A definition is given for influence functionals of parameter estimates in time-series models. The definition involves the use of a contaminated observations process of the form $y^\gamma_t=(1-z^\gamma_t){x_t+z^\gamma_tw_t}$, $p=1,2,...,0\leq\gamma\leq1$ where $x_t$ is a core process (usually Gaussian), $w_t$ is a contaminating process, and $z^\gamma_t"$ is a zero-one process with $P(z^\gamma_t=1)={\gamma+0(\gamma)}$. This form is sufficiently general to model such diverse contamination types as isolated outliers and patches of outliers. Let $T(\mu^\gamma_y)$ denote the functional representation of a given estimate, where the measures $\mu^\gamma_y, 0\leq\gamma\leq1$ for $y^\gamma_t$ are in an appropriate subset of the family of stationary and ergodic measures on $(R^\infty,\beta^\infty)$. The influence functional IF is a derivative of T along "arcs" traced by $\mu^\gamma_y$ as $\gamma\rigtharrow0$, and correspondingly $\mu^\gamma_y\rigtharrow\gamma_x$. Although this influence functional is similar in spirit to Hampel's influence curve ICH for the i. i.d. setting, it is not the same as ICH. However, a simple relationship between the IF and the ICH is established. Results are given which aid in the computation of IF and insure that IF is bounded. We compute the IF for some robust estimates of the first-order autoregressive and first-order moving average parameters using various contamination processes. A definition of gross-error sensitivity (GES) for the IF is given, and some estimates are compared in terms of their GES's. Also the IF is used to show that a class of generalized RA estimates has a certain optimality property. Finally, some possible generalizations of the IF are indicated.

In the differential geometric approach to parametric statistics, developed by Chentsov, Efron, Amari, and others, the parameter space is set up as a differentiable manifold with expected information as metric tensor and with a family of affine connections, the a-connections, determined from the expected information and the skewness tensor of the score vector. The usefulness of this approach is particularly notable in connection with Edgeworth expansions of estimators. Motivated by the conditionality viewpoint, an "observed" parallel to that theory is established in the present paper using obseved information and an "observed skewness" tensor instead of the above expected quantities. The formula $c\|\overset{hat}{j}\|^{1/2}\overset{-}{L}$ for the conditional distribution of the maximum likelihood estimator is expanded (to third order) asymptotically and the "observed geometries" are shown to have a role in this type of expansion similar to that of the "expected geometries" in the Edgeworth expansions mentioned above. In these new developments "mixed derivatives of the log model function," defined by means of an auxiliary statistic complementing the maximum likelihood estimator, take the place of moments of derivatives of the log likelihood function.

Rectangular lattice designs are shown to be generally balanced with respect to a particular decomposition of the treatment space. Efficiency factors are calculated, and the analysis, including recovery of interblock information, is outlined. The ideas are extended to rectangular lattice designs with an extra blocking factor.

This paper gives a method for constructing conservative Scheffe-type simultaneous confidence bands for curvilinear regression functions over finite intervals. The method is based on the use of a geometric inequality giving an upper bound for the uniform measure of the set of points within a given distance from y, an arbitrary piecewise differentiable path with finite length in $S^{k-1}$, the unit sphere in $R^k$. The upper bound is obtained by "straightening" the path so that it lies in a great circle in $S^{k-1}$.

Suppose $u_1,\cdots,u_n$ are fixed points on the sphere, $v_1,\cdots,v_n$ are random points such that the distribution of $v_i$ depends only upon $v^t_iAu_i$ for some unknown rotation A. This paper provides asymptotic tests and confidence regions for A and for the axis of rotation of A. Results are given in arbitrary dimension.

Consider a supercritical Galton-Watson process $(Z_n)$ with offspring distribution a member of the power series family, and having unknown mean $\theta$. The conditional asymptotic normality of the suitably normalized maximum likelihood estimator of $\theta$ given the conditional information is established. The conditional information here is proportional to the total number of ancestors $V_n$, and it is also seen that this statistic is asymptotically ancillary for $\theta$ in a local sense. The proofs are via a detailed analysis of the joint characteristic function of $(Z_n, V_n)$, and the derivation serves to highlight the difficulties involved in establishing such conditional results generally.

It is desired to estimate a real valued function F on the unit square having observed F with error at N points in the square. F is assumed to be drawn from a particular Gaussian process and measured with independent Gaussian errors. The proposed estimate is the Bayes estimate of F given the data. The roughness penalty corresponding to the prior is derived and it is shown how the Bayesian technique can be regarded as a generalisation of variance components analysis. The proposed estimate is shown to be consistent in the sense that the expected squared error averaged over the data points converges to zero as $N\rightarrow\infty$. Upper bounds on the order of magnitude of magnitude of the expected average squared error are calculated. The proposed technique is compared with existing spline techniques in a simulation study. Generalisations to higher dimensions are discussed.

The risk in a trial to compare two medical treatments is borne by the patients who receive the inferior treatment during the experimental phase and by those remaining after the experiment who will all receive the inferior treatment if the results are misleading. The Bayes rule indicates, for the observed progression of successes and failures, when it is optimal to stop this experimental phase. This stopping rule can be described exactly, or nearly so, for symmetric two-point priots. Less precise descriptions are possible for other types of priors. An admissible stopping rule is described which is best possible, among symmetric Bayes rules, in that it minimizes the probability of choosing the inferior treatment no matter what the values are for the probabilities of success.

Consider finite sequences $X_1, X_2,\cdots, X_m$ and $Y_1, Y_2,\cdots, Y_n$ where the letters ${X_i}$ and ${Y_i}$ are chosen i.i.d. on a countable alphabet with $p=P(X_1=Y_1)\in(0,1)$ We study the distribution of the longest contiguous run of matches between the X's and Y's allowing at most k mismatches. The distribution is closely approximated by that of the maximum of (1 - p)mn i.i.d. negative binomial random variables. The latter distribution is in turn shown to behave like the integer part of an extreme value distribution. The expectation is approximately $\log(qmn)+k\log\log(qmn)+k\log(q/p)-\log(k!)+\gamma\log(e)-\frac{1}{2}$, where q = 1 - p, log denotes logarithm base 1/p, and y is the Euler constant. The variance is approximated by $(\pi\log(e))2/6+\frac{1}{2}$. The paper concludes with an example in which we compare segments taken from the DNA sequence of the bacteriophage lambda.

Recent interest in skewness has tended to separate two aspects of the concept. Two distributions may be compared with respect to skewness, or a distribution may be self-compared, that is, the distributions of the random variables of X and - X may be compared. This paper uses the unification of these two aspects to attemp to complete a skewness structure of orderings that identifies the roles of various skewness and scale measures and enables classification of the skewness properties of any distribution. The structure is also used to propose measures of asymmetry. Some skewness properties of the Weibull and Johnson systems are examined.

Let $X_1,X_2,\cdots$ be iid with density $f_y$ with respect to a sigma finite measure $\mu$ where ${f_y}_(y\in\omega$, $\omega\subseteqR$ is an exponential family. Let F be a probability measure on $\omega$ and let $\theta_0\in\omega$. Define $T(B,F)=\min \big\{n \left| \int_omega \frac{f_y(X_1) \1dots f_y(X_n)} {f_{\theta_0}(X_1)\dots f_{\theta_0 (X_1) \1dots f_{\theta_0} (X_n)} dF(y)\geq B \big\}$, $T (B,F) =\infty$ if no such n exists. Previous studies have found that if F has a positive and continuous density with respect to Lebesgue measure on $\omega$, then $BP_\theta\0(t(B,F)<\infty)\rigtharrow_{B\rigtharrow\infty}\int_\omega\ int^\infty_0\exp\{-x\}dH_\theta(x)dF(\theta)$, where $H_\theta$ are certain measures arising in a renewal-theoretic context. Here we show that in a nonlattice context, this convergence holds for general probability measures F. We also show that the convergence is uniform for all probability measures F whose support is contained in an arbitrary interval [a,b] interior to $\omega$, if the distribution of $X_1$ is strongly nonlattice for all $y\in\omega$.

The problem of determining Bayes tests of power one (without an indifference zone) is considered for Brownian motion with unknown drift. When we let the unit sampling cost depend on the underlying parameter in a natural way, it turns out that a simple Bayes rule is approaximately optimal. Such a rule stops sampling when the posterior probability of the hypothesis is too small.

Asymptotic expansions are derived for a class of averages of the coverage probabilities for some sequential confidence bounds, when the data consist of i.i.d. observations from a one-parameter exponential family. These expansions show the effect of the optional stopping on the coverage probabilities quite clearly and provide a method for changing the confidence limits to reduce this effect.

In the present paper we will improve the results concerning the rate of convergence of the error of second kind of the Neyman-Pearson test if the Kullback-Leibler information $K(P_0,P_1)$ is infinite. It is pointed out that in certain cases the sequence $exp(-q_\infty,n)$ is the correct rate of convergence if $-q_\infty,n$ denotes the logarithm of the critical value of the Neyman-Pearson test of level and sample size n. Therefore we generalize the classical results of Stein, Chernoff, and Rao which deal with the error probability of second kind and state that $q_\infty,n\tilde nK(P_0,P_1)$ if the Kullback-Leibler information is finite. Moreover the relation between $q_\infty,n$ and the local behavior of the Laplace transform of the log-likelihood distribution with respect to the hypothesis is studied. The results can be applied to one-sided test problems for exponential families if the hypothesis consists of a single point. In this case it may happen that $q_\infty,n$ is of the order $n^{1/p}$ for some p, 0<p<1.

As a modification of the notion of algorithmic complexity, the stochastic complexity of a string of data, relative to a class of probabilistic models, is defined to be the fewest number of binary digits with which the data can be encoded by taking advantage of the selected models. The computation of the stochastic complexity produces a model, which may be taken to incorporate all the statistical information in the data that can be extracted with the chosen model class. This model, for example, allows for optimal prediction, and its parameters are optimized both in their values and their number. A fundamental theorem is proved which gives a lower bound for the code length and, therefore, for prediction errors as well. Finally, the notions of "prior information" and the "useful information" in the data are defined in a new way, and a related construct gives a universal test statistic for hypothesis testing.

The asymptotic optimality of Mallows' $C_L$ and generalized cross-validation is demonstrated in the setting of ridge regression. An application is made to spline smoothing in nonparametric regression. A counterexample is given to help understand why sometimes GCV may not be asymptotically optimal. The coefficient of variation for the eigenvalues of the information matrix must be large in order to guarantee the optimality of GCV. The proff is based on the connection between GCV and Stein's unbiased risk estimate.

We construct nonparametric estimators of the distribution function F and its hazard function in the class of all increasing failure rate (IFR) distribution. Denoting the empirical distribution and empirical hazard function by $F_n$ and $H_n$ , respectively, let $C_n$ be the greatest convex minorant of $H_n$, and $G_n$ the distribution with hazard function $C_n$. The estimator $G_n$ is itself IFR. We prove that under suitable restrictions on F, and for any fixed $\lambda$ with $F(\lambda)<1$, $sup_{x\leq\lambda {n^{1/2}}}|C_n(x)-H_n(x)|$ and $sup_{x\leq\lambda{n^{1/2}}}|G_n(x)-F_n(x)|$ both tend to zero in probability. This means that $G_n$ and $F_n$ are asymptotically $n^{1/2}$ equivalent. It follows from Millar (1979) that $F_n$ is asymptoticallyminimax among the class of all IFR distributions for a large class of loss functions. This property extends to our estimator $G_n)$ under some restrictions.

This paper deals with the problem of efficiently estimating (asymptotically minimax) a distribution function when essentially nothing is known about it except that it is unimodal. The sample distribution function $F_n$ is shown to be asymptotically minimax among the family $\mathscr{E}$ of all unimodal distribution functions. Since $F_n$ does not belong to this family, estimators belonging to this family are constructed and are shown to be asymptotically minimax relative to the collection of subfamilies of $\mathscr{E}$.

A general method for the construction of asymptotically efficient estimates in semiparametric models is presented. It improves and modifies Bickel's (1982) construction of adaptive estimates and obtains asymptotically efficient estimates under conditions weaker than those in Bickel.

Consider a regression model $Y_i=x'_i\beta+R_i$, i=1,\cdots,n, where ${R_i}$ are i.i.d. with c.d.f., F; $x_i\inR^p$ and $\beta\inR^p$. Let $\hat{\beta}$ be a M-estimator defined using kernel, $\psi$; let $\hat{F}_n(x)$ denote the empiric distribution of the residuals, $Y_i-x'_i\hat{\beta}$ and let $\hat{F}^\ast_n$ be the empiric c.d.f. of the errors, ${R_i}$. Under suitable smoothness conditions on $\psi$, F, and the density F' = f and conditions requiring essentially that ${x_i}$ behave like a random sample from some distribution in $R_p$, it is show that, for fixed x, $\sqrt n (\hat{F}_n(x) - \hat{F}^\ast_n(x)-H_n(x))-\frac{P}{\sqrt{n}} g(x) \rightarrow_p 0$, where $g(x) = af(x)\psi(x) + bf'(x)$ and $H_n(x) = (1/nd)f(x)\sum^n_{i=1}\psi(R_i)$ if the design has a constant term [and $H_n(x)$ vanishes otherwise]. A tightness result shows that if $p/\sqrt n\rigtharrow c$ $\sqrt n(\hat{F}_n(x)-F(x))$ converges weakly to a Gaussian process with drift given by the bias term cg(x), and covariance function strongly affected by $H_n(x)$ and different from that for the usual Brownian bridge. In the course of the proof, an expansion for the fitted values, $x'_i\hat{\beta}$, is obtained, with error $O_p(p^{11/4}ln^2n/n^2)=o_p (1/\sqrt{n}) if p^2/n$ is bounded.

Let $Z_i:-\infty<i <+\infty}$ be a strictly stationary $\infty$ -mixing sequence. Without specifying the dependence model giving rise to ${Z_i}$ and without specifying the marginal distribution of $Z_i$ , we address the question of variance estimation for a general statistic $t_n=t_n (Z_1,...,Z_n)$. For estimating $Var{t_n}$ from just the available data $(Z_1,...,Z_n)$ we propose computing subseries values: $t_m(Z_{i+1},Z_{i+2}, Z_{i+m})$, $0\leq i<i+m\leq n$. These subseries values are used as replicates to model the sampling variability of $t_n$. In particular, we use adjacent nonoverlapping subseries of length $m = m_n$, with $m_\rightarrow\ infty$ and $m_n/n\rigtharrow 0$. Our variance estimator is just the usual sample variance computed amongst these subseries values (after appropriate standardization). This estimator is shown to be consistent under mild integrability conditions. We present optimal (i.e.,minimum m.s.e.) choices of $m_n$ for the special case where $t_n=\overset{-}{Z}_n$ and ${Z_i}$ is a normal AR(1) sequence. A simulation study is conducted, showing that those same choices of $m_n$ are effective when $t_n$ is a robust estimator of location and ${Z_i}$ is subject to contamination.

In this paper we construct a class of minimum distance Cramer-von Mises-type estimators for the parameter in the first-order stationary autoregressive time series. The estimator is proved to be asymptotically normal under appropriate assumptions. The proofs involve some results of independent interest.

This paper gives a class of minimum $L_2$-distance estimators of the autoregression parameter in the first-order autoregression model when the errors have an unknown symmetric distribution. Within the class an asymptotically efficient estimator is exhibited. The asymptotic efficiency of this estimator relative to the least-squares estimator is the same as that of a certain signed rank estimator relative to the sample mean in the one sample location model. The paper also discusses goodness-of-fit tests for testing for symmetry and for a specified error distribution.

In a regression model with an error that is a general linear process, the second-order expansion of the risk matrix of GLSE or MLE is obtained. a set of sufficient conditions for the effect of estimating the structural parameter of the linear process to vanish in the above expasion is obtained. The relation of the covariance matrix of SLSE with those of GLSE and MLE up to $O(T^-2)$ is elucidated.

Bayesian statistical inference for sampling a finite population is studied by using the Dirichlet-multinomial process as prior. It is shown that if the finite population variables have a Dirichlet-multinomial prior, then the posterior distribution of the inobserved variables given a sample is also Dirichlet-multinomial. If the population size tends to infinity (the sample size is fixed), sampling without replacement from a Dirichlet multinomial process is equivalent to the iid sampling from a Dirichlet process. If both the population size and sample size tend to infinity, then given a sample, the posterior distribution of the population empirical distribution function converges in distribution to a Brownian bridge. The large-sample Bayes confidence band interval are given and shown to be equivalent to the usual ones obtained from simple random sampling.

Results on the total time on test plot are usually obtained on the assumption that the number of events to be observed is fixed in advance. Here it is shown that the same large sample results hold when the number of events is random if a simple condition is satisfied.

In this paper, we study the local deviations of the empirical measure defined by the Kaplan-Meier (1958) estimator for the survival function. The results are applied to derive best rates of convergence for kernel estimators for the density and hazard rate function in the random censorship model.

We investigate the asymptotic behaviour of the median deviation and the semi-interquartile range based on the residuals from a linear regression model by deriving weak asymptotic representations for the estimators. These representations may be used to obtain a variety of central limit theorems and yield conditions under which the median deviation and the semi-interquartile range are asymptotically equivalent. The results justify the use of the estimators as concommitant scale estimators in the general scale equivariant M-estimation of a regression parameter problem. Finally, the results contain as a special case those obtained by Hall and Welsh (1985) for independent and identically distributed random variables.

In Boos (1985) equicontinuity conditions are given which ensure the uniform convergence of densities in $\mathbb{R}^k$, given convergence in distribution. In the present note we show that such equicontinuity conditions in fact characterize uniform local convergence with no additional assumptions on the sequence of densities, or on the limit density. Versions of these results are also given when the distributions depend on an unknown parameter; these forms will be relevant for the uniform approximation of likelihood functions.

The asymptotic efficiency of Good's nonparametric coverage estimator is obtained relative to the best estimator derived under the assumption that all classes are equally likely. Even when that assumption is true, Good's estimator is quite efficient, with an asymptotic relative efficiency of greater than 85% in all cases, and greater than 95% if the expected coverage is less than one-half.