Derivations are given of the size and power of a test of a hypothesis for an incompletely specified fixed linear model. Such a testing procedure involves a test of a hypothesis of main interest subsequent to a preliminary test to determine the inclusion or not of a term in the tentatively specified complete fixed linear model. An evaluation of the formulas of the size and power was made over a wide range of values of the parameters involved, and recommendations concerning the choice of a proper significance level are given. One important objective of this study for the fixed model was to parallel as nearly as feasible the study given by Bozivich, H., Bancroft, T. A. and Hartley, H. O. (1956), Ann. Math. Statist. 27 1017-1043 for the random model, and to compare the two. It should be noted that the results and recomendations for the fixed model are quite different from those given in the 1956 paper for the random model.

This paper introduces a new mixing condition for stationary processes which is weaker than $\phi$-mixing but stronger than strong mixing. Many processes arising in applications, e.g., first order autoregressive processes, obey the conditions. The main result is that the empiric cdf of a sample from such processes converges to a Gaussian process.

In this paper, we shall represent a generalized Chernoff-Savage statistic as the sum of i.i.d. random variables plus a remainder term and analyze the order of magnitude of the remainder term. While Chernoff and Savage have proved that the remainder term, when suitably normalized, converges to O in probability, we obtain a stronger form of convergence in this paper. Our result gives an invariance principle and a law of the iterated logarithm for generalized Chernoff-Savage statistics. We also use our result to obtain asymptotic approximations for the stopping rules of certain sequential rank tests.

In many situations one is faced with the task of constructing a linear order, or ranking, of n objects based on data derived from a paired comparison experiment. Alternatively, one may desire to estimate the preference relation on the set of objects. Numerous criteria appear in the literature, and in practice, under which rankings or preferences may be selected. The main emphasis of this paper is on developing a general approach to maximum likelihood estimation of rankings and preferences. Utilizing what we term f criteria, our results unify and extend both the theory of constrained maximum likelihood ranking estimation and the work of Singh and Thompson on preference estimation. In addition, we show that a specific mathematical programming problem subsumes the problem of finding a maximum likelihood ranking and also that an efficient branch search algorithm can be used to find maximum likelihood preferences. Four specific f criteria are selected for illustration and each is applied to three examples from the paired comparison literature.

A point estimator and a set of confidence intervals based on the Kolmogorov-Smirnov statistic are proposed for the shift parameter in the two-sample problem. Asymptotic distibution of the etimator as well as asymptotic bounds for the lengths of the intervals are derived. The two-sample results are then adapted to the one-sample problem to define an estimator and a set of confidence intervals for the center of a symmetric population.

Recent results of Shorack (Ann. Math. Statist. (1972) 412-427) on the asymptotic normality of functions of order statistics are extended to stationary $\phi$-mixing processes and to a class of strong mixing processes. The results of this paper are based on the weak convergence of empirical processes relative to the metric $d_q$ as developed in Fears and Mehra (Ann. Statist. (1974) 586-596). Some remarks on trimmed and Winsorized means in the strong mixing case are also included.

Let P be a convex set of finitely additive probability measures defined on a Boolean algebra of subsets of a set X of consequences. Axioms are specified for a preference relation $\prec$ on P which are necessary and sufficient for the existence of a real-valued utility function u on X for which expected utility E (u, p) is finite for all p in P and for which $p\precq$ iff E (u, p) > E (u, q), for all p and q in P. A slightly simpler set of axioms yields the same results when the algebra is a Borel algebra and every measure in P is countably additive. The axioms allow P to contain nonsimple probability measures without necessarily implying that the utility function u is bounded.

Let ${P_n}^{\infty_n=1}$ denote a sequence of probability measures on $(R^k,B^k)$, where $R^k$ is k-dimensional Euclidean space and $B^k$ the Borel subsets. For $A\inB^k$, let $e(A)=lim_{n\rigtharrow\infty}(1/n) log P_n(A)$, if the limit exists. Sufficient conditions are given for expressing e(A) as the supremum of e(B) for certain "rectangular" sets $B=X^k_{j=1}(\alphaj,\betaj)$ with either $\alphaj=-\infty or \betaj=+\infty$ for each j = 1, ..., k. Also, some k-dimensional generalizations of the density theorem of Killeen, et al. (1972) are given for expressing e(A) in terms of certain limits of the sequence of density (or probability) functions. Finally, an example is considered where $P_n$ is the distribution of k order statistics from a sample of size n.

Suppose N objects are hidden multinomially in m boxes, where m is known and N is random. The boxes are to be searched sequentially. Asssociated with a search of box k is a cost $c_k>0$ and a conditional probability ${alpha k}$ of finding a specific object in box k, given that it is hidden there. An optimal strategy is one which minimizes the total expected cost required to find at least one object. If N has a positive-Poisson distribution, then an optimal strategy is shown to take a simple form. Conversely, if for all possible ${c_k}$ and ${\alpha_k}$ an optimal strategy takes this simple form, then N has a positive-Poisson distribution.

The asymptotic non-null distributions of four statistics, used for testing the hypothesis Ho: $\Sigma_1=\Sigma_2$ , are developed. These expansions are obtained by the partial differential equations method of Muirheard (Ann. Math. Statist. 41 1002-1010). The procedure permits a simple extension of the results of Pillai and Nagarsenker (J. Multivariate Anal. 2 96-114).

It is well known at by identifying the points of an affine geometry AG (t, q) with treatments and identifying the $\mu$-flats $(1\leqq\mu<t)$ of AG (t, q) with blocks, a BIB design denoted by AG (t, q): $\mu$ is derived from AG (t, q) where q is a prime or a prime power. In this paper, we introduce a new association scheme called an affine geometrical association scheme and show that the dual of the BIB design AG (t, q): $\mu$ is an affine geometrical type PBIB design with $m=min{2\mu+1,2(t-\mu)}$ associate classes. It is also shown that in the case $\mu=1$ and $t\geqq3$, the number of the associate classes of this dual design can be reduced from three to two but it is not reducible except for the above case. From those results, we can get a new series of PBIB designs.

Let $X_1,X_2,...$ be a Bernoulli sequence with parameter p. An algorithm $T_{n=1}=\\f(T_n,X_n,n)$; $d_n = d(T_n); \\f:\{1,2,\1dots,8\} \times \{0,1\} \times \{0,1, \1dots}\rightarrow \{1, \1dots, 8\}; d:\{1,2,\dots,8\} \rigtharrow \{H_0,H_1\}$; is found such that $d(T_n)= H_0$ all but a finite number of times with probability one if p is rational, and $d(T_n)= H_1$ all but a finite number of times with probability one if p is irrational (and not in a given null set of irrationals). Thus, an 8-state memory with a time-varying algorithm makes only a finite number of mistakes with probability one on determining the rationality of the parameter of a coin. Thus, determining the rationality of the Bernoulli parameter p does not depend on infinite memory of the data.

Exact limiting Bahadur efficiencies of linear rank tests for scale are derived using a minor extension of the results of Hoadley. Efficiency curves are derived for the linear rank statistics of Mood, Freund and Ansari, and Capon and Klotz relative to the F test for normal scale alternatives when the null hypothesis is that of common normality.

Sharp bounds on the variance of the Wilcoxon-Mann-Whitney U-statistic are obtained for the case of symmetric distributions and shift alternatives.

The chi-square test, likelihood ratio test, and other goodness of fit tests in a family are shown to have monotonic power functions, and hence are unbiased, for testing a simple hypothesis that all cell probabilities are equal. The chi-square test is also shown to be Type D for this situation.

Moore (1971) derives the limiting distribution of chi-square statistics for testing goodness of fit to k-variate parametric families, where the cell boundaries are allowed to be functions of the estimated parameter values. The only point at which random cells, multivariate observations etc. require a deviation from methods of proof given by Cramer (1946) is in the proof of the asymptotic negligibility of the remainder terms. Attention is drawn to an alternative proof of this asymptotic negligibility, which turns out to be an immediate consequence of a modification of Lemma 1 by Bahadur (1966) in more dimensions.

Let $S(b) = \Sigma r\sigma r^2Xr^2(nr,br^2)$ be a positive linear combination of independent noncentral chi-square random variables. This note derives two representations for the tail probabilities P[S(b) >x], a Taylor series in the noncentrality parameters and a limiting form of this series for large x. An application of the latter result to statistical tests of Cramer-von Mises type is discussed.

In estimating a vector model, $\Sigma B(j)x(n-j)=\Sigma A(j)\epsilon(n-j), A(0)=I_r, E(\epsilon(m)\epsilon(n)')=\delta_{mn}K$ it is suggested that attention be confined to cases where $g(z) =\Sigma A(j)z^j, h(z)=\Sigma B(j)z^j$ have determinants with no zeroes inside the unit circle and have $I_r$ as greatest common left divisor and where $\1brack A(p)\vdots B(q) \rbrack$ is of rank r, where p, q are the degrees of g, h, respectively. It is shown that these conditions ensure that a certain estimation procedure gives strongly consistent estimates and the last of the conditions is probably necessary for this to be so, when the first two are satisfied. The strongly consistent estimation procedure may serve to initiate an iterative maximisation of a likelihood.

A simple expression is developed for the difference between the least squares and minimum variance linear unbiased estimators obtained in linear models in which the covariance operator of the observation vector is nonsingular. Bounds and series expansion for this difference are obtained, and bounds for the efficiency of least squares estimates are also obtained.

Sequential tests for one-sided hypotheses are compared, asymptotically, with non-sequential counterparts. An analog of Pitman efficiency is obtained, as is another comparison that has no purely non-sequential analog. With these methods of comparison, the limiting relative efficiency of the sequential test is never less than one and for most parameter values, it is infinite. An asymptotic notion of minimal relative efficiency is also considered.

In this paper, we prove certain theorems about the first exit time $N=inf{n\geqq1:S_nT_n+R_n\not\in(-a,b)}$, where $S_n$ is the partial sum of i.i.d. random variables with zero mean and finite positive variance, and $R_n,T_n$ are two sequences of random variables satisfying certain conditions. Such exit times arise in the analysis of the stopping rules of invariant sequential probability ratio tests, and our theorems are then applied to study the stopping rules of these tests.

Let N be the stopping time of the sequential t-test, based on the i.i.d. sequence $Z_1,Z_2,...$, for testing that the ratio of mean to standard deviation in a normal population equals $\gamma_1$ agaist the alternative that it equals $\gamma_2$. Let P be the actual distribution of the $Z_i$ (not necessarily normal). It is proved that if $\gamma_1^2\neq\gamma_2^2$ and P is an arbitrary unbounded distribution, then there exist constants c > 0 and $\rho<1$ such that $P(N > n) < c\rho^n, n =1,2,\cdots$.

Let nS be an $m\times m$ matrix having the Wishart distribution $W_m(n,\Sigma)$. For large n and simple latent roots of $\Sigma$, it is known that the latent roots of S are asymptotically independently normal. In this paper an expansion, up to and including the terms of order $n^-1$, is given for the joint density function of the roots of S in terms of normal density functions. Expansions for the marginal distributions of the roots are also given, valid when the corresponding roots of $\Sigma$ are simple.

Prediction sufficiency (adequacy), as it is usually defined in terms of conditional expectations, does imply "real" prediction sufficiency; i.e. sufficiency in terms of risk functions. The converse holds provided we permit the loss to depend on the unknown parameter. This is no longer true if we insist on loss functions which do not involve the unknown parameter. Conditional independence still holds but ordinary sufficiency may fail. If, however, we require equivalence of risk functions, then ordinary sufficiency and, consequently, prediction sufficiency follows.

The rate at which the mean square error decreases as sample size increases is evaluated for general $L^1$ kernel estimates and for the Fourier integral estimate for a probability density function. The estimates are then compared on the basis of these rates.