bluuebunny/crossref_metadata_2025 huggingface dataset:
Using the Observations to Estimate the Prior Distribution
Authors: Clutton-Brock M.
Published: December 2018

A parameter μ is to be estimated from an unbiased, unit-variance measurement x. The prior distribution is symmetrical about zero with unknown scale parameter σ. It is claimed that the observation x gives information about σ which is contained in the likelihood distribution p(dx|σ). An estimator μ̂ = γx is found, based on the posterior distribution of μ conditional on the most probable value of σ. For a prior distribution of arbitrary shape, γ is zero when x2 ≤ 1 and approaches 1 – x–2 when x2 ≥ 1. A minimum mean-square estimator is found based on an estimate of the prior variance. This gives the same result, independently of the shape of the prior distribution, as the previous estimator for a normal prior.
Applications of a Ballot Theorem in Physics and in Order Statistics
Authors: Takács Lajos
Published: December 2018

The author finds the distribution of the number of gas molecules ionized by an electron passing through an electron tube, and the distribution of the deviation of empirical and theoretical distribution functions, when the sample size is a random variable having a Poisson distribution.
Some General Results in Sequential Design
Authors: Whittle P.
Published: December 2018

We consider a sequential decision problem in which one has a choice of several different types of observation (or “experiment”) of differing costs and degrees of informativeness. In the most general case considered it is assumed that the cost of an observation of type s is crs if the last observation used was of type r, so allowance is made for transition cost (the cost of a change in type of experiment). The equations governing the loss function are discussed in Sections 2 and 3. In Theorem 4 the optimal choice of observation type is completely determined for the case of sequential discrimination between two simple hypotheses in the absence of transition costs, overshoot being neglected. The optimal switching values for the case when there are positive transition costs are determined in Theorem 5. In Section 7 partial results, embodied in Theorem 6, are obtained for the determination of the optimal number of observations to take at a time in the case when sampling costs are not simply proportional to the number of observations.
Plane Truncation in Normal Populations
Authors: Tallis G. M.
Published: December 2018

This paper considers the truncation of normal distributions by means of planes. The moment-generating function for the truncated distribution is obtained, and it is shown that, by suitable transformations, the problem reduces to the case of rectangular truncation.
Application of Stochastic Approximation to Process Control
Authors: Comer J. P.
Published: December 2018

The objective of process control may be to hold a response Y as close to some specified target Y0 as possible by adjusting an independent variable X. In the simplest cases, this objective may be met by choosing a suitable fixed value of X as quickly as possible after the process is started up or after an unusual disturbance occurs during operation. In more complicated cases, it may be advantageous to continue the adjustment of X during normal operation. Two stochastic approximation procedures are presented for handling these two different control requirements. The advantage of the stochastic approximation procedures over more conventional procedures is that relatively little prior knowledge of the process is required to achieve reasonably good results.
Note on the Confidence-Prior of Welch and Peers
Authors: Hartigan J. A.
Published: December 2018

Welch and Peers have shown that it is possible to determine a prior distribution for a one-dimensional parameter by requiring that the associated one-sided Bayes intervals are asymptotically confidence intervals. It is shown in this note that their procedure does not apply if certain natural two-sided Bayes intervals are used.
A Test for “Intrinsic Correlation” in the Theory of Accident Proneness
Authors: Subrahmaniam K.
Published: December 2018

We consider a model in accident theory which is a modified version of the one proposed and studied by Bates and Neyman (1952a). The model envisages a correlation between the two kinds of accident in the model of Bates and Neyman. We set up a test of significance that this correlation is zero. Some properties of the distribution arising from the modified approach are presented.
The Decision Theory Approach to Sampling Inspection
Authors: Wetherill G. B. , Campling G. E. G.
Published: December 2018

Some decision theory models are outlined for acceptance inspection of large batches of items. Firstly, the effects of errors in the algebraic formulation of the models and of errors in parameters used in the models are studied. Secondly, the amount of improvement in utility through use of double and sequential plans is investigated.
On Sharpening Scheffe Bounds
Authors: Bohrer Robert
Published: December 2018

An extension of Scheffé's (1959) confidence bounds gives sharp confidence bounds for a linear function of non-negative arguments. The length of these bounds is compared with that of the conservative Scheffé bounds which would usually be used in this situation. A table facilitating application of the sharpened bounds is provided.
Inferences Concerning a Population Correlation Coefficient from One or Possibly Two Samples Subsequent to a Preliminary Test of Significance
Authors: Srivastava S. R. , Bancroft T. A.
Published: December 2018

It is shown in this paper that under certain restrictions on population parameters, estimates formulated on the basis of a preliminary test of significance concerning inferences for a population correlation coefficient are better than the usual unbiased estimators.
Estimation of the Parameter of an Exponential Distribution
Authors: El-Sayyad G. M.
Published: December 2018

Some new estimators, which are unbiased with respect to some loss functions, are derived for the parameter in the exponential distribution. The corresponding Bayes estimators are also obtained. The comparison between these two kinds of estimation is discussed.
Asymptotically Efficient Tests by the Method of n Rankings
Authors: Sen Pranab Kumar
Published: December 2018

The problem of finding asymptotically most powerful analysis of variance tests by the method of n rankings is studied here. In this context, characterizations of the optimality of the well-known tests by Friedman (1937) and Brown and Mood (1951) are also considered.
A General Approach to Some Screening and Classification Problems
Authors: Marshall Albert W. , Olkin Ingram
Published: December 2018

This paper is concerned with the problem of classifying items which belong to one of several classes. Each item is characterized by a variable Y which may lie in one of several sets; the actual class to which the item belongs is determined by the set which contains Y. We suppose that Y is costly, inconvenient, or impossible to observe, but that an observation X=(X1,⋯,XK) is available which is correlated with Y. The problem considered is the optimal use of screening variables in the classification problem. We assume that the conditional distribution of Y given x is known, and that optimality is to be judged in terms of a given loss function.

A simple characterization of minimum risk procedures is given which suggests useful computational methods. These are illustrated by a number of examples for which specific procedures are obtained. The possibility of observing X1,…, Xk sequentially is also considered.
A Multi-Stage Test for a Normal Mean
Authors: Scott A.
Published: December 2018

Grundy, Healy and Rees (1956) have examined the problem of deciding how much experimentation to carry out when trying to choose the best of two processes. This paper extends their results to situations in which more than one additional stage of observations can be taken.
Least-Squares Efficiency for Vector Time Series
Authors: Hannan E. J.
Published: December 2018

A theorem due to Rosenblatt (1959) concerning the circumstances under which least squares is asymptotically efficient for the estimation of the matrix of regression parameters in a system of regressions is re-formulated in a simpler form and is extended. A theorem is proved relating to the efficiency of least squares under linear constraints which is not specifically of a time series nature.
Confidence Properties of Bayesian Interval Estimates
Authors: Peers H. W.
Published: December 2018

Coverage probabilities, in the confidence theory sense, of Bayes interval estimates constructed according to different criteria are examined. Suggestions are made for the choice of a prior density in the case of complete ignorance concerning the parameter under estimation. These densities are members of the family of relatively invariant densities discussed by Hartigan (1964). Confidence properties of a related family are also discussed.
On the Superposition of Point Processes
Authors: Çinlar E. , Agnew R. A.
Published: December 2018

Given point processes N1, …, Nm their superposition is the point process N defined by N(t) = N1(t) + … + Nm(t), t ≥ 0. An equivalent description of the system (N1, …, Nm) is by the process (Xn, Tn) where the Tn are the points of N, and Xn = k if and only if Tn is a point of Nk. The use of (X, T) process enables one to study the dependence of N1, …, Nm. Necessary and sufficient conditions are obtained for N, …, Nm to be independent, and for the superposition N to be a renewal process. For example, if N1 and N2 are renewal processes and X is independent of N, then N is a renewal process only if X is a homogeneous Markov chain.
A Simplified Monte Carlo Significance Test Procedure
Authors: Hope Adery C. A.
Published: December 2018

The use of Monte Carlo test procedures for significance testing, with smaller reference sets than are now generally used, is advocated. It is shown that, for given α = 1/n, n a positive integer, the power of the Monte Carlo test procedure is a monotone increasing function of the size of the reference set, the limit of which is the power of the corresponding uniformly most powerful test. The power functions and efficiency of the Monte Carlo test to the uniformly most powerful test are discussed in detail for the case where the test criterion is N(γ, 1). The cases when the test criterion is Student's t-statistic and when the test statistic is exponentially distributed are considered also.
On the Asymptotic Behaviour of Posterior Distributions
Authors: Walker A. M.
Published: December 2018

Let a random sample of size n be taken from a distribution having a density depending on a real parameter θ, and let θ have an absolutely continuous prior distribution with density π(θ). We give a rigorous proof that, under suitable regularity conditions, the posterior distribution of θ will, when n tends to infinity, be asymptotically normal with mean equal to the maximum-likelihood estimator and variance equal to the reciprocal of the second derivative of the logarithm of the likelihood function evaluated at the maximum-likelihood estimator, independently of the form of π(θ).
On Cox and Snell's General Definition of Residuals
Authors: Loynes R. M.
Published: December 2018

Cox and Snell have introduced a definition of residuals, in a situation more general than the usual linear hypothesis. They have also discussed briefly the distribution of these residuals, and the complementary aspect of transforming them so that they have (approximately) the same distribution as the random variables which they are effectively estimating. Here it is shown how a complete discussion can be given.
Constrained Discrimination between K Populations
Authors: Anderson J. A.
Published: December 2018

The Bayes rule approach to discrimination, assigning a sample point to the population whose posterior probability is largest, can lead to large misallocation probabilities. There have been several empirical attempts to overcome this difficulty but an objective method is suggested here. This is to find the allocation rule that maximizes the probability of correct allocation subject to the misallocation probabilities satisfying individual upper bounds. Necessary and sufficient conditions for the optimal rule are given and it is shown that a similar approach enables the Neyman–Pearson lemma to be generalized. Analytic methods of finding the optimal rule are discussed but these are of limited application so an approximate, iterative solution is suggested which can be used quite generally.
Subjective Bayesian Models in Sampling Finite Populations
Authors: Ericson W. A.
Published: December 2018

A general and basic model for inference about characteristics of a finite population of distinguishable elements is presented from a subjectivistic–Bayesian point of view. A subjectivist analogue to simple random sampling, based on the notion of exchangeable random variables, is discussed and the inputs and assumptions underlying the model are shown to involve nothing more than is required for inference under Bayesian models for infinite populations. The model is illustrated by a number of particular examples including one based on the multinomial distribution which incorporates a prior distribution representing an extreme position of initial ignorance. Inferences under this particular model are shown to agree closely in several respects with usual “classical” results. Finally, an extension of the results is presented involving the use of concomitant measurements, and under this Bayesian model several common ratio and regression estimators are shown to arise as means of posterior distributions.
Tests for the Exponential Distribution Using Kolmogorov-Type Statistics
Authors: Seshadri V. , Csörgő M. , Stephens M. A.
Published: December 2018

A critical examination is made of the relative merits of two techniques for testing for the exponential distribution, involving transformations of the original observations to random variables uniformly distributed on the null hypothesis.
Beta-Optimal Tests and an Application to the Summary Evaluation of Experiments
Authors: Davies Robert B.
Published: December 2018

A concept of optimality of a test, based on the speed with which its power function reaches a pre-assigned value, is introduced and conditions for a test to have this property are considered. The concept is applied to the weighted combination of independent normal test statistics; tests for the presence of effects and for the equality of effects are deduced.
The Measurement of Association of Rows and Columns for an r × s Contingency Table
Authors: Altham Patricia M. E.
Published: December 2018

The generalization of Edwards's argument for the measure of association of the rows and columns of 2 × 2 table, to that of an r × s table whose rows and columns are assumed unordered, shows, not surprisingly, that association ought to be measured by some function of the (r − 1)(s − 1) cross-ratios. Such a function is suggested by the introduction of a metric on certain equivalence classes. The properties of such metrics are examined, and in particular comparisons are made with Good's suggestion of the use of the algebraic rank of the contingency table, and with Lindley's significance test for association in the r × s table.
A Probability Plotting Procedure for General Analysis of Variance
Authors: Gnanadesikan R. , Wilk M. B.
Published: December 2018

This paper describes a generalization of probability plotting to supplement general analysis of variance procedures. The mean squares in a general orthogonal analysis of variance are ordered and plotted against the corresponding expected values of standardized ordered mean squares. Since the mean squares may have differing degrees of freedom, alternate conceptions are possible with respect to association of the ordered mean squares with their parent distributions.

In considering the statistical distribution of the ith standardized ordered mean square, the view adopted here is that of complete conditioning, that is, repeated sampling so constrained that the order relationships of the sample mean squares is such that the ith ordered mean square comes from a χ2(vi)/vi distribution, for i = 1, …, K, where K denotes the total number of mean squares in the collection and v1, v2, …, vK are the respective degrees of freedom of the ordered mean squares as observed. Using this completely conditioned distribution, methods are described for computing the required plotting positions, viz. the expected values of the standardized ordered mean squares.

Some illustrative examples of use of the proposed procedure are given.
On Bounds Useful in the Theory of Symmetrical Factorial Designs
Authors: Gulati B. R. , Kounias E. G.
Published: December 2018

Consider a finite n-dimensional projective space PG(n, s) over a Galois field of order s = ph (where p, h are positive integers and p is a prime characteristic of the field). A set of k distinct points in PG(n, s), no four coplanar, is said to be complete if there exists no other set with kx points with k1 > k. The number of points in a maximal complete set is denoted by m4(n + 1, s). The exact value of m4(n + 1, 2) is known for n ≤ 7. When n ≥ 8, the best upper bound on m4(n + 1, 2) is due to Seiden (1964). It is the purpose of this paper to show that m4(4, s) = s + 1 for s > 4 and to obtain bounds for m4(n + 1, s), n > 3.
An Accuracy Criterion for Bayesian Tolerance Intervals
Authors: Easterling Robert G. , Weeks David L.
Published: December 2018

An accuracy criterion for a Bayesian approach to γ tolerance intervals for a proportion β is proposed and illustrated for the exponential and normal densities. In particular, the sample size required to attain a specified accuracy is investigated.
Some Problems of Fitting for Contingency-Type Bivariate Distributions
Authors: Mardia K. V.
Published: December 2018

Plackett (1965) has given a family of contingency-type (C-type) bivariate distributions. He has studied the median regressions for the C-type distributions and we obtain the scedastic curves for these distributions based on the semi-interquartile range and derive various regressions and scedastic curves for the uniform C-type distribution. We use these results to construct general methods of fitting C-type distributions. One of the methods is illustrated by using Johannsen's data of beans and our results are compared with the fit given by some other distributions.
The Saddle Point of the Likelihood Surface for a Linear Functional Relationship
Authors: Sprent P.
Published: December 2018

In a recent paper Solari has shown that there is no maximum-likelihood solution to the problem of estimating a linear functional relationship when both variates are subject to errors that have an unspecified bivariate normal distribution. The estimators purporting to be maximum-likelihood estimators were shown by Solari to determine a saddle point. This discovery clears up some anomalies that have been noted in the past.
Some Tests of Independence for Stationary Multivariate Time Series
Authors: Wahba Grace
Published: December 2018

Let X(t)=(X1(t)X2(t)⋮XP(t)),t=⋯−1,0,1…, be a p-dimensional zero-mean stationary Gaussian time series possessing a spectral density matrix F(ω) = {fij(ω)}, i, j = 1, 2, …, p, satisfying some mild regularity conditions. Let F^(ωl), l=1, 2, … ,M, F^(ωl)={f^ij(ωl)} be suitably defined sample spectral density matrices for M values of ω, based on a record of length T ≫ M. We consider the following (null) hypotheses: H1: Xi(t), Xi(s) independent if i ≠ j, all s, t, H2: X1(t) independent of Xi(s), j = 2, …, p, all s, t, H3: X1(t), X1(t + τ) independent, all τ ≠ 0 (“white noise”).

Approximate likelihood-ratio tests are found and the test statistics λi, i = 1, 2, 3, are found to be functions of the {F^(ωl), l=1, 2, … ,M}. The {F^(ωl)} converge in mean square to a family of independent complex Wishart matrices. As a consequence of this fact, log λ1 and log λ2 converge in first mean to random variables whose null densities can be given explicitly, being distributed as the logarithms of products of independent beta random variables with integer indices. Under the null hypothesis λ3 is distributed as λ˜3 where λ˜3 is Bartlett's statistic for homogeneity of variances. The asymptotic distributions of log λi, i = 1, 2, 3, under the alternative are discussed. Under the alternative, log λ3 tends in first mean to log λ˜3. A power series expansion for the characteristic function of log λ˜3 under the alternative is given.
Bias in Nonlinear Estimation
Authors: Box M. J.
Published: December 2018

Although it is widely recognized that maximum-likelihood estimates of the parameters in non-linear models are generally biased, little work appears to have been done on quantitatively assessing these biases. In this paper the difficulties of exact calculation of the bias for a simple example are first illustrated, after which a general method of calculating the biases in a class of nonlinear least-squares problems is presented. The bias in Bayesian estimation is also considered, although the illustrative examples are all for the case of a uniform prior, i.e. the estimation is maximum likelihood. In the most important of the subsidiary results, a generalized ratio of the bias to the variance–covariance matrix of the parameter estimates is defined, and shown to be closely related to Beale's (1960) measures of nonlinearity. Finally the philosophy is applied to a somewhat different problem, namely determining the biases in the maximum-likelihood estimates of the parameters of the gamma distribution.
Arbitrary Event Initial Conditions for Branching Poisson Processes
Authors: Lawrance A. J.
Published: December 2018

Arbitrary event initial conditions for the branching Poisson process (the Bartlett–Lewis cluster process) are obtained. These correspond to the process starting at an arbitrary event, where an arbitrary event is defined in the Khintchine fashion. The conditions give the joint distribution of the number and structure of the initial subsidiary processes; in particular, the marginal distribution of the number of initial subsidiary processes is shown to be the mixture of a Poisson variable and a Poisson variable plus one. It is shown that these conditions marginally repeat themselves at each subsequent event, and thus that the sequence of intervals between events forms a stationary sequence of random variables.
The Moments of the Maximum of Correlated Normal and t-Variates
Authors: Afonja Biyi
Published: December 2018

For a k-dimensional random variable x, the moments of max (x1, …, xk) = max(x) are expressed in terms of the moments of the corresponding truncated standardized distributions. The normal and t-distributions are treated as special cases. When the variables xi have equal means, the moments of max (x) for the t-distribution are expressed in terms of the moments of the truncated normal distribution. In the normal case, explicit expressions are given for the mean and the second moments of max (x), corresponding ones being given for the case of equal means in the t-distribution. The results are applied to the selection and estimation of the maximum of a set of parameters.
The Likelihood Surface in the Linear Functional Relationship Problem
Authors: Copas J. B.
Published: December 2018

In a recent paper, Solari has shown that no maximum likelihood solution exists in the linear functional relationship problem. When account is taken of the rounding-off errors in the observations, however, the appropriate likelihood function is bounded and a solution can be found which is approximately maximum likelihood, in the sense that the value of the likelihood at that solution is close to the global supremum. The novel form of this solution is shown and the likelihood surface in a numerical example is discussed.
Order Invariant Methods for Data Analysis
Authors: Sibson Robin
Published: December 2018

Analysis of objects-by-attributes matrices is often carried out by first calculating a measure of pairwise similarity or dissimilarity between the objects. This paper investigates order invariance criteria which can be applied to methods for analysing such measures, and applies these criteria to methods for classification, scaling, and sequencing.
A Comparison of Methods of Sampling from a Poisson Process
Authors: El-Sayyad G. M.
Published: December 2018

Comparison of fixed sampling schemes from a Poisson process is discussed by means of minimum average risk. The best scheme depends upon the loss function, the prior distribution and the cost of sampling which plays a fundamental role in this problem. Also optimum schemes are considered.
Comparison of Some Two-Sample Location Tests for Non-Normal Alternatives
Authors: Afifi A. A. , Kim P. J.
Published: December 2018

Small to moderate sample size performance of the two-sample χ2, Kolmogorov–Smirnov, Wilcoxon and F-tests of location is investigated in a class of non-normal distributions through Monte Carlo sampling. The class of distributions is that of mixtures of two normal distributions with different means and/or variances. The cases of mixtures considered represent part of the regions for types I, II, IV, VI and VII of the Pearson system of continuous distributions. The mixture distributions from which the samples are drawn differ only in means, and possess varying degrees of skewness and kurtosis. Based on the Monte Carlo results, the Wilcoxon and Kolmogorov–Smirnov tests generally have higher power than the F-test, except for near normal underlying distributions where the F-test is optimal. The power of the χ2 test is less than optimal except when the sample size is large.
The Performance of a Sequential Procedure for a Slippage Problem
Authors: Srivastava M. S.
Published: December 2018

The basic problem known as a “Slippage problem” is to decide that none of the k experimental categories is better than the control or decide that a certain category is better. In this paper, a class of asymptotically efficient sequential procedures is proposed when both the error probabilities are controlled at a specified level (see also Srivastava, 1966). Assuming that these populations are normal with unknown variance σ2, the performance of the sequential procedure is studied for moderate sample sizes. The results of the calculation show that these error probabilities are controlled at the specified level for all values of the parameters and d > 0, where d is the amount of slippage. A similar study for the ranking problem has been done by Srivastava and Ogilvie (1968).
Asymptotic Results on Matching Distributions
Authors: Lanke Jan
Published: December 2018

The generalized matching distribution is investigated. A necessary and sufficient condition for asymptotic Poissonity is given. The case of two identical decks is given special attention.
On the Analysis of Bivariate Non-Stationary Processes
Authors: Priestley M. B. , Tong H.
Published: December 2018

In this paper, we propose a general definition of the evolutionary (time-dependent) cross-spectrum between two non-stationary processes and describe its physical interpretation. We also study the estimation of the evolutionary cross-spectrum at each time instant t from a single realization of a bivariate process. Further, we propose a definition (and a method of estimation) for the coherency (spectrum) between the two components of the bivariate process and show that the notion of residual variance bound first introduced in the analysis of bivariate stationary processes can be extended to that of non-stationary processes. As an application of the evolutionary cross-spectral analysis of bivariate non-stationary stochastic processes, we consider the estimation of the transfer function of a linear open-loop time-dependent system. Numerical illustrations of the estimation of a time-dependent transfer function are included.
Simultaneous Confidence Intervals for Product-Type Interaction Contrasts
Authors: Gabriel K. R. , Putter J. , Wax Y.
Published: December 2018

The set of product-type interaction contrasts, which contains the subsets of interaction residuals, tetrad contrasts, double-dichotomy contrasts and pooled-tetrad contrasts, is discussed. In the Normal case with equal and known variances, simultaneous confidence intervals for a given set of such contrasts can be constructed in several ways. Five such methods (the Scheffé method, the Tukey method, the Dunn method, a method based on Roy's maximum-root statistic and a modification of Tukey's method) are presented and compared in terms of the widths of the resulting 95 and 99 per cent confidence intervals. The Dunn method is found to yield the shortest intervals for interaction residuals, for tetrad contrasts and for double-dichotomy contrasts. On the other hand, for the set of pooled-tetrad contrasts, and therefore also for any larger set of product-type interaction contrasts, the maximum-root method yields the shortest intervals. A table of the half-widths of the intervals given by the recommended methods is provided.
Two-Sample Tests with Unordered Pairs
Authors: Hinkley D. V.
Published: December 2018

Pairs of observations, x and y, are taken on n individuals such that all observations are independent normal with a common variance, and x and y have possibly different means. The ordering of each pair (x, y) is unknown, so that only min (x, y) and max (x, y) are available. Some simple tests of equality of means are discussed for known and unknown variance.
Minimax Designs for Fourier Series and Spherical Harmonics Regressions : A Characterization of Rotatable Arrangements
Authors: Kupper Lawrence L.
Published: December 2018

In the framework of the approximate theory of the optimal design of experiments, Hoel (1965) represented minimax designs for Fourier series and spherical harmonics regressions of order r in terms of certain normalizing coefficients which are difficult to interpret. In this paper it is shown that these designs are intimately related to rotatable arrangements of points of order r in two and three dimensions, respectively. This result is used to construct exact minimax designs for Fourier series regression of all orders and for spherical harmonics regression of orders one and two. Sequential designs and infinite classes of designs depending on only one parameter are given.
Some Discrete Approximations to Continuous Time Stochastic Models
Authors: Sargan J. D.
Published: December 2018

The paper considers the asymptotic biases in various econometric estimators when data generated by a stochastic differential equation system of the form

d y d t − A y ( t ) = B z ( t ) + a ( t )

are estimated by approximating these equations in the form

{ y ( t + δ ) − y ( t ) } / δ = 1 2 A { y ( t + δ ) + y ( t ) } + 1 2 B { z ( t + δ ) + z ( t ) } + v ( t ) ,

where y(t) is a vector of endogenous variables, z(t) is a vector of exogenous variables and A and B are matrices of fixed coefficients, assumed estimated subject to certain a priori restrictions. It is shown that the asymptotic biases are O(δ2) as δ→0, on reasonable assumptions.
Generalizations of Mean Square Error Applied to Ridge Regression
Authors: Theobald C. M.
Published: December 2018

Hoerl and Kennard (1970) have proposed a method of estimation for multiple regression problems which involves adding small positive quantities to the diagonal of XT X. They use a type of mean square error to justify the procedure. This paper considers some generalizations of mean square error and provides a stronger defence of the method.
Cross-Validatory Choice and Assessment of Statistical Predictions
Authors: Stone M.
Published: December 2018

A generalized form of the cross-validation criterion is applied to the choice and assessment of prediction using the data-analytic concept of a prescription. The examples used to illustrate the application are drawn from the problem areas of univariate estimation, linear regression and analysis of variance.
Allocation to Strata and Relative Efficiencies of Stratified and Unstratified πPs Sampling Schemes
Authors: Ramachandran Geetha , Rao T. J.
Published: December 2018

The problem of optimum allocation to strata has been earlier examined in the light of a priori distributions. In this context, under the criterion of minimum expected variance, the sampling strategy consisting of an unstratified πPS sampling scheme together with the Horvitz–Thompson (HT) estimator was shown to be inferior to the strategy consisting of a stratified πPS sampling scheme with the corresponding HT estimator with this optimum allocation. In this paper, when stratification is based on the auxiliary information, we study whether a stratified πPS sampling strategy with various non-optimal allocations is likely to be worth while and whether it should be attempted at all. For populations commonly met in practice, we derive sufficient conditions for unstratified πPS sampling to be preferable to non-optimal stratified πPS sampling. An illustrative example is provided towards the end of the paper.
Construction and Comparison of Statistical Models
Authors: Lindsey J. K.
Published: December 2018

A general method of constructing mathematical models for probability distributions other than the normal, closely related to that of Nelder and Wedderburn (1972), is briefly outlined. A likelihood procedure for comparing statistical models based on different distributions is described and numerical examples, using models based on the exponential family, given to illustrate the techniques.
Projectors, Generalized Inverses and the Blue'S
Authors: Rao C. Radhakrishna
Published: December 2018

It is well known that in the Gauss–Markov model (Y, Xβ, σ2 V) with | V | ≠ 0, the BLUE (best linear unbiased estimator) of Xβ is Y1, the orthogonal projection of Y on M(X), the space spanned by the columns of X, with inner product defined as (x, y) = x'V-1 y. A quadratic function of Y2, the projection of Y on the orthogonal complement of M(X), provides an estimate of σ2. It may be seen that Y = Y1 + Y2. When V is singular, the inner product definition as in non-singular case is not possible. In this paper a suitable theory of projection operators is developed for the case | V | = 0, and a decomposition Y = Y1 + Y2 is obtained such that Y1 is the BLUE of Xβ and a quadratic function of Y2 is the MINQUE (Minimum Norm Quadratic Unbiased Estimator) of σ2 in the sense of Rao (1972).
Components of Cramér-Von Mises Statistics. Ii
Authors: Durbin J. , Knott M. , Taylor C. C.
Published: December 2018

Let θ^ be an asymptotically efficient estimator of parameter vector θ based on an ordered sample of n independent observations y  1, ..., y  n from a distribution with distribution function F(y, θ) and let x^j=F(yj,θ^), j = 1, ..., n. Let Ŵn  2 be the Cramér–von Mises statistic calculated from ◯  1, ..., ◯  n. This paper shows how to construct components ẑ  n1, ẑ  n2, ... such that Ŵ  n  2 = ∑j=1  ∞ μj  ẑ  nj  2 where ẑ  n1, ẑ  n2, ... are asymptotically independent N(0, 1) variables and considers the use of these components for testing goodness of fit in the presence of unknown parameters. Asymptotic significance points are given for tests of normality and exponentiality for Ŵn  2, Anderson and Darling's Â  n  2 and Watson's Û  n  2. The asymptotic powers of these statistics and of the components are studied against alternatives of interest.
Synchronous and Asynchronous Distributions for Poisson Cluster Processes
Authors: Oakes David
Published: December 2018

A simple relation between the synchronous and asynchronous distributions of a Poisson cluster process is proved. This is used in conjunction with the Palm–Khintchine equations to derive an expression for the counting distribution of such a process. It is shown that the synchronous inter-event intervals have coefficient of variation greater than or equal to one, and that the forward recurrence time has a non-increasing hazard function. An expression is derived for the joint survivor function for two consecutive synchronous intervals of a Bartlett–Lewis cluster process.
Joint Probability Distributions in the Meixner Classes
Authors: Lancaster H. O.
Published: December 2018

The classes of statistical distributions, for which exp [xu(t)]/E(exp [xu(t)]) generates a system of orthogonal polynomials, have been characterized by Meixner (1934). His proof is now simplified. The theory is applied to obtain joint distributions in the normal, binomial, gamma and Poisson Meixner classes. The pre-eminent importance of those joint distributions with the polynomial biorthogonal property is established. Their simple regression properties are demonstrated. Some general theorems on convolutions of the biorthogonal distributions are proved.
Maximum Likelihood Estimation in the Three-Parameter Lognormal Distribution
Authors: Giesbrecht F. , Kempthorne O.
Published: December 2018

Several authors (Hill, 1963; Box and Cox, 1964; Harter and Moore, 1966) have discussed the problem of estimating α, μ and σ2 where observations xi, i = 1, 2, ..., n, are available such that y = log (x + α) has a normal distribution with mean μ and variance σ2. In this paper it is shown that the difficulties encountered because the “likelihood” becomes unbounded are avoided if one takes into account the grouping error inherent in the data and uses the “correct” likelihood (Kempthorne and Folks, 1971; Copas, 1972). The behaviour of the maximum likelihood estimators is investigated using asymptotic theory and Monte Carlo simulations.
Optimal Incomplete Block Designs
Authors: John J. A. , Mitchell Toby J.
Published: December 2018

The class of regular graph (RG) designs, conjectured to be optimal among all incomplete block designs, is defined. The results of a systematic search of the class of RG designs are given for all cases in which v ≤ 12, r ≤ 10 and v ≤ b. In nearly every case, the optimum design is determined for each of the three optimality criteria: A, D and E. For v > b, we present the duals of the optimal designs for v < b. Many of the designs are new and these are tabulated in a back-up report.
Qualitative Probability and Improper Distributions
Authors: Goodman T. N. T.
Published: December 2018

Axioms are given for qualitative probability that are sufficient for an exact description in a quantitative manner which admits, in particular, improper probability distributions.
Exponential Families and Conditioning on Statistics Which are Not Minimal Sufficient
Authors: Berman Mark
Published: December 2018

The likelihood function for each of k independent sets of data is assumed to belong to the two-parameter exponential family, the two parameters for the ith data set being φi, which is a nuisance parameter, and θ, which is common to all the data sets and is the parameter of interest. The theory of similar tests suggests that the appropriate method for testing θ is to condition on the statistic which is minimal sufficient for Ψ = (φ1 …, φk) when θ is known. This minimal sufficient statistic will be different depending on whether or not the φi‘s are known to be equal. The effect of assuming that the φi‘s are not equal when in fact they are (and vice versa) is investigated and some examples are given. The broad conclusion is that, provided the k samples are balanced in a certain sense, no great harm is done by conditioning on the incorrect statistic.
Finding the Outliers that Matter
Authors: Andrews David F. , Pregibon Daryl
Published: December 2018

A method is proposed for examining a linear model for the presence of one or more important outliers—deviant observations which have a potentially large influence on the resulting parameter estimates. For this purpose, a new statistic is proposed and its distributional properties are discussed. The method itself is exploratory in nature, but exact significance tests are available. Several examples illustrate the method. Computational aspects are discussed.
Non-Parametric Bayesian Analysis of Survival Time Data
Authors: Kalbfleisch John D.
Published: December 2018

A Bayesian analysis of the semi-parametric regression and life model of Cox (1972) is given. The cumulative hazard function is modelled as a gamma process. Both estimation of the regression parameters and of the underlying survival distribution are considered. The results are compared to the results obtained by other approaches.
Some Tests of Randomness Based Upon the Variance-Time Curve of the Poisson Process
Authors: Liebetrau Albert M.
Published: December 2018

In an earlier paper, the author shows that a suitably normalized estimator η^0 of the variance-time curve of a Poisson process is asymptotically normal as the time of observation of the process increases. In this paper, the distributions of the functionals ζ^1c=∫01η^c(t) and ζ^2c=∫01[η^c(t)]2 are obtained, and computing formulae for the sample analogues are provided. The first is normally distributed. For the second, which is reminiscent of the Cramer-von Mises statistic, the characteristic function, the first two moments and approximations to the upper tail of the distribution function are given. These approximations are compared with the results of a Monte Carlo study.
Inference About the Intraclass Correlation Coefficient in the Beta-Binomial Anova for Proportions
Authors: Crowder Martin J.
Published: December 2018

The paper is concerned with the analysis of sets of proportions where there is strong variation within sets. In one parameterization (Crowder, 1978) the underlying mean proportions are π1,…, πk in the k sets, and variation within sets is associated with a parameter σ2. The last of these is the intraclass correlation coefficient for the basic binary data and is of independent interest, rather than just a nuisance parameter. We propose a technique for inference about σ2.
Finitely Additive Priors, Coherence and the Marginalization Paradox
Authors: Sudderth William D.
Published: December 2018

The marginalization paradox, which arises in a Bayesian theory of inference based on improper, countable additive prior distributions, is resolved when proper, finitely additive distributions are used instead. The resolution is related to the equivalence of coherent inferences to inferences from proper, finitely additive priors under a certain definition of “coherence”.
A Trimmed Jackknife
Authors: Hinkley David , Wang Hai-Li
Published: December 2018

The jackknifed version of a regular estimate is the average of the sample pseudo-values, which in general are not independent and often have long-tailed distributions. We examine the simple alternative of the trimmed average of the pseudo-values, which is a naive type of robust estimate. The asymptotic theory is developed and two simple examples are discussed.
A Note on Maximum Likelihood Estimation for Regression Models Using Grouped Data
Authors: Burridge J.
Published: December 2018

The estimation of parameters for a class of regression models using grouped or censored data is considered. It is shown that with a simple reparameterization some commonly used distributions, such as the normal and extreme value, result in a log-likelihood which is concave with respect to the transformed parameters. Apart from its theoretical implications for the existence and uniqueness of maximum likelihood estimates, this result suggests minor changes to some commonly used algorithms for maximum likelihood estimation from grouped data. Two simple examples are given.
On the Non-Parametric Estimation of Mixture Proportions
Authors: Hall Peter
Published: December 2018

Non-parametric estimates of mixing proportions based on kernel-type density estimators are badly suited to several types of data. They suffer from aberrations due to rounding or truncation of the measurements, and their construction involves the crucial choice of the “window size”, or smoothing parameter. In many circumstances estimators based on the empiric distribution function would be more suitable, and in this paper we investigate their properties. The estimators we introduce lead in a natural way to non-parametric forms of well-known parametric estimators. Their efficiency approaches 100 per cent as the distances between the component distributions increase.
A Small Sample Property of the Cliff-Ord Test for Spatial Correlation
Authors: King Maxwell L.
Published: December 2018

The Cliff-Ord test for spatial correlation in regression disturbances is found to be a locally best invariant test.
On the Existence of Maximum Likelihood Estimators for the Binomial Response Models
Authors: Silvapulle Mervyn J.
Published: December 2018

Necessary and sufficient conditions are given for the existence of maximum likelihood estimators of the linear regression parameter in binomial response (this includes Logit and Probit) models.
Invariant Distributions Associated with Matrix Laws Under Structural Symmetry
Authors: Jensen D. R. , Good I. J.
Published: December 2018

Let Y(n x m) be a random matrix in the space F  n x m of real matrices having location-scale parameters (θ, Ω) such that (i) θ belongs to a linear subspace M of F  n x m; (ii) the distribution L(Y) exhibits a type of structural symmetry; and (iii) a certain transformation T(Y) is translation-invariant with respect to M and is invariant under changes of scale. Then the distribution L{T(Y)} is invariant for all underlying symmetric distributions L(Y). The types of symmetry are spherical symmetry and symmetry under multiplication on the left by orthogonal and similar matrices; the scale changes are multiplication by a scalar and multiplication on the right by a non-singular matrix. Applications yield the invariance under structural symmetry of distributions usually associated with normal-theory inferences in multivariate regression analysis and in testing equality of dispersion matrices.
Exact and Approximate Distributions of the Maximum Likelihood Estimator of a Slope Coefficient
Authors: Anderson T. W. , Sawa Takamitsu
Published: December 2018

When the errors are normally independently distributed with equal variance, the maximum likelihood estimator of the slope of a linear functional relationship is the slope of the line minimizing the sum of squared deviations orthogonal to the line. The exact density and distribution of this estimator are obtained. Approximate distributions are obtained, and their accuracies are discussed.
The Decomposition of Treatment Degrees of Freedom in Quantitative Factorial Experiments
Authors: Bailey R. A.
Published: December 2018

In an experiment in which every treatment factor has the same number, p, of levels, where p is prime, there is a classical breakdown of the treatment degrees of freedom into components such as AB, AB  2, AB  2  C,…, each of (p - 1) degrees of freedom. The disadvantages of this system are (i) the lack of canonical single degrees of freedom and (ii) the fact that the breakdown bears no relation to the contrasts typically of interest if all treatment factors are quantitative. In this note we decompose the treatment degrees of freedom into single degrees of freedom, and use these to show how appropriate choice of levels of quantitative factors enables quantitative contrasts to be completely unconfounded in classical designs.
Some Statistical Methods for Bivariate Circular Data
Authors: Rivest Louis-Paul
Published: December 2018

A function of the smallest singular value of the cross product matrix between two circular unit vectors is suggested as a measure of correlation. It possesses most of the desirable properties for a correlation coefficient given by Jupp and Mardia (1980). The concept of cluster dependence is introduced. A good predictor of one unit vector given the other is shown to depend on the type of dependence observed. The wrapped bivariate normal is characterized.
Saddlepoint Approximations for Permutation Tests and Confidence Intervals
Authors: Robinson J.
Published: December 2018

Asymptotic approximations for the tail probabilities of certain statistics with complex moment generating functions are obtained by the usual methods of large deviation theory. It is pointed out that this approximation should be adequate over the entire range but of particular use for extreme tails. Approximations to the significance levels for one and two sample permutation tests are obtained and these tests are inverted to obtain approximate confidence intervals. These methods are applied to some numerical examples and the results are compared with the exact values and with approximations obtained from Edgeworth expansions.