [10]). Implications for stochastic processes (such as those studied in [19] and for the theory of goodness of fit tests (compare [1]) are not considered. In [1] also the statistic $F_n(x) - F(x)$, divided by its standard deviation, has been proposed. Certain functionals of this statistic may be studied by the method of the present paper. On the other hand, the method is not immediately applicable to the study of probabilities that the sample df lies, e.g., between two lines (for a recent paper on this compare [16]). The vast existing literature on Kolmogorov-Smirnov type statistics (for recent surveys compare [20] and the appendix of [12a]) is not being surveyed for possible applications other than the few above-mentioned representative examples.

A particular noiseless, discrete channel with memory (called a transducer) is made to correspond to a function in the unit square by associating the infinite sequences of symbols of the transducer with the expansions of points in the unit interval. It is shown that the Hausdorff dimension of the set of points received over the transducer is equal to the transducer capacity. A definition of ambiguity is given which has a geometric interpretation in the square, and it is shown that the transducer has a homogeneity property by proving that the ambiguity is almost everywhere the same.

Let $(X(t): t \geqq 0)$ be a Markov process with lifetime $\zeta$, let $g$ be a bounded continuous nonnegative function on the state space of the process, and let $P_x$ (respectively, $E_x$) be the probability measure (respectively, expectation operator) associated with paths starting at $x$. For any (extended real-valued) Markov time $T$ let $f(x, T) = \int_{T < \zeta} g(X(T))dP_x$ and let $f(x) = \sup_Tf(x,T)$. For any nonnegative $\varepsilon$, a Markov time $T^\ast$ is called (i) $\varepsilon$-optimal at $x$ if $f(x, T^\ast) \geqq f(x) - \varepsilon$; (ii) optimal at $x$ if $f(x, T^\ast) \geqq f(x)$; (iii) $\varepsilon$-optimal if $f(x, T^\ast) \geqq f(x) - \varepsilon$ for all $x$; and (iv) optimal if $f(x, T^\ast) \geqq f(x)$ for all $x$. We interpret $g(X(T))$ as a reward associated with stopping at time $T$ in state $X(T)$ and we are searching for stopping rules or Markov times $T^\ast$ which maximize or nearly maximize the expected value of this reward. Since the process is Markov and the reward depends only on the state in which one stops and not on the time nor the previous process history, one would anticipate that an optimal stopping time (provided one exists) would be of the form "Continue as long as $f(X(t)) > g(X(t))$ and stop when first $f(X(t)) \leqq g(X(t))$." That is, at time $t$ in state $X(t) = x$, one continues if the optimal reward from continuing $f(x)$ exceeds the reward from stopping $g(x)$; otherwise, one stops. This reasoning leads one to hope that the search for optimal rules can be restricted without loss to rules specified by a partition of the state space into two sets, one of continuation states and one of stopping states. Thus we ask under what conditions \begin{equation*}\tag{1} f(x) = \sup f(x, T(\Gamma))\end{equation*} where the supremum is over all appropriately measurable sets $\Gamma$ and $T(\Gamma)$ is the process entry time to $\Gamma$: \begin{align*} \tag{2} T(\Gamma) &= \infty\quad \text{if} X(t) \notin \Gamma \text{for all} t \geqq 0; \\ &= \inf\{t:t \geqq 0 \text{and} X(t)\in \Gamma\},\quad \text{otherwise}.\end{align*} Dynkin (1963) shows that if $X$ is a standard process then for $\varepsilon \geqq 0, \Gamma_\varepsilon = \{x:f(x) - \varepsilon \leqq g(x)\}$ is closed in the fine or intrinsic topology and for $\varepsilon > 0, T(\Gamma_\varepsilon)$ is $\varepsilon$-optimal, so that in particular (1) holds. When $g$ is unbounded, Dynkin (1968) interprets a result of Chow and Robbins (1967) to state that for $\varepsilon \geqq 0$, if $P_x \lbrack T(\Gamma_\varepsilon) < \infty \rbrack = 1$ for all $x$, then $T(\Gamma_\varepsilon)$ is $\varepsilon$-optimal. Taylor (1968) shows that if in addition the transition semigroup of the process is Feller (leaves invariant the space of bounded continuous functions) then $\Gamma_0$ is closed, and if there exists an optimal Markov time, then $T(\Gamma_0)$ is optimal at all points $x$ for which $f(x) < \infty$. In this paper we present a Markov process on continuous path space and a bounded continuous nonnegative $g$ such that for some positive $\varepsilon$ and some point $x$, no entry time is $\varepsilon$-optimal at $x$. It follows that no entry time can satisfy any of the other, more stringent, criteria for optimality. Our process is Markov but not strong Markov (hence, not Feller) and was suggested to me by a related example I learned from David Freedman (1969). Our construction is based on a transformation by a singular continuous strictly increasing function. A local time transformation serves the same purpose in Freedman's example. The idea behind the example is quite simple, but the details are many and tedious and tend to obscure the basic picture, which is this: Let $y(t)$ be a Brownian motion process starting at the origin. Our process $x(t)$ is a distorted version of $y(t)$. In particular, begin with $x(t) = y(t)$ and continue until first $y(t)$ hits $+1$. Beginning then, distort the process by setting $x(t) = G^{-1}\lbrack y(t) \rbrack$, where $G$ is strictly increasing, continuous and $G(1) = -G(-1) = 1$. Continue until $y(t)$ next reaches $-1$, at which time, revert to $x(t) = y(t)$. Repeat, again waiting until $y(t)$ hits $+1$, then switching to $x(t) = G^{-1}\lbrack y(t)\rbrack$, and so on. For the $x(t)$ process it is clear that a good decision on whether or not to stop should be based in part on whether currently $x(t) = y(t)$ or $x(t) = G^{-1}\lbrack y(t) \rbrack$, or equivalently, whether the process is on a $-1$ to $+1$ section, or on a $+1$ to $-1$ section. In particular, entry times cannot be good because they do not include this information. Granted, to nail down an example, a sufficiently simple stopping problem must be chosen so that a variety of calculations can be made, but it is intuitively clear why entry times cannot be good and how to improve them. This part of our example is presented in Section 3. It is not clear, however, that we can distort the Brownian motion as we have described and yet maintain the Markov property. Here is where we require $G$ to be singular, so that $G^{-1}$ carries a set of full Lebesgue measure into a set, call it $\Lambda$, which is Lebesgue null. At a fixed time $t, y(t)$ is not in $\Lambda$ with probability one, since $\Lambda$ is Lebesgue null. Thus, if we observe $x(t)$ in $\Lambda$ we may infer, with probability one, that $x(t) = G^{-1}\lbrack y(t) \rbrack$. Similarly, still for fixed $t$, if we observe $x(t)$ not in $\Lambda$, we infer, with probability one, that $x(t) = y(t)$. This feature preserves the Markov property. A careful development of this idea is given in Section 2.

Raj and Khamis [2], and Raj [3], have shown that in sampling with replacement the average over distinct units possesses lower variance than the average over the entire sample including repetitions. But no usable results are available regarding the amount of reduction in variance. The relevant expression given by Raj and Khamis is prohibitively cumbersome even for numerical computations. The present note provides the simplest possible exact expression as well as a convenient approximation. In particular, it follows readily that the relative benefit due to averaging over distinct units only is basically an increasing function of the sampling fraction.

For non-identically distributed random variables, some inequalities on the median and the tails of the distribution of a sample order statistic are derived, and a simple condition for the existence of its moments is studied.

For a broad class of one-sample rank order statistics, almost sure (a.s.) convergence and exponential bounds for the probability of large deviations, when the basic random variables are not necessarily identically distributed, are established here. In this context, extending a result of Brillinger (1962) to the case of non-iidrv (independent and identically distributed random variables), a result on the a.s. convergence of sample means for a double sequence of random variables is derived. These results are of importance for the study of the properties of sequential tests and estimates based on rank order statistics.

Asymptotic normality of a class of rank statistics based on random number of observations, called random rank statistics, is proved. Underlying rv's are assumed to be i.i.d.

For a logistic distribution which is defined by $x = \ln \{F/(1 - F)\}$, where $F$ is the probability of a value less than $x$, Plackett [4], Birnbaum and Dudman [1], and Gupta and Shah [2], obtained the expressions for the moments of the ordered statistics in terms of digamma functions. Once the digamma functions are tabulated and their values are stored in the computer memory, it is easy to calculate the moments (first and second). This procedure was adopted by Birnbaum and Dudman and the author (jointly with others) in the past to calculate the moments of these order statistics for which some tables are available in the literature. In this note a recurrence relation for the moments of all order statistics for a logistic distribution is derived. This result not only generates higher moments of order statistics but also generates the moments of all order statistics for a logistic distribution, and this eliminates problems such as errors involved in computing and machine time etc. Thus using a computer language such as Fortran or APL, this simple relation is easily programmable for robustness studies, [3] for a logistic distribution.

Let $X_1, X_2, \cdots$ be independent and identically distributed. We give a simple proof based on stopping times of the known result that $\sup(|X_1 + \cdots + X_n|/n)$ has a finite expected value if and only if $E|X| \log |X|$ is finite. Whenever $E|X| \log |X| = \infty$, a simple nonanticipating stopping rule $\tau$, not depending on $X$, yields $E(|X_1 + \cdots + X_\tau|/\tau) = \infty$.

This paper is a sequel to the earlier paper, "Maximum Likelihood Estimation of a Unimodal Density Function." The MLE of a unimodal density with unknown mode is shown to agree, for sufficiently large $n$ and on certain regions, with the MLE of a unimodal density with known mode. The asymptotic distributions of the MLE's then agree. Also a geometrical interpretation of the MLE of a unimodal density with unknown mode is given.

Consider a saturated main effect plan for the parametric vector $\beta_1$, then this paper shows that when this plan is optimum (in the sense of maximum determinant of the design matrix) then the complementary plan is also optimum for the complementary vector $\beta_2$. Some additional corollaries are derived for special cases.