This paper describes an analysis of cell membrane noise. The physiological nature of cell membrane noise is outlined, and the experimental methods and data processing methods used are described. The statistical methods used to make inferences about interesting physiological parameters are discussed and a summary of some of the experimental findings is given. Since cell membrane noise is modelled as a form of shot noise, background material on shot noise is reviewed.

This paper deals with $M$-estimators for the linear model $y_i = \mathbf{x}'_i\mathbf{\theta} + u_i1 \leqslant i \leqslant n$, where the $\mathbf{x}_i$ are fixed $p$-dimensional vectors, and the $u_i$ are i.i.d. random variables with distribution $F$. The estimators considered are solutions $\hat\mathbf{\theta}$ of the equation $\sum^n_{j = 1}\psi(y_j - \mathbf{x}'_i\hat{\mathbf{\theta}})\mathbf{x_j = 0}$ for some function $\psi$. Let $\mathbf{X}$ be the matrix whose $i$th row is $\mathbf{x}'_i$. Then it is proved that $(\mathbf{\hat{\theta} - \theta)'X'X(\hat{\theta} - \theta)}$ is bounded in probability assuming that $\psi$ satisfies a set of conditions which include $\psi$ to be monotone and $X$ to have full rank. This implies that a sufficient condition for consistency is that the smallest eigenvalue of $\mathbf{X'X}$ tends to infinity. For the case in which $p = p_n \rightarrow \infty$ it is proved that $p^{-1}(\mathbf{\hat{\theta} - \theta)'X'X(\hat{\theta} - \theta)}$ is bounded in probability, assuming that $p\varepsilon\rightarrow 0$ where $\varepsilon = \max_{1\leqslant i \leqslant n}(\mathbf{x'_iX'Xx_i})$. The asymptotic normality of these estimators is proved for both the cases of $p$ fixed and $p \rightarrow\infty$. The proof of the former is an easy consequence of a result of Bickel on one-step $M$-estimators. In the case of $p\rightarrow \infty$ we assume that $\psi$ has a bounded derivative and that $p^{3/2}\varepsilon\rightarrow 0$. This improves an analogous result by Huber, who requires $p^2\varepsilon\rightarrow 0$.

Let $X$ be a random vector distributed according to an exponential family with natural parameter $\theta \in \Theta$. We characterize conjugate prior measures on $\Theta$ through the property of linear posterior expectation of the mean parameter of $X : E\{E(X|\theta)|X = x\} = ax + b$. We also delineate which hyperparameters permit such conjugate priors to be proper.

This paper deals with the estimation of the parameters (cell probabilities) of a multinomial distribution. For the binomial distribution it is known that the maximum likelihood estimator (MLE) is admissible with respect to the squared error loss. It is shown that the MLE is admissible also in the case of the multinomial distribution.

Admissible and minimax estimation is discussed for estimating the parameters in the (a) multinomial distribution and in (b) $k$ independent binomial distributions. In (a) the loss function is $\sum^n_0\lbrack\delta_i(x) - \theta_i\rbrack^2/\theta_i$, where $\theta_0, \cdots, \theta_k(\sum\theta_i = 1)$ are the parameters in the multinomial distribution, and the estimators are restricted to $\sum^k_0\delta_i(x) = 1$. In (b) the loss functions considered are the weighted sum of quadratic losses. The method of proof is based on a multivariate analog of the Cramer-Rao inequality, and uses the divergence theorem in a novel way.

In comparing $K(\geqslant 2)$ treatments in a medical trial, suppose that the response of the patient to treatment is dichotomous. We propose and analyze a class of simple and nondeterministic treatment assignment schemes which tend to put more patients on better treatments. These schemes are also applicable when we have delayed responses from patients.

The strong consistency of the maximum likelihood parameter estimation method is established for multivariate Gaussian stochastic processes possessing autoregressive moving average (ARMA) representations. The demonstration in this paper exploits the ergodic theorem together with results from linear prediction theory.

The density estimator, $f^\ast_n(x) = n^{-1}\sum^n_{j = 1}h^{-1}_jK((x - X_j)/h_j)$, as well as the closely related one $f^\dagger_n(x) = n^{-1}h_n^{-\frac{1}{2}}\sum^n_{j = 1}h_j^{-\frac{1}{2}}K((x - X_j)/h_j)$ are considered. Expressions for asymptotic bias and variance are developed. Using the almost sure invariance principle, laws of the iterated logarithm are developed. Finally, illustration of these results with sequential estimation procedures are made.

Let $X_1, X_2, \cdots, X_n$ be i.i.d. random variables with common density function $f$. A method of density estimation based on "delta sequences" is studied and mean square rates established. This method generalizes certain others including kernel estimators, orthogonal series estimators, Fourier transform estimators, and the histogram. Rates are obtained for densities in Sobolev spaces and for densities satisfying Lipschitz conditions. The former generalizes some results of Wahba who also showed the rates obtained are the best possible. The rates obtained in the latter case have been shown to be the best possible by Farrell. This is shown independently by giving examples for which the rates are exact. Finally, a necessary and sufficient condition for asymptotic unbiasedness for continuous densities is given.

This paper deals with a general class of recursive estimates of the density function in a continuous-time stationary Markov process. Under the condition $G_2$ of Rosenblatt sufficient conditions for almost sure convergence of such estimates are given.

Let $P_{ab}$ be the probability that the outcome of a paired comparison involving $a$ and $b$ is favorable to $a$. This paper discusses composition rules that generate $p_{ac}$ given $p_{ab}$ and $p_{bc}$. The basic properties of composition rules are developed via an axiomatic approach. If $p_{ab} = F(w_a - w_b)$, where $F$ is a distribution function for a distribution that is symmetric about zero, then the paired comparison model is a linear model that is based on the distribution function $F$. It is shown that given any composition rule, which obeys certain basic axioms, there exists a linear model that generates an identical composition rule. The behavior of the composition rules are used to place a partial ordering on the paired comparison models and in particular on the linear models. This partial ordering is denoted as the extreme partial ordering. It is shown that linear models based on distributions with short tails tend to be more extreme than those based on distributions with long tails. The resulting partial ordering includes the result that the Thurstone-Mosteller model is more extreme than the Bradley-Terry model. The extreme ordering can also be used to place a partial ordering on distributions according to the lengths of their tails. The relation of this ordering to the $s$-ordering and $r$-ordering is examined.

For the coupon collector's problem, invariance principles for the partial sequence of bonus sums after $n$ coupons as well as the waiting times to obtain the bonus sum $t(\geqslant 0)$ are studied through a construction of a triangular array of martingales related to these sequences and verifying the invariance principles for these martingales.

The commutation matrix $K$ is defined as a square matrix containing only zeroes and ones. Its main properties are that it transforms vecA into vecA', and that it reverses the order of a Kronecker product. An analytic expression for $K$ is given and many further properties are derived. Subsequently, these properties are applied to some problems connected with the normal distribution. The expectation is derived of $\varepsilon' A\varepsilon\cdot\varepsilon' B\varepsilon\cdot\varepsilon'C\varepsilon$, where $\varepsilon \sim N(0, V)$, and $A, B, C$ are symmetric. Further, the expectation and covariance matrix of $x \otimes y$ are found, where $x$ and $y$ are normally distributed dependent variables. Finally, the variance matrix of the (noncentral) Wishart distribution is derived.

We consider a finite set of units, a population. With each unit is associated a real value (unknown to us) and a label (identifying the unit). Based on the labels we may select a sample, i.e., a subset of the population, to estimate the mean of the real values. In simple random sampling (not necessarily of fixed size) the selection probabilities of all samples are not affected by a permutation of the labels. It is assumed that we have to choose both a sampling design and a linearly invariant estimator, i.e., a linear function of the observed values with the property: equality of the observed values implies that the estimate is equal to this common value. Under these conditions we should use simple random sampling together with the sample mean as an estimator. This follows from the minimax criterion.

A general system of inference which leads to upper and lower posterior distributions based on sample data has been proposed by Dempster (1967). This general theory of inference is applied to the two-parameter logistic function, given the data from independent binomial populations. Inferences are developed for fixed regions about the two parameters and about interesting combinations of these parameters. The resulting upper and lower probabilities are generated by a random polygonal-type region, or more exactly by specific extreme points of this region. For these extreme points, the exact marginal and joint distributions are derived; approximate distributions are also derived.

For progressive censoring schemes pertaining to a general class of (parametric as well as nonparametric) testing situations, one encounters a (partial) sequence of linear combinations of functions of order statistics where the coefficients are themselves stochastic variables. Weak convergence of such a quantile process to an appropriate Gaussian function is studied here, and the same is incorporated in the formulation of suitable (time-) sequential tests based on these quantile processes.

The main aim of the present work is to establish connections between the theory of dynamic programming and the statistical decision theory. The paper deals with a nonMarkovian dynamic programming decision model that includes Markovian decision models and Markov renewal decision models as special cases. The analysis is based on the total cost criterion where the convergence condition on the expected total cost is such that the discounted and the negative (unbounded) case are included. The striking feature of the present model is the fact that the law of motion is not completely known, which leads to a treatment of the model by the approach of statistical decision theory. The assumptions of the present paper are discussed for a sequential statistical decision problem.

The union-intersection method of test construction can be used to derive from the Mann-Whitney rank test a test statistic for testing whether two samples of observations from circular distributions come from the same population. The null distribution of this test statistic will be investigated here. The approach followed is to use the principle of inclusion-exclusion to obtain an expression for the number of partitions of a positive integer which satisfy certain conditions. This enables the probabilities for values of the circular Mann-Whitney test statistic to be expressed explicitly in terms of the probabilities for values of the usual Mann-Whitney test statistic. Recurrence formulae enabling computation of the distribution for small sample sizes are given. There is a clear relationship between our work and results obtained by Steck for the Kolmogorov and Smirnov statistics. These results can also be derived from the present approach.

The exact distribution of a nonparametric test statistic for ordered alternatives, the so-called $\bar{\chi}^2_{\operatorname{rank}}$ statistic, is discussed. Two tables are given.

It is shown that in the MANOVA situation, certain dependent tests of hypothesis may be combined to obtain an overall conservative significance level. The result holds whether one uses the Wilks, Roy, Lawley-Hotelling or Pillai test criterion. This generalizes a result due to Kimball in univariate ANOVA.

This article corrects some additional errors in Aroian's article and preseents some new results on two associated continued fraction representations of the incomplete beta function.