This is a selective review on robust statistics, centering on estimates of location, but extending into other estimation and testing problems. After some historical remarks, several possible concepts of robustness are critically reviewed. Three important classes of estimates are singled out and some basic heuristic tools for assessing properties of robust estimates (or test statistics) are discussed: influence curve, jackknifing. Then we give some asymptotic and finite sample minimax results for estimation and testing. The material is complemented by miscellaneous remarks on: computational aspects; other estimates; scale, regression, time series and other estimation problems; some tentative practical recommendations.

Huber, in his fundamental paper [1] and in [2], has considered the robust estimation of a location parameter and has obtained results which he applied to some examples including the $\varepsilon$-normal model, $\{F|\sup_x|F(x) - \Phi(x)\mid \leqq \varepsilon, F \text{symmetric}\}$, when $\varepsilon$ is sufficiently small $(\varepsilon \leqq \varepsilon_0 \sim .03)$. In this note we show how his methods work for the family of distributions $\{F \mid \int^A_{-A} dF \geqq p, F \text{symmetric}\}$ and then use this to solve the $\varepsilon$-normal problem when $\varepsilon > \varepsilon_0$.

The limiting behavior of the one-dimensional RM procedure is investigated when the prescribed $x$-levels of sequential experiments can be realized when random errors only. The errors of constant variance and of variance decreasing to zero are studied in Sections 2 and 3, respectively.

The proof of the existence of the optimal stopping rule for $S_n/n$ for the case where the i.i.d. random variables $X_i$ have a moment of order greater than one has been obtained by B. Davis. In the present paper the asymptotic growth of the boundary of the optimal stopping region is studied. The method used generalizes one of Shepp (1969), and involves comparison with the corresponding problem for an infinitely divisible process, obtained as a limit of processes $(S_{\lbrack nt\rbrack}/a_n, t \geqq 0)$ for properly chosen norming constants $a_n$. When the $X_i$ are in the domain of attraction of a random variable which is stable with exponent greater than one, an explicit asymptotic expression for the curve defining the boundary is obtained.

Asymptotic normality of linear rank statistics for testing the hypothesis of independence is established under fixed alternatives. A generalization of a result of Bhuchongkul [1] is obtained both with respect to the conditions concerning the orders of magnitude of the score functions and with respect to the smoothness conditions on these functions.

A $U$-statistic $J_n$ is proposed for testing the hypothesis $H_0$ that a new item has stochastically the same life length as a used item of any age (i.e., the life distribution $F$ is exponential), against the alternative hypothesis $H_1$ that a new item has stochastically greater life length $(\bar{F}(x)\bar{F}(y) \geqq \bar{F}(x + y)$, for all $x \geqq 0, y \geqq 0$, where $\bar{F} = 1 - F). J_n$ is unbiased; in fact, under a partial ordering of $H_1$ distributions, $J_n$ is ordered stochastically in the same way. Consistency against $H_1$ alternatives is shown, and asymptotic relative efficiencies are computed. Small sample null tail probabilities are derived, and critical values are tabulated to permit application of the test.

A sequential procedure for estimating the mean of an exponential distribution is proposed. It is shown to perform well for large values of the mean, and the results of a Monte Carlo study indicate that it also performs well for moderate values of the mean.

We consider, simultaneously, $N$ statistical decision problems with identical generic structure: state space $\Omega$, action space $A$, sample space $\mathscr{X}$ and nonnegative loss function $L$ defined on $\Omega \times A \times \mathscr{X}$. With $x = (x_1, \cdots, x_N)$ distributed according to $\prod^N_{i=1} \mathbf{P}_{\theta_i} = \mathbf{P}_\theta$, a compound procedure is a vector, $\mathbf{\phi} = (\phi_1, \cdots, \phi_N)$, such that $\phi_i(\mathbf{x}) \in A$ for each $i$ and $\mathbf{x}$. The risk of the procedure $\mathbf{\phi}$ is $\mathbf{R}(\mathbf{\theta}, \mathbf{\phi}) = N^{-1} \sum^N_{r=1} \mathbf{\int L} (\theta_r, \phi_r(\mathbf{x}), x_r)\mathbf{P_\theta} (d\mathbf{x})$ and the modified regret is $\mathbf{D}(\mathbf{\theta, \phi}) = \mathbf{R}(\mathbf{\theta, \phi}) - R(G)$ where $G$ is the empirical distribution of $\theta_1, \cdots, \theta_N$, and $R(G)$ is the Bayes risk against $G$ in the component problem. We discuss quite wide classes of procedures, $\mathbf{\phi}$, which consist of using $\mathbf{x}$ to estimate $G$, and then playing $\varepsilon$-Bayes against the estimate in each component problem. For one class we establish a type of uniform convergence of the conditional risk in the $m \times n$ problem (i.e. $\Omega$ has $m$ elements, $A$ has $n$), and use this to get $\mathbf{D}(\mathbf{\theta, \phi}) < \varepsilon + o(1)$ for another class in the $m \times n$ and $m \times \infty$ problems. Similar, but weaker, results are given in part II for the case when $\Omega$ is infinite.

The sequence of partial maxima for i.i.d. random variables is considered. Two theorems concerning the sample behaviour of the maxima are proved. Also a large deviations result is given, connected with weak convergence to the double exponential distribution.

Let $\{X_t\}$ be a $p$-dimensional stationary autoregressive series. The main result is the determination of the autoregressive matrices of the series which is reversed in time with respect to $\{X_t\}$. The series which is reversed with respect to itself is called symmetric. The conditions for the symmetry of $\{X_t\}$ are given in the paper. The inverse of the covariance matrix is evaluated for the finite part of the symmetric autoregressive series.

Rates of convergence for linear combinations of order statistics are obtained. The work is in the spirit of those authors who have used in one form or another the weak convergence of the sample empirical process to a tied-down Wiener process, except that the Skorokhod embedding is explicitly used to obtain a rate of convergence via control on the tail-behavior of the stopping times. The paper concludes with a remark on the limitations of the technique as far as getting the best possible rate is concerned.

Let $\{X_t; t \geqq 0\}$ be a stochastic process in $R^N$ defined on the probability space $(\Omega, \mathscr{F}, \mathbf{P})$ which has stationary independent increments. Let $\nu$ be the Levy measure for $X_t$ and let $\beta = \inf\{\alpha > 0: \int_{|x| < 1}|x|^\alpha\nu(dx) < \infty\}$. For each $\omega \in \Omega$, let $V_\gamma(\mathbf{X}(\bullet, \omega); a, b) = \sup \sum^m_{j=1} |X(t_j, \omega) - X(_{t-1}, \omega)|^\gamma$ where the supremum is over all finite subdivisions $a = t_0 < t_1 < \cdots < t_m = b$. Then if $\gamma > \beta, \mathbf{P}\{\mathbf{V}_\gamma(\mathbf{X}(\bullet, \omega); a, b) < \infty\} = 1$.

Taking as our point of departure the methods and results in a 1960 paper of Bahadur and Ranga Rao, we derive asymptotic representations of large deviation probabilities for weighted sums of independent, identically distributed random variables. The main theorem generalizes the Bahadur-Ranga Rao result in the absolutely continuous case. The method of proof closely parallels that of the 1960 paper, a major component of which was the use of Cramer's 1923 theorem on asymptotic expansions. For our result, we need an extension of Cramer's theorem to triangular arrays, and that extension is also developed in the paper. We then show that the main theorem implies a logarithmic result which generalizes a 1952 theorem of Chernoff and is of more precision but less generality than a 1969 result of Feller. Finally, we note that in the exponential case the theorem can be used to estimate large deviation probabilities for linear combinations of exponential order statistics.

Let $\{X(s, t; \omega): (s, t) \in \lbrack 0, \infty) \times \lbrack 0, \infty)\}$ be a two parameter Gaussian process with mean function zero and covariance function $R(s_1, t_1; s_2, t_2) = \min (s_1, s_2) \min (t_1, t_2)$. This paper derives a multiparameter law of the iterated logarithm and modulus of continuity for the process $X(s, t; \omega)$. Estimates are also given which enable the author to define an Ito type integral for a suitable class of functions and to solve a diffusion equation involving the process.

Let $X(t), t \geqq 0$, be a centered separable Gaussian process with stationary increments. Put $\sigma^2(t) = EX^2(t)$, and suppose that $\sigma(0) = 0$. Let $Y(t)$ be the normalized process $X(t)/\sigma(t)$, and $B$ an arbitrary bounded closed subinterval of the positive real axis. Under general conditions on $\sigma$ we find (1) an explicit asymptotic formula for $P\{\max_B Y > u\}$ for $u \rightarrow \infty$ in terms of $\sigma$ and various functions derived from it, and (2) the limiting conditional distribution of the time spent above the level $u$ (for $u \rightarrow \infty$) given that the time spent is positive. This limiting distribution is a scale mixture of the corresponding distribution previously obtained under comparable conditions in the case of the stationary Gaussian process.

In this paper weak laws of large numbers are proved for random elements (function-valued random variables) in separable normed linear spaces. One result states that for identically distributed random elements $\{V_n\}$ such that the Pettis integral $EV_1$ exists and $E\|V_1\| < \infty$ $\|n^{-1} \sum^n_{k=1} V_k - EV_1\| \rightarrow 0 \text{in probability}$ if and only if $|n^{-1} \sum^n_{k=1} f(V_k) - Ef(V_1)| \rightarrow 0 \text{in probability}$ for each continuous linear functional $f$. The condition of identically distributed random elements $\{V_n\}$ can not be relaxed by just assuming a bound on the moments of $\{\|V_n\|\}$, but a weak law of large numbers is obtained for random elements which need not be identically distributed. Both of these weak laws can also be obtained by assuming only that the space has a Schauder basis such that the weak law of large numbers holds in each coordinate. An application of these results yields a uniform weak law of large numbers for separable Wiener processes on [0, 1].

Feller (1966) claims that a regenerative stochastic process with a non-lattice interarrival-time distribution has a limiting distribution. This is true if (i) the class of interarrival-time distributions is restricted, or (ii) regularity conditions are imposed on the sample paths of the process.

Let $(\Omega, \mathscr{F}, P)$ be a probability space, $\{X_n: n \geqq 1\}$ an inhomogeneous Markov chain assuming a finite number of states defined on this space, $E = \{a_1, \cdots, a_s\}$ the set of its states, $p_j^{(n)} = P\{X_n = a_j\}, p^{(k, n)}_{ij} = P\{X_n = a_j\mid X_k = a_i\}$ for $n = 2,3, \cdots, n > k, a_i, a_j \in E, \{f_n: n \geqq 1\}$ a sequence of real valued functions defined on $E$ and $S_n = f_1(X_1) + \cdots + f_n(X_n)$. To study the Markov chains which are not subjected to "asymptotic independent" restrictions, the author proposes the coefficients $a_{k,n} = \max'_{i\in\{1, \cdots, s\}} \sum^s_{j=1} (pj^{(n)} - p^{(k,n)}_{ij})^+ (n = 2,3, \cdots, n > k)$ where the dash indicates that the max is taken over those $i$ such that $p_i^{(k)} > 0$. Some limit properties of the sums $\{S_n: n \geqq 1\}$ suitably normed, as the behavior of the series of random variables and the strong law of large numbers are investigated. In the end some examples are given and it is proved that the arbitrary homogeneous Markov chains satisfy most of the conditions imposed in the paper.

A stopping time $T$ for the Wiener process $W(t)$ is called minimal if there is no stopping time $S \leqq T$ such that $W(S)$ and $W(T)$ have the same distribution. In the first section, it is shown that if $E\{W(T)\} = 0$, then $T$ is minimal if and only if the process $W(t \wedge T)$ is uniformly integrable. Also, if $T$ is minimal and $E\{W(T)\} = 0$ then $E\{T\} = E\{W(T)^2\}$. In the second section, these ideas are used to show that for any right continuous martingale $M(t)$, there is a right continuous family of minimal stopping times $T(t)$ such that $W(T(t))$ has the same finite joint distributions as $M(t)$. In the last section it is shown that if $T$ is defined in the manner proposed by Skorokhod (and therefore minimal) such that $W(T)$ has a stable distribution of index $\alpha > 1$ then $T$ is in the domain of attraction of a stable distribution of index $\alpha/2$.

The locally best invariant test for testing sphericity in a multivariate normal distribution is obtained. It is different from the likelihood ratio criterion. Some other known results are remarked. The limiting distributions of the test statistics are investigated. In particular Pitman efficiency of the likelihood ratio test with respect to the locally best invariant test is shown to be one for normal alternatives.

Consider $p \geqq 2$ random variables, and let $A_1, \cdots, A_p$ denote the hyperplanes corresponding to the linear regression of each variable onto the other $(p - 1)$ variables. Let $A_0$ denote the hyperplane which passes through the centroid of the distribution and is spanned by the direction vectors defining the first $(p - 1)$ principal components. A new optimality property of $A_0$ is established; $A_0$ is the best single approximation to $A_1, \cdots, A_p$ when each regression hyperplane is given a certain weighting inversely proportional to the variability associated with its orientation and its prediction rescaling. When $p > 2$ and $k = 1, \cdots, p - 2$, certain $k$-dimensional linear subspaces of $A_0$ are also shown to have regression optimality properties.

$\{P_\theta\}$ is a set of probabilities on a countable set $_\chi$ such that $P_\theta(x) > 0$ for each $x$ and $\theta$. We prove that if $\{P_\theta\}$ is not an exponential family, then each sufficient statistic for $n$ independent observations must be one-to-one, modulo permutations, on an infinite product set (which does not depend on the sufficient statistic).

A condition on the sampling design which previously [2] was shown to be a sufficient condition for the Horvitz-Thompson estimator to be the unique hyperadmissible estimator of the population total is now shown to be a necessary condition also save for trivial exceptions. The exceptions are the sampling designs in which the only samples with positive probability are such that they include the whole population.

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.

In this paper a class of multidimensional experimental designs said to have partial factorial balance is introduced. These designs are shown to belong to the more general class of multidimensional partially balanced designs. The analysis of designs with partial factorial balance is given in detail and several series of three, four and five dimensional designs are presented.

Using two distinct BIB designs with the same number of treatments, a method of construction of balanced binary and ternary designs with unequal block sizes and unequal replications is given. The method is a generalization of the method of construction of balanced designs, given by John [6].

The authors previously presented a randomized procedure for nonorthogonal saturated main effect fractional replicates in an $s^n$-factorial and presented an unbiased estimator of the main effect parameter vector. However, the explicit expression of the variance of the estimator remained an unsolved problem. In this paper our attention is restricted to a $2^n$-factorial, and the randomized procedure is extended to any preassigned parameters in a $2^n$-factorial system. An explicit expression of the variances of unbiased estimators of the parameters is presented. Also, in a $2^n$-factorial, some invariant properties of the information matrices and variances of the estimators in the randomized fractional replicates and a semi-invariant property of alias schemes of the fractional replicates are obtained.

There exist well-known necessary and sufficient conditions for a distribution function to belong to the domain of attraction of the double exponential distribution $\Lambda$. For practical purposes a simple sufficient condition due to von Mises is very useful. It is shown that each distribution function $F$ in the domain of attraction of $\Lambda$ is tail equivalent to some distribution function satisfying von Mises' condition.

The results of Dwass (1964) [Ann. Math. Statist. 35 1718-1725] and Lamperti (1964) [Ann. Math. Statist. 35 1726-1737] on the weak convergence of extremal processes (in the Skorokhod $J_1$-topology) to appropriate Markov processes are extended here for random sample sizes.

We determine $\sup E\lbrack r(S_T)\rbrack$, where $S_n$ is a sequence of partial sums of independent identically distributed random variables, for two reward functions: $r(x) = x^+$ and $r(x) = (e^x - 1)^+$. The supremum is taken over all stop rules $T$. We give conditions under which the optimal expected return is finite. Under these conditions, optimal stopping times exist, and we determine them. The problem has an interpretation in an action timing problem in finance.

The purpose of this note is to show that the usual metric on the set of complete $\operatorname{sub} \sigma$-algebras of a probability space is very tight; a recent result from E. B. Boylan on equi-convergence of martingales follows and is thereby, we believe, better understood.

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 $P_n$.

Examples are provided of Markovian martingales that: (i) converge in distribution but fail to converge in probability; (ii) converge in probability but fail to converge almost surely. This stands in sharp contrast to the behavior of series with independent increments, and settles, in the negative, a question raised by Loeve in 1964. Subsequently, it is proved that a discrete, real-valued Markov-chain with stationary transition probabilities, which is at the same time a martingale, converges almost surely if it converges in distribution, provided the limiting measure has a mean. This fact does not extend to non-discrete processes.