Rank analogues of the integrated periodogram spectral process are introduced and used to generate distribution-free tests for independence of a set of random variables. Under simple autoregressive alternatives, the rank spectral process with normal scores yields a test of Kolmogorov-Smirnov type whose local asymptotic efficiency relative to the analogous test based on the integrated periodogram is at least one. Moreover, the same rank test has good local asymptotic efficiency relative to tests based on optimally lagged rank serial correlation coefficients.

Order of dependence is defined in a normally distributed two-way series. Under certain stationarity and symmetry conditions it is shown that when the extents of the series in both directions are large in comparison with the order of dependence, the joint density function reduces to a simple form with a small number of parameters after some adjustments. In this form a central role is played by the Kronecker products of some matrices having common eigenvectors. Maximum likelihood estimates of the parameters and likelihood ratio test criteria for certain hypotheses on order of dependence are derived.

In 1965 Marshall and Proschan (1965) (see also Grenander (1956)), considered the maximum likelihood estimation for life-distributions with monotone failure rate over the support of the distribution functions. They considered data arising from a testing plan which does not allow censoring, time-truncation or replacements. In the present paper we consider the maximum likelihood estimation of life-distributions with monotone failure rates over the interval $\lbrack 0, T)$, where $T$ is a fixed positive real number, and no other assumptions about the distribution or its failure rate are given outside that interval. The following renewal type testing plan is used, which allows for time-truncation and replacement. At time zero, the beginning of the testing, $n$ new items from the population to be tested are put on test. When an item fails it is instantaneously replaced with another new item from the same population and at time $T$ all testing is stopped. The maximum likelihood estimates of the distribution function and its failure rate over $\lbrack 0, T)$ are given and shown to be uniformly strongly consistent as $n$ tends to infinity.

For the one-sample independence problem, the one-sample symmetry problem, and the two-sample problem it is shown that every one-sided rank test is asymptotically optimal for a certain nonparametric subclass of contiguous alternatives, provided the test and the associated subclass of alternatives are generated by certain square-integrable functions defined on the unit square. Then the asymptotic normality of the respective rank statistics under every alternative contiguous to the hypothesis is used in order to give necessary and sufficient conditions for local asymptotic unbiasedness of such tests. Finally, for locally asymptotically unbiased tests there are given necessary and sufficient conditions for having bounds for their asymptotic relative efficiency under contiguous alternatives.

Let $\mathscr{P}_n$ be the space of $n \times n$ positive definite symmetric matrices. If $S_1$ and $S_2$ are random matrices in $\mathscr{P}_n, S_1$ is a better $\alpha$ denominator than $S_2$ (written $S_1 \prec_{(\alpha)} S_2$) $\operatorname{iff} U(x'S_1^{-1}x)^{\alpha/2} \ll_{st} U(x'S_2^{-1}x)^{\alpha/2}$ for all $x \in R^n$ where $U$ is uniform on [0, 1], independent of $S_1$ and $S_2, \alpha > 0$, and "$\ll_{st}$" means stochastically smaller than. A principal result is this. THEOREM. Let $S_1, \cdots, S_m$ be exchangeable random matrices in $\mathscr{P}_n$. If $0 < \alpha \leqq 2$, then $\sum^m_{i=1} \eta_i S_i \prec_{(\alpha)} \sum^m_{i=1} \psi_iS_i$ provided $(\psi_1, \cdots, \psi_m)$ majorizes $(\eta_1, \cdots, \eta_m)$. This has applications in establishing probability inequalities for certain common test statistics. The results in this paper extend those of Lawton. (Some inequalities for central and non-central distributions. Ann. Math. Statist. (1965) 36 1521-1525; Concentration of random quotients. Ann. Math. Statist. (1968) 39 466-480.)

Sequential probability ratio tests are defined for testing a simple hypothesis against a simple alternative for the mean value function of a real Gaussian process with known covariance kernel. Exact formulas are obtained for the error probabilities and the OC function using the fact that the $\log$ likelihood ratio process is a Gaussian process with independent increments and have continuous sample paths. An identity of a familiar nature holds for the expected value of the $\log$ likelihood ratio process at a random stopping time. In certain situations this identity yields an exact formula for the ASN function. Two examples are given. The analysis employs the theory of reproducing kernel Hilbert spaces.

Let $X_1, X_2, X_3, \cdots$ be independent random variables and $a_1, a_2, a_3, \cdots$ positive real numbers. Define $F(t) = \sup_k P\{|X_k| > t\}$ and $S_m = \sum^m_{k=1} a_k X_k.$ Inequalities of the form $P\{\sup_m|S_m| > \delta\} \leqq C \sum_k \int^1_0 \varphi'(u)F(u/a_k) du$ are given for a large class of functions $\varphi$, as well as inequalities of a somewhat different form that are appropriate for considering exponential convergence rates. Examples of how the inequalities can be used to prove rate theorems are also given.

A coin is tossed repeatedly at a fixed cost per toss. The payoff is the length of the terminal run of heads, less the cost of tossing. Properties of the dynamic programming solution are derived; the exact optimal policy and value of the game are obtained when the game has an infinite horizon, and the rate at which this solution is approached by the sequence of truncated strategies is analyzed numerically.

Finite state and action discrete time Markov decision processes with discounting are considered under the criterion of moment optimality. The case of small interest rates is studied, in particular the behavior of optimal policies as the interest rate approaches zero. Laurent expansions in the interest rate are developed for all moments of return for stationary policies, and a proof is given that there is a stationary policy which is moment optimal for all sufficiently small interest rates.

A necessary condition is given for the convergence of distributions of the sums of a random number of independent random variables. This is made on the basis of a theorem which gives sufficient conditions for the convergence of distributions of randomly stopped stochastic processes. The random indices are supposed to be independent of the sequence of summands.

Let $\rho(x, y)$ be a positive definite symmetric kernel defined over the unit square such that $\rho(x, y) = K(x, y) - \sum^k_{i=1} \psi_i(x)\psi_i (y), 0 \leqq x, y \leqq 1$, where $K(x, y)$ is a bounded symmetric positive definite kernel defined over the unit square, and $\psi_i(x) \in L_2(0, 1)$. Methods of finding Fredholm determinant $D(\lambda)$ of $\rho(x, y)$ in terms of the eigenvalues and the eigenfunctions of $K(x, y)$ are given. A kernel of the type of $\rho(x, y)$ arises as the covariance function of a Gaussian process in the limiting distribution of the modified Cramer-Smirnov test statistic in the $k$-parameter case which may be described as follows: Let $X_1, \cdots, X_n$ be $n$ independent observations (random variables) from a population with a continuous distribution function $G(x)$. Suppose for every $\theta = (\theta_1, \cdots, \theta_k) \in \mathbf{I,I}$ being an open interval in the $k$-dimensional Euclidean space $R^k, F(x, \theta)$ is a continuous distribution function. Let $\hat{\theta}_n$ be an estimate of $\theta$ obtained from the sample. A test of the hypothesis $H: G(x) = F(x, \theta)$ for some unspecified $\theta \in \mathbf{I}$ based on the statistic $C_n^2 = n \int^{+\infty}_{-\infty} \lbrack F_n(x) - F(x, \hat{\theta}_n) \rbrack^2 dF(x, \hat{\theta}_n),$ is considered and the characteristic function of the asymptotic distribution of $C_n^2$ is shown to be the Fredholm determinant of a kernel of the type of $\rho(x, y)$ with $K(x, y) = \min (x, y) - xy$ whose eigenvalues and eigenfunctions are known. Results are also used to obtain the limiting distribution of $l$-sample analogue of $C_n^2$.

A direct proof is given for an inequality relating the expected absolute value of stopped Brownian motion to the expected time to stopping. This inequality was originally proved by means of the martingale square function inequality. The latter is then derived from the former through use of a Skorokhod embedding. The first inequality is also applied to prove a martingale strong law of large numbers.

Several explicit expressions are presented for the minimum mean square error in linear causal filtering, prediction and interpolation of weakly stationary discrete-time processes corrupted by additive noise. A general procedure for deriving error expressions of this kind is established.

The subject of this paper is the study of the correspondence between Gaussian processes with paths in linear function spaces and Gaussian measures on function spaces. For the function spaces $C(I), C^n\lbrack a, b\rbrack, AC\lbrack a, b\rbrack$ and $L_2(T, \mathscr{A}, \nu)$ it is shown that if a Gaussian process has paths in these spaces then it induces a Gaussian measure on them, and, conversely, that every Gaussian measure on these spaces is induced by a Gaussian process with paths in these spaces.

Real-valued stochastic processes of the form $x(t) = \int A(t, \lambda)Z(d\lambda)$ are considered, where $Z(\lambda)$ is a zero mean Gaussian process with independent increments and $\int\int |A(t, \lambda)|^2F(d\lambda) dt < \infty$, where $F(d\lambda) = E|Z(d\lambda)|^2$. It is shown that the energy of $x(t), \int x^2(t) dt$, is a well-defined random variable and an exponential bound for $P(\int x^2(t) dt - E\int x^2(t) dt \geqq \varepsilon)$ is derived. This bound is used to obtain an exponential bound for crossing probabilities $P(|y(t)| > a$ for some $t)$ where $y(t) = \int h(t - \tau)x(\tau) d\tau, \int h^2(t) dt < \infty$.

An elementary proof of an existence theorem for infinite particle systems interacting through a finite range interaction is given. The results are then used to prove that the system preserves ergodicity of the initial measure.

Given a Hunt process $X(t)$, we investigate the consequences of the assumption that $\mathscr{G}(T+) = \sigma(X(T))$ for every finite stopping time $T$, where $\mathscr{G}(T+) = \bigcap_{\varepsilon > 0} \mathscr{F}^0\lbrack T, T + \varepsilon)$. Such processes constitute a simple extension of the right-continuous Markov chains without instantaneous states.

Two conservative sample size approximations are given for the Bechhofer formulation of the problem of selecting the population with the largest mean, when the populations have a common known variance. A table of numerical comparisons of these approximations with the exact sample size is included. In addition, both of these results are applied to the problem of selecting the factor with the largest mean from a single multivariate normal population with known covariance matrix.

A single-sample procedure is given for selecting the least probable event in a multinomial distribution. Given two numbers $(c, P^\ast)$, it is required to determine the smallest sample size for which the probability of a correct selection is at least as large as $P^\ast$ when the (unknown) difference between the smallest and the next smallest cell probabilities is greater than or equal to $c$. A table is given showing the required sample size for specified values of $c$ and $P^\ast$.

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.

Let $X_1, X_2, \cdots$ be independent identically distributed random variables taking on values in the positive integers with a family of possible probability distributions indexed by $G \in \mathscr{G}$, the class of all probability distribution functions on $\lbrack 0, + \infty)$. Under the assumption that the family is identifiable we wish to estimate the true but unknown $G_0$. This is done by constructing a prior probability distribution on $\mathscr{G}$ and showing that the Bayes estimate corresponding to the prior is consistent.

Let $u_n$ denote the sample correlation coefficient for $n$ observations from a bivariate, equal variance, complex Gaussian distribution. In this note we derive the exact distribution of $u_n$ by extending a method of Mehta and Gurland to the complex case. The asymptotic behavior of $E|u_n|^k$ as $n \rightarrow \infty$ is determined via the method of steepest descent. Applicability of the results to the analysis of certain estimators of spectral parameters of stationary time series is discussed.

Based on $k(\geqq 2)$ independent samples, a class of aligned rank order tests for the hypothesis of homogeneity of intercepts of the $k$ (simple) regression lines is considered here. The alignment procedure is similar to the one in Sen [Ann. Math. Statist. (1969) 19 1668-1683], and the theory is developed with the aid of the fundamental results of Jureckova [Ann. Math. Statist. (1969) 19 1889-1900] on the asymptotic linearity of rank statistics in regression parameters. Local asymptotic optimality of the proposed tests is also studied.

The paper deals with problems of rank tests of symmetry when samples are drawn from purely discrete distributions so that ties of zero and non-zero observations may occur. Zero observations are considered in the same way as nonzero ones. Two ways of treatment of ties are used in the paper, randomization of ties and the method of averaged scores. The asymptotic distributions of the statistics are derived under hypothesis of symmetry and under contiguous alternatives of location. The asymptotic power and efficiency of tests are established.

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.

This paper presents results obtained in efforts to generalize ideas reported during the past 15 years on the subject of variance components. The paper applies the result of Newcomb (1960) to the mixed effects model $Y = \sum^t_{j=1} X_{j\tau j} + \sum^h_{k=1} Z_kb_k$ where the $\tau_j$ are fixed parameters and the $b_k$ are random vectors distributed normally with covariance matrix $\sigma k^2I$. The restrictions made on the model are far less severe than those imposed by other authors. Through a nonsingular transformation of the data vector $Y$, minimal sufficient statistics are obtained. Theorems are presented which give conditions under which minimum variance unbiased estimates exist and these estimates are displayed. Properties of the model and the estimates are discussed in both the complete and noncomplete density cases. Perhaps the most important contribution relates to the simplicity with which the theoretical methods treat the general variance components situation.

In successive sampling on two occasions, the theory developed so far aims at providing the optimum estimate by combining (i) a double-sampling regression estimate from the matched portion of the sample and (ii) a simple mean based on a random sample from the unmatched portion of the sample on the second occasion. Theory has been generalized in the present note by using a double-sampling multivariate ratio estimate using $p$ auxiliary variates $(p \geqq 1)$ from the matched portion of the sample. Expressions for optimum matching fraction and of the combined estimate and its error have been derived and results are presented for some special cases which have practical applications.

A Cramer von-Mises type statistic is proposed for testing the symmetry of a continuous distribution function. Its asymptotic null distribution is found explicitly, and its asymptotic distribution under a sequence of local alternatives is described. A Monte Carlo study indicates that the asymptotic formulae are accurate for sample sizes as small as twenty.

There exist independent random variables $X_1$ and $X_2$ such that $X_1$ is symmetric, $X_2$ is not symmetric, but $X_1 + X_2$ is symmetric. If $X_1$ and $X_2$ are i.i.d. random variables with a fractional moment and if for all real $\alpha P\lbrack X_1 + \alpha X_2 > 0\rbrack = \frac{1}{2}$ then they are symmetric.

A distribution function $G(x)$, on the real line, is called unimodal if there exists a value $x = a$, such that $G(x)$ is convex for $x < a$ and concave for $x > a$. Given that $G(x)$ is unimodal, a condition is given for the unimodality of $G^r(x)$, where $r$ denotes a positive integer. $G^r(x)$ represents the distribution function of the largest observed value in a sample of $r$ observations from the distribution $G(x)$. Some of the standard distributions, such as, the normal, gamma, Poisson and binomial distributions satisfy the given condition. An application of the given result to a problem of estimating the largest parameter is given.

In this paper there are solved two problems raised by Bahadur (1954) which concern the relation between the existence of a minimal sufficient $\sigma$-field and the existence of a minimal sufficient statistic. Two examples show that the existence of a minimal sufficient $\sigma$-field is neither necessary nor sufficient for the existence of a minimal sufficient statistic.

In this note we show that the renewal function $H$ corresponding to a random walk with positive mean $\mu$ and finite variance $\sigma^2$ satisfies the inequality $H(x) < \mu^{-1} x + 3(1 + \mu^{-2}\sigma^2)$.

Let $a_n(i), b_n(i), i = 1, \cdots, n$, be $2n$ numbers defined for every $n$ and let $\bar{A}(k) = \sum^n_{i=1} |a_n(i)|^k$ and $\bar{B}(k) = \sum^n_{i=1}|b_n(i)|^k$. Let $(I_{n1}, \cdots, I_{nn})$ be a random permutation of $(1, \cdots, n)$ and let $S_n = \sum^n_{i=1} b_n(i)a_n(I_{ni})$. If $\bar{A}(k)/\lbrack\bar{A}(2)\rbrack^{\frac{1}{2}k} \rightarrow 0\quad \text{and}\quad \bar{B}(k)/\lbrack\bar{B}(2)\rbrack^{\frac{1}{2}k} \rightarrow 0.$ then it is known that the condition of Hoeffding, $n^{\frac{1}{2}k-1} \bar{A}(k)\bar{B}(k)/\lbrack\bar{A}(2) \bar{B}(2)\rbrack^{\frac{1}{2}k} \rightarrow 0,\quad k = 3,4, \cdots,$ is sufficient for the standardized moments of $S_n$ to tend to the moments of a standard normal variate. It is shown here that these conditions are also necessary. The relationship of these conditions to the Liapounov conditions is pointed out.

In this paper it is shown how martingale theorems can be used to appreciably widen the scope of classical inferential results concerning autocorrelations in time series analysis. The object of study is a process which is basically the second-order stationary purely non-deterministic process and contains, in particular, the mixed autoregressive and moving average process. We obtain a strong law and a central limit theorem for the autocorrelations of this process under very general conditions. These results show in particular that, subject to mild regularity conditions, the classical theory of inference for the process in question goes through if the best linear predictor is the best predictor (both in the least squares sense).

Suppose that $G$ is the distribution function $(\operatorname{df})$ of a (non-negative) $\mathrm{rv} Z$ satisfying the integral-functional equation $G(x) = b^{-1} \int^{lx}_0 \lbrack 1 - G(u)\rbrack du$, for $x > 0$, and zero for $x \leqq 0$, with $l \geqq 1$. Such a $\operatorname{df} G$ arises as the limit $\operatorname{df}$ of a sequence of iterated transformations of an arbitrary $\operatorname{df}$ of a $\operatorname{rv}$ having finite moments of all orders. When $l = 1, G$ must be the simple exponential $\operatorname{df}$ and is unique. It is shown, for $l > 1$, that there exists an infinite number of $\operatorname{df}$'s satisfying this equation. Using the fact that any $\operatorname{df} G$ which satisfies the given equation must have finite moments $\nu_k = K! b^kl^{k(k-1)/2}$ for $k = 0, 1, 2, \cdots$, it is shown that the $\operatorname{df}$ of the $\operatorname{rv} Z = UV$, where $U$ and $V$ are independent rv's having log-normal and simple exponential distributions, respectively, satisfies the integral functional equation. It is then easy to exhibit explicitly a family of solutions of the equation.

A theorem of de Finetti states that if odds are posted on each set in a finite partition of a probability space, then either the odds are consistent with a finitely additive probability measure or a sure win is possible. A generalization of this result is proved which in turn implies a generalization of Von Neumann's theorem on the existence of the value of a game. Also, two horse race examples are considered.

In denumerable state, denumerable action sequential decision problems in which the reward function has uniformly bounded 2nd moment, the optimal reward for the decisionmaker who restricts himself to the countable set of stationary policies consisting of those which choose some arbitrary action at all but a finite number of states will be the same as the optimal reward for the decisionmaker who optimizes over all stationary policies. Under some further restriction, he can do almost as well simply by solving a large finite state truncation of the original problem.

Some properties of the multiplicity of a class of multivariate random processes are examined with particular emphasis on the derivation of a necessary and sufficient condition in order that the multiplicity of such a process be equal to one. Discussion is limited to processes with mutually orthogonal univariate components, but possible applications of the results to more general situations are suggested.

Let $T_r$ be the time of first passage to the level $r > 0$ by a random walk with independent and identically distributed steps and mean $\nu \geqq 0$. Estimates are given for the rate at which the distribution of $T_r$, suitably scaled and normalized, converges to the stable distribution with index $\frac{1}{2}$ when $\nu = 0$ and to the normal distribution when $\nu > 0$ as $r \rightarrow \infty$.

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.

It is shown that the number of cells alive at time $t$, denoted by $Z_0(t)$, of a critical age dependent branching process with immigration satisfies $t^{-1}Z_0(t)$ approaches a gamma law with specified constants in distribution.

In this paper the author considers the two methods that Feller discusses in [3] and [4] which find a sequence $b_n$ so that $\lim \sup S_n(b_ns_n)^{-1} = 1$ a.s. where $S_n = \sum^n_{i=1}X_i$ and $X_i$ are independent random variables with $EX = 0, EX^2 < \infty$ and $E\lbrack\exp(hX_i)\rbrack < \infty$ for all $h < 0$. The more elementary and general method, which is not developed by Feller in [3], is used in a most elementary manner to derive a theorem general enough to include: $(l(n) \equiv (2lnlns_n)^{\frac{1}{2}})$. (A) Kolmogorov's classical law of the iterated logarithm and the result of Egorov [2]: $X_i$'s bounded and $\sup(X_i)l(n)s_n^{-1} = O(1)$ implies $0 < \lim \sup S_n(l(n)s_n)^{-1} < \infty$. (B) A slightly different version of a result of Feller [3]: $X_i$ bounded above, $\sup (X_i)l(n)/s_n = O(1)$ and two other conditions then $0 < \lim \sup S_n(l(n)s_n)^{-1} < \infty$ (the "slightly different version" is to replace one of the "two other conditions" with a different condition). (C) A generalization of a Thompson [5]: $X_i = a_i Y_i$, where $Y_i$'s are identically distributed with common negative exponential distribution, then $a_il(n)/s_n = O(1)$ implies $\lim \sup S_n(s_nl(n))^{-1} = 1$ (the generalization is to require only that $Y_i$'s be identically distributed with $E\lbrack\exp(hY_i)\rbrack < \infty$ for all $h > 0$). Also under these conditions the theorem includes: $a_1l(n)/S_n = O(1)\quad \text{implies}\quad 0 < \lim \sup S_n(s_nl(n))^{-1} < \infty.$