Some new test criteria are proposed for testing various hypotheses concerning covariance matrices. Asymptotic expansions of their null distributions are derived in terms of the $\chi^2$-distribution.
This paper considers some empirical Bayes procedures which have been discussed by H. Robbins, E. Samuel, and M. V. Johns. These procedures are shown to be inadmissible relative to a class of priors and by using some of the results of Rolph [6] admissible procedures are found for two examples. For an introduction to the empirical Bayes approach see Robbins [5].
In this paper there are given new proofs of two convergence theorems for conditional expectations, concerning convergence in measure and convergence almost everywhere of a sequence of conditional expectations $P_n^\mathscr{F}0f$ of a bounded function $f$, given a $\sigma$-field $\mathscr{F}_0$, with respect to varying probability measures $P_n$.
It is well known that the first passage times for Brownian motion have stable laws with exponent $\frac{1}{2}$. It is shown here that first passage times for random walks have distributions in the domain of attraction of a stable law with exponent $\frac{1}{2}$.
The finite sequence of $n$ random variables $U_1 U_2,\cdots, U_n$ is divided into two complementary groups of random variables in one of $2^n$ ways. The random variables in each group are summed and the two sums are compared. Let $|S_n|$ be the minimum of the difference of the sums out of all the $2^n$ possible divisions. A lower bound to all sequences $\{\epsilon_n\}$ such that $P\{|S_n| < \varepsilon_n\} \rightarrow 1$ as $n \rightarrow \infty$ is found in two cases: - $U_i = X_i i = 1,2,\cdots n$ and $U_i = X_i/\sum^n_{i=1} X_i, i = 1,2\cdots n$ where the $X_i$ are independent and identically distributed random variables which have densities and satisfy certain regularity conditions. The results lead to the solution of the particular problem of minimising the difference between two sums formed from segments of a fractured unit interval.
Let $X_1, X_2, \ldots, X_N$ be independent identically distributed random variables with common continuous distribution function $F$. Designate by $\mathscr{J}$ a nonempty set of subsets of the integers $\{ 1, 2,\ldots, N\}$ and by $\mathscr{Y} = \mathscr{Y}(\mathscr{J})$ the mapping which assigns to each set $I \in \mathscr{J}, I = \{t_1, t_2, \ldots, t_k\}$ the partial sum $\sum{t_j \in I}X_{t_j}$. Define the random variable $N = N(\mathscr{J})$ as the number of positive sums in the range of $\mathscr{Y}. N(\mathscr{J})$ has been shown to be distribution free when $F$ is the distribution function of a symmetric random variable if $\mathscr{J} = \{1,2, \ldots, N\}$ or $\mathscr{J} = \text{power set of} \{1,2, \ldots, N\}$. Several other nontrivial examples of this phenomenon have been discovered--all by different methods. This paper presents a unified method that derives all previously known results, provides a constructive method for obtaining infinitely many essentially different sets $\mathscr{J}$ with this property, and finally provides a powerful necessary condition on any such set $\mathscr{J}$ that yields a complete characterization of those sets $\mathscr{J}$ for which $N(\mathscr{J})$ is distribution free and $\mathscr{J}$ contains all $k$ element subsets of $\{1,2, \ldots, N\}$ where $k = 2,3, \ldots, N - 1$.
In a paper by Abrahamson [1], it is shown that the Kuiper test generally performs better than the Kolmogorov-Smirnov (K-S) test according to exact Bahadur relative efficiency. The present note concerns the Bahadur efficiency of a related test statistic $U_n$ whose exact null probability distribution is available in the two-sample case with equal sample sizes. It is shown that $U_n$ is often more efficient than the K-S test and may even be as efficient as the Kuiper test.
In sampling with replacement from a finite population, the sample mean is known to get a smaller variance if repetitions are deleted before forming the mean. The asymptotic behavior of the variance just mentioned is studied.
This paper deals with two different problems. The first one deals with asymptotic normality of simple linear rank statistics based on random number of observations $X_i$, henceforth called random rank statistics, under the alternative where each $X_i$ has a different distribution $F_i$. The second problem deals with showing that the random rank statistics as a function of the regression parameter in the simple linear regression model is asymptotically uniformly linear (hence continuous) in that parameter. Obviously the two problems are different and could be solved in separate papers but for certain lemmas which are common to the solution of both of these problems. It is suggested not to try to apply the result of Section 3 to Section 4, unless mentioned explicitly. The results of Section 2 are the results which are common, to some extent, to the solution of these two problems. Pyke and Schorack [11] proved asymptotic normality of a class of two sample random rank statistics under two sample alternatives. Our theorem 3.1 could be thought of as a generalization of the result of [11] to more than two samples situation. Our score function $\varphi$ is in smaller class than that of [11]. On the other hand our methods yield the asymptotic normality for random rank-sign statistics. This is also contained in Section 3. Section 2 proves a basic lemma about weak convergence of random weighted empirical cumulatives to a tied down continuous Gaussian process. In [5] and [7] asymptotic uniform linearity of rank statistics based on nonrandom number of observations was proved. In [5] conditions are very general on $\varphi$ and underlying distribution $F$ whereas conditions in [7] are quite stringent. But in [7] we do not need any artificial condition like (2.1) of [5] on underlying regression constants. However in both of these references, regression scores were assumed to be bounded. In Section 4 here we extend the results of [7] to random rank and random rank-sign statistics and to the case where regression scores need not be bounded. In Section 5 we show how the results of Section 4 can be used to construct a bounded length confidence interval for a regression parameter using rank sign statistics with asymptotic (as length $\rightarrow 0$) coverage probability achieved. Apart from applying Theorem 3.1 to the i.i.d. case, as is mentioned in the remark at the end of Section 5, it is hoped that Theorem 3.1 can be found applicable in some other interesting situations.
We first show a way of constructing a path of length $2n$ from a pair of paths of length $n$ by means of which one may arrive at many results on pairs of paths of length $n$, simply by examining properties of paths of length $2n$. Secondly, for two random walk paths of length $n, A$ and $B$, with vertical coordinates $A(i)$ and $B(i)$ respectively, at times $i = 0,1,\cdots, n$, and such that for some $m A(m) > B(m)$ but $A(i) = B(i)$ when $i < m$, we define $d_{A,B}(i) = \frac{1}{2}(A(i) - B(i))$. For obvious reasons $A(i) - B(i)$ is always even, which incidentally, implies that the intersection of two paths are points with integral coordinates. We find that $d_{A,B}$ can be graphed against time by a three-valued random walk path, i.e. a path which may have horizontal steps. Questions about the pair consisting of $A$ and $B$ may then be answered by observing the path described by $d_{A,B}$. Results in the theory of three-valued random walk paths can thus be translated into results about pairs of random walk paths of equal length.
That the Kolmogorov-Smirnov statistics obey iterated logarithm laws is well known. For the purpose of developing nonparametric tests with power one it has become of interest to find accurate upper bounds for the probability that a sequence of Kolmogorov-Smirnov statistics ever exceeds a given boundary sequence. This paper is concerned with finding such probability bounds for a wide class of boundary sequences.
The problem of partitioning a set of normal populations with respect to one control has been considered by Tong (1969). This note points at the possibility of using two controls which gives a partitioning into three disjoint groups. The procedure utilizes Tong's tables. It is also stated that the procedure can be used for mixed randomized block designs with a certain symmetry. An extension to more than two controls is also mentioned.
Let $(S, \mathscr{I})$ be a measurable space, $M$ the set of all finite measures on $\mathscr{I}, \mathscr{F}_M$ the $\sigma$-algebra generated by the family of all measurable cylindrical sets $\cap^k_{i=1} \{\mu \in M: \mu(A_i) \leqq a_i\}$. With each probability measure $P$ on $\mathscr{F}_M$ the family $\{P_{A_1}, \cdots, A_k\}$ of all finite-dimensional probability measures of the cylidrical sets is associated. The following problem is considered: Given a sequence $P^{(n)}$ of probability measures on $\mathscr{F}_M$ such that each sequence $P^{(n)}_{A_1}, \cdots, A_k$ converges weakly to a $k$-dimensional probability measure $P_{A_1}, \cdots, A_k$, does the family $\{P_{A_1}, \cdots, A_k\}$ generate a probability measure $P$ on $\mathscr{F}_M?$ It is proved that the answer is affirmative if $(S, \mathscr{I})$ is the Euclidean $n$-space with the $\sigma$-algebra of Borel sets.
