This paper extends and unifies the theory of simultaneous estimation for the discrete exponential family. We discuss construction of estimators which theoretically dominate the uniformly minimum variance unbiased estimator (UMVUE) under a weighted squared error loss function, and show by means of computer simulation results that new simultaneous Poisson means estimators perform more favorably than those previously proposed. Our improved estimators shift the UMVUE towards a possibly nonzero point or a data-based point.

Interpoint-distance-based graphs can be used to define measures of association that extend Kendall's notion of a generalized correlation coefficient. We present particular statistics that provide distribution-free tests of independence sensitive to alternatives involving non-monotonic relationships. Moreover, since ordering plays no essential role, the ideas are fully applicable in a multivariate setting. We also define an asymmetric coefficient measuring the extent to which (a vector) $X$ can be used to make single-valued predictions of (a vector) $Y$. We discuss various techniques for proving that such statistics are asymptotically normal. As an example of the effectiveness of our approach, we present an application to the examination of residuals from multiple regression.

Ten invariant multivariate testing problems involving the real, complex, or quaternion structure of covariance matrices are considered. In each problem the maximal invariant statistic and its distribution are described, as well as the maximum likelihood estimators and likelihood ratio test statistics. These results are obtained by means of a new, unified method based on invariance arguments.

An earlier introduced estimation principle, which calls for minimization of the number of bits required to write down the observed data, has been reformulated to extend the classical maximum likelihood principle. The principle permits estimation of the number of the parameters in statistical models in addition to their values and even of the way the parameters appear in the models; i.e., of the model structures. The principle rests on a new way to interpret and construct a universal prior distribution for the integers, which makes sense even when the parameter is an individual object. Truncated real-valued parameters are converted to integers by dividing them by their precision, and their prior is determined from the universal prior for the integers by optimizing the precision.

Asymptotic lower bounds for estimation of the parameters of models with both parametric and nonparametric components are given in the form of representation theorems (for regular estimates) and asymptotic minimax bounds. The methods used involve: (i) the notion of a "Hellinger-differentiable (root-) density", where part of the differentiation is with respect to the nonparametric part of the model, to obtain appropriate scores; and (ii) calculation of the "effective score" for the real or vector (finite-dimensional) parameter of interest as that component of the score function orthogonal to all nuisance parameter "scores" (perhaps infinite-dimensional). The resulting asymptotic information for estimation of the parametric component of the model is just (4 times) the squared $L^2$-norm of the "effective score". A corollary of these results is a simple necessary condition for "adaptive estimation": adaptation is possible only if the scores for the parameter of interest are orthogonal to the scores for the nuisance function or nonparametric part of the model. Examples considered include the one-sample location model with and without symmetry, mixture models, the two-sample shift model, and Cox's proportional hazards model.

The kernel function method developed during the last twenty-five years to estimate a probability density function essentially is a way of smoothing the empirical distribution function. This paper shows how one can generalize this method to estimate counting process intensities using kernel functions to smooth the nonparametric Nelson estimator for the cumulative intensity. The properties of the estimator for the intensity itself are investigated, and uniform consistency and asymptotic normality are proved. We also give an illustrative numerical example.

There is growing interest in statistical inference under order restrictions. A major demand in this subject is to have a fast, direct method to solve the least squares problem of partially ordered isotonic regression. The Min-Max algorithm is such a method in which the user searches for the global minimum and the local maximum successively. A comparison of algorithms for partially ordered isotonic regression is included. As an application, using this efficient algorithm, it is feasible to approximate critical values of isotonic tests by simulation.

For a first order non-explosive autoregressive process with unknown parameter $\beta \in \lbrack -1, 1 \rbrack$, it is shown that if data are collected according to a particular stopping rule, the least squares estimator of $\beta$ is asymptotically normally distributed uniformly in $\beta$. In the case of normal residuals, the stopping rule may be interpreted as sampling until the observed Fisher information reaches a preassigned level. The situation is contrasted with the fixed sample size case, where the estimator has a non-normal unconditional limiting distribution when $|\beta| = 1$.

The conditional score function is found to be generally fully informative concerning a parameter of interest when the conditioning statistic $S$ is sufficient for the nuisance parameter and has an exponential family distribution. Information is here measured by assuming the nuisance parameter to have been generated by an unknown mixing distribution and then computing the minimal Fisher information. The solution depends upon a study of the geometry of centered likelihood ratios within the space of zero-unbiased functions of $S$. The two-by-two table model is considered in detail.

The performance of a sequence of estimators $\{T_n\}$ of $\theta$ can be measured by the probability concentration of the estimator in an $\varepsilon_n$-neighborhood of $\theta$. Classical choices of $\varepsilon_n$ are $\varepsilon_n = cn^{-1/2}$ (contiguous case) and $\varepsilon_n = \varepsilon$ fixed for all $n$ (non-local case). In this article all sequences $\{\varepsilon_n\}$ with $\lim_{n\rightarrow\infty} \varepsilon_n = 0$ and $\lim_{n\rightarrow\infty} \varepsilon_nn^{1/2} = \infty$ are considered. In that way the statistically important choices of small $\varepsilon$'s are investigated in a uniform sense; in that way the importance and usefulness of classical results concerning local or non-local efficiency can gather strength by extending to larger regions of neighborhoods; in that way one can investigate where optimality passes into non-optimality if for instance an estimator is locally efficient and non-locally non-efficient. The theory of moderate deviation and Cramer-type large deviation probabilities plays an important role in this context. Examples of the performance of particularly maximum likelihood estimators are presented in $k$-parameter exponential families, a curved exponential family and the double-exponential family.

A nonparametric estimate $\beta^\ast$ is presented for the slope of a regression line $Y = \beta_0X + V$ subject to the truncation $Y \leq y_0$. This model is relevant to a cosmological controversy which concerns Hubble's Law in Astronomy. The estimate $\beta^\ast$ corresponds to the zero-crossing of a random function $S_n(\beta)$, which for each $\beta$ is a Mann-Whitney type of statistic designed to measure heterogeneity among the calculated residuals $Y - \beta X$. The asymptotic distribution of $\beta^\ast$ is derived making extensive use of $U$-statistics to show that $S_n(\beta_0)$ is asymptotically normal and then showing that $S_n(\beta)$ behaves like $S_n(\beta_0)$ plus a deterministic term which is locally linear. Results on asymptotic efficiency are compared with finite sample size results by simulation.

The normal, Poisson, gamma, binomial, negative binomial, and NEFGHS distributions are the six univariate natural exponential families (NEF) with quadratic variance functions (QVF). This sequel to Morris (1982) treats certain statistical topics that can be handled within this unified NEF-QVF formulation, including unbiased estimation, Bhattacharyya and Cramer-Rao lower bounds, conditional distributions and moments, quadratic regression, conjugate prior distributions, moments of conjugate priors and posterior distributions, empirical Bayes and $G_2$ minimax, marginal distributions and their moments, parametric empirical Bayes, and characterizations.

Limit theorems giving rates of convergence of nonparametric regression estimates obtained from smoothing splines are proved. The main emphasis is on nonlinear, robust smoothing splines, but new results are obtained for the usual (linear) case. It is assumed that the knots become asymptotically uniform in a vague sense. Convergence of derivatives is also investigated. The main mathematical tools are a linearization of the robust smoothing spline, and an approximation of the linear smoothing spline utilizing the Green's function of an associated boundary value problem.

Suppose that there is a population of $N$ identifiable units each with two associated values $x_i$ and $y_i$. All $N$ values of $x$ are given but $y_i$ is determined only after the $i$th unit is selected and observed. The objective is to estimate the population total $\sum^N_{i=1} y_i$. It is assumed that $y_i = \theta x_i + \delta_ig(x_i), 1 \leq i \leq N$, where $(\delta_1, \cdots, \delta_N)$ is in some bounded neighborhood of $(0, \cdots 0)$. The Rao-Hartley-Cochran and Hansen-Hurwitz strategies are shown to be approximately minimax under certain models with $g(x) = x^{1/2}$ and with $g(x) = x$, the latter relating to a problem considered by Scott and Smith (1975). These two strategies are then compared with some commonly-used strategies and are found to perform favorably when $g^2(x)/x$ is an increasing function of $x$. The problem of estimating $\theta$ is also considered. Finally, some exact minimax results are obtained for sample size one.

In the random walk stock market model, a stock is purchased at price $x$ and is sold at time $t$ for the price $x + S_t$ where $S_t = \sum^t_0 X_i, X_i$ is the price change during the $i$th epoch, and $X_1, X_2, \cdots$ are i.i.d. random variables with $\mu = E(X_1) > 0$ and finite $\sigma^2 = E(X^2_1) - \mu^2 > 0$. Discounting the future by a factor of $\gamma$ per epoch, $0 < \gamma < 1$, a selling or stopping policy $t$ has expected payoff or utility $u(t) = E\{\gamma^t(x + S_t)\}$. This article determines second order asymptotic properties of the optimal selling policy $s$, the first passage time of $S_n$ across a straight line boundary $c$, whose utility is equal to the value $V(x) = \sup_tu(t)$ of the stock purchased at price $x$. Specifically, as $\gamma \rightarrow 1$, renewal theory is utilized to evaluate the limiting distribution of $s, E(s), V(x)$, and the first passage boundary $c$ up to second order terms.

We provide a method for inverting a general Edgeworth expansion, so as to correct a statistic for the effects of non-normality. This technique is applied to the special case of the "Studentized" mean. Explicit formulae are given for the correction terms.

Let $T(\cdot)$ be a suitably regular functional on the space of distribution functions, $F$, on $R^s$. A method is given for obtaining the derivatives of $T$ at $F$. This is used to obtain asymptotic expansions for the distribution and quantiles of $T(F_n)$ where $F_n$ is the empirical distribution of a random sample of size $n$ from a distribution $F$ with an absolutely continuous component. One- and two-sided confidence intervals for $T(F)$ are given of level $1 - \alpha + O(n^{-j/2})$ for any given $j$. Examples include approximate nonparametric confidence intervals for the mean and variance of a distribution on $R$.

In the general two-sample testing problem $F = G$ versus $F \leq G, F \neq G$ the shift-model score function $f' \circ F^{-1}/f \circ F^{-1}$ has to be replaced by the nonparametric score function $b = \bar{f} - \bar{g}$, where $\bar{f} = d(F \circ H^{-1})/dx, \bar{g} = d(G \circ H^{-1})/dx, H = (mF + nG)/(m + n)$, and adaption of linear rank tests should be based on rank estimators of $b$. We consider an easy but rough and inconsistent rank estimator of $b$. The resulting rank test turns out to be a generalization of Galton's test. A formula for local asymptotic power under arbitrary local alternatives is derived which allows for comparison of Galton's test with every linear rank test. For various types of alternatives the Galton test is compared with the optimal linear rank test and with the Wilcoxon test. In order to get an impression of the validity of extrapolation to finite sample sizes, we included a Monte Carlo study under the same types of fixed alternatives.

How often does the empirical distribution pass over an acceptance band around the hypothetical distribution function? Sheffer polynomials are applied to derive the exact distribution of this goodness-of-fit distribution and its two-sample analogue. This method leads to new proofs and extensions of Takacs's (1971) results.

An Edgeworth expansion with remainder $o(N^{-1})$ is established for simple linear rank statistics under the null-hypothesis. The theorem is proved for a wide class of scores generating functions which includes the normal quantile function.

A randomization model for a two-sample situation with additive treatment effects is considered. Edgeworth expansions for the power of the usual permutation test are derived, under some conditions on the unit errors, from previously obtained expansions under the null hypothesis of no treatment effect. A general error structure is considered and conditions for the validity of the expansions for both conditional and unconditional power are examined. The results are shown to generalise expansions obtained earlier by different methods for the special case of independent and identically distributed random variables.

We derive an explicit expression for the Bayes risk (using weighted squared error loss) of Dalal's Bayes estimator of a symmetric distribution under a $\mathscr{G}$-invariant Dirichlet process prior. We compare this risk to the risk of Ferguson's estimator of an arbitrary distribution under the $\mathscr{G}$-invariant prior. This enables us to (i) assess the savings in risk attained by incorporating known symmetry structure in the model and (ii) provide information about the robustness of Ferguson's estimator against a prior for which it is not Bayes.

Consider the problem of sequentially testing the hypothesis that the mean of a normal distribution of known variance is less than or equal to a given value versus the alternative that it is greater than the given value. Impose the linear combination loss function under which the risk becomes a constant $c$, times the expected sample size, plus the probability of error. It is known that all admissible tests must be monotone--that is, they stop and accept if $S_n$, the sample sum at stage $n$, satisfies $S_n \leq a_n$; stop and reject if $S_n \geq b_n$. In this paper we show that any admissible test must in addition satisfy $b_n - a_n \leq 2\bar{b}(c)$. The bound $2\bar{b}(c)$ is sharp in the sense that the test with stopping bounds $a_n \equiv -\bar{b}(c), b_n \equiv \bar{b}(c)$ is admissible. As a consequence of the above necessary condition for admissibility of a sequential test, it is possible to characterize all sequential probability ratio tests (SPRT's) regarding admissibility. In other words necessary and sufficient conditions for the admissibility of an SPRT are given. Furthermore, an explicit numerical upper bound for $\bar{b}(c)$ is provided.

In the context of a normal model, testing problems with missing data are considered. Tests on means are treated when independent extra data on the first $p_1$ variates of $p$ variates is available in addition to complete data. For testing that the mean of the first $p_1$ variates is zero, the LRT is UMP invariant, but for testing that the whole mean is zero, no LMPI (locally most powerful invariant) test exists. Second, tests for independence are treated in similar situations, and an LMPI test for each situation is derived. In some situations it is found that the LRT for independence ignores the extra data.

Let $M \in M(m, p)$ be the matrix of means of interest in the GMANOVA problem. Our main results characterize all confidence sets for $M$ in a given class (invariant plus a weak additional restriction) that are exact for the families of parametric functions $a'Mb$ for all $a \in \mathbb{R}^m, b \in \mathbb{R}^p$ and $\operatorname{tr} N'M$ for all $N \in M(m, p)$. The corresponding families of smallest exact simultaneous confidence intervals are also given. Similar results are obtained for the MANOVA problem under triangular group reduction.

This paper gives conditions under which $L$- and $M$-estimators of a location parameter have asymptotically the same distribution, conditions under which they are asymptotically equivalent and conditions under which $L$-estimators are asymptotically efficient relative to the Cramer-Rao lower bound. Our results differ from analogous results of Jung (1955), Bickel (1965), Chernoff, Gastwirth, Johns (1967), Jaeckel (1971) and Rivest (1978, 1982) in that we do not require that the derivative of the density of the observations is absolutely continuous nor that the function defining the $M$-estimator is absolutely continuous.

We consider the problems of (i) covariance adjustment of an unbiased estimator, (ii) combining two unbiased estimators, and (iii) improving upon an unbiased estimator. All these problems consist in determining a minimum dispersion linear unbiased combination of given two statistics, one of which is an unbiased estimator of a vector parameter $\mathbf{\theta} \in \mathscr{H}$, and the expectation of the other is a zero vector in the problem of covariance adjustment, is equal to $\mathbf{\theta}$ in the problem of combining, and is equal to a subvector of $\mathbf{\theta}$ in the problem of improving. The solutions obtained are substantial generalizations of known results, in the sense that they are valid for an arbitrary joint dispersion matrix of the given statistics as well as for the parameter space $\mathscr{H}$ being an arbitrary subspace of $\mathscr{R}^k$.

A $K - S$ type statistic computed from sequential ranks has been proposed in the astrophysics literature for testing the slope of a truncated regression. There is an easy heuristic justification for the test in the nontruncated case, but it fails under truncation. This paper extends the heuristic justification to the truncated case and outlines a more complete proof of the asymptotic property.

The permutation distribution induced upon a finite population by the order of selection under successive sampling is closely related to the order statistics of independent exponentially distributed waiting times. This characterization is applied to obtain necessary and sufficient conditions for asymptotic normality of the sum of characteristics observed in a successive sample from a finite population. The necessary and sufficient conditions generalize previous results for simple random sampling without replacement, and apply to sampling fractions close to 0 or 1.

This paper is concerned with the problem of finding Bayes sequential designs for successively choosing between two given Bernoulli variables so as to maximize the total discounted expected sum. Simple hypotheses concerning the success probabilities are assumed and dynamic programming methods are used to characterize optimal designs. Explicit solutions are described for certain special cases.