bluuebunny/crossref_metadata_2025 huggingface dataset:
On the Theory of Bilinear Time Series Models
Authors: Rao T. Subba
Published: December 2018

The theory of bilinear time series models is considered in this paper. The sufficient conditions for asymptotic stationarity of the bilinear time series models are derived, and the expressions for the variance and covariance are obtained. The conditions for the invertibility of the model are also included. The estimation of the parameters of the scalar bilinear time series model is considered. The bilinear models are fitted to sunspot numbers and also to a P-wave of a nuclear explosion. The forecasting of sunspot numbers is also considered.
Another Look at the Symmetric M-Dimensional Simple Epidemic Process
Authors: Lefevre Cl.
Published: December 2018

Simple probabilistic arguments are used to reobtain and complete results of Billard, Lacayo and Langberg (1979) for the symmetric m-dimensional simple epidemic process.
The Statistical Analysis of Compositional Data
Authors: Aitchison J.
Published: December 2018

The simplex plays an important role as sample space in many practical situations where compositional data, in the form of proportions of some whole, require interpretation. It is argued that the statistical analysis of such data has proved difficult because of a lack both of concepts of independence and of rich enough parametric classes of distributions in the simplex. A variety of independence hypotheses are introduced and interrelated, and new classes of transformed-normal distributions in the simplex are provided as models within which the independence hypotheses can be tested through standard theory of parametric hypothesis testing. The new concepts and statistical methodology are illustrated by a number of applications.
Inference for the Linear Model Using a Likelihood Based on Ranks
Authors: Pettitt A. N.
Published: December 2018

An approximation to the marginal likelihood of the ranks of a set of observations is used to analyse data. The observations, after some arbitrary monotone transformation, are assumed to be independently normally distributed with means related by a linear model. An approximate posterior distribution for the parameters of the linear model is given and used to find predictive distributions for the ordering of future observations. A numerical example is given. Appendices give mathematical details of the approximation.
Methods for Analysing Spatial Processes of Several Types of Points
Authors: Lotwick H. W. , Silverman B. W.
Published: December 2018

Two approaches are described to the analysis of spatial patterns consisting of several types of points. The first approach uses a method of asymptotically unbiased estimation of the second moment distribution; the second uses methods based on regions of empty space in the patterns. Some distributional results are given in each case, and a method of Monte Carlo testing conditional on the marginal structure is described. The methods are illustrated by being applied to some physiological data.
Assessing Sensitivity to an Unobserved Binary Covariate in an Observational Study with Binary Outcome
Authors: Rosenbaum P. R. , Rubin D. B.
Published: December 2018

This paper proposes a simple technique for assessing the range of plausible causal conclusions from observational studies with a binary outcome and an observed categorical covariate. The technique assesses the sensitivity of conclusions to assumptions about an unobserved binary covariate relevant to both treatment assignment and response. A medical study of coronary artery disease is used to illustrate the technique.
Epidemics in Competition Ii: The General Epidemic
Authors: Kendall W. S. , Saunders I. W.
Published: December 2018

Previous results on competition of epidemics are extended to the case of the general epidemic. Coupling techniques are used for the stochastic model. A result of Kurtz is used to carry the result over to the deterministic case.
On Assessing Multivariate Normality
Authors: Koziol James A.
Published: December 2018

We discuss procedures for assessing multivariate normality based on properties of radii and angles of the multivariate normal distribution, and suggest that different procedures may be efficiently combined in order to achieve omnibus tests.
On Sprent's Generalized Least-Squares Estimator
Authors: Mak Tak K.
Published: December 2018

This paper studies the asymptotic properties of Sprent's (1966) generalized least-squares estimator of the slope parameter in a functional relationship model which allows errors at different data points be correlated. Consistency is established and the asymptotic variance derived. When the Joint distribution of the errors is normal, in which case Sprent's estimator coincides with the maximum likelihood estimator, the asymptotic variance formula is further simplified and the limiting distribution of the estimator is also shown to be normal. The variance expression obtained under the normality assumption differs from that of Dolby (1972) derived by inverting the information matrix, and the latter procedure is known to be generally invalid in functional relationship problems due to the presence of incidental parameters (Patefield, 1978). An estimator of the asymptotic variance is also suggested.
A Unified Method for Constructing PBIB Designs Based on Triangular and L
2 Schemes
Authors: Cheng Ching-Shui , Constantine G. M. , Hedayat A. S.
Published: December 2018

A general and simple method for constructing triangular and L  2 type partially balanced incomplete block (PBIB) designs is presented. This method unifies several well-known methods and produces many new series of designs. The idea is based on a well-known relation between triangular and L  2 type PBIB designs and line graphs. Triangular or L  2 type PBIB designs can often be obtained by suitably “symmetrizing” a given initial block.
Robust Non-Parametric Function Fitting
Authors: HÄrdle W. , Gasser T.
Published: December 2018

A robust non-parametric function fitting method is introduced. The estimate is motivated from the theory of M-estimation and of kernel estimation of regression functions. Consistency and asymptotic normality are shown. Bias and variance rates are the same as those previously obtained by Gasser and Müller (1979) for linear smoothers. The estimate satisfies a minimax property, i.e. it minimizes the maximal asymptotic variance as the error distributions vary over a suitable contamination neighbourhood.
Estimation of the Mean of a Selected Negative Exponential Population
Authors: Sackrowitz Harold , Samuel-Cahn Ester
Published: December 2018

Let Xi be independent, negative exponentially distributed with unknown expectation λi, i = 1, …, k. Let J and M denote the (random) indices of the maximal and minimal observation, respectively. The paper considers estimation of the random quantities λJ and λM. The loss function is (a – λM)2/λ2  M for λM, and correspondingly for λJ. UMVUE and minimax estimators are shown to exist and are exhibited, and certain other reasonable estimators are discussed.
On the Admissibility of the Regression Estimator
Authors: Bellhouse D. R. , Joshi V. M.
Published: December 2018

Admissibility of the regression estimator is established for two types of sampling designs: single stage sampling using probability proportional to a size variable and double sampling.
A Predictive Model Selection Criterion
Authors: Martini A. San , Spezzaferri F.
Published: December 2018

A procedure for model selection is presented which chooses the model that gives the best prediction of the future observation. The predictive distributions associated with each model are compared by means of the logarithmic utility function. For two nested normal linear models, the choice criterion is the product of the posterior odds ratio and a factor depending on the design point of the future observation. Taking appropriate mean values, a criterion is obtained which is independent of the particular design point. This average criterion differs from the ones proposed by Akaike, Schwarz and others in that it adjusts the likelihood ratio statistic by taking into account not only the difference in dimensionality, but also the estimated distance of the two models.
Quasi-Complete Latin Squares: Construction and Randomization
Authors: Bailey R. A.
Published: December 2018

A general method for constructing quasi-complete Latin squares based on groups is given. This method leads to a relatively straightforward way of counting the number of inequivalent quasi-complete Latin squares of side at most 9. Randomization of such designs is discussed, and an explicit construction for valid randomization sets of quasi-complete Latin squares whose side is an odd prime power is given. It is shown that, contrary to common belief, randomization using a subset of all possible quasi-complete Latin squares may be valid while that using the whole set is not.
A Conflict between Two Axioms for Combining Subjective Distributions
Authors: Genest Christian
Published: December 2018

The problem of how to combine the probability distributions of a number of individuals is examined. When two intuitively appealing axioms of group rationality are used to determine the pooling procedure, a generally unacceptable consensus arises which ignores all but one of the opinions expressed.
Jackknifing Differentiable Statistical Functionals
Authors: Parr William C.
Published: December 2018

The purpose of this paper is to give a general unified view of “how smooth” a statistic needs to be for the jackknife to work. We give general sufficient conditions for the jackknife to produce consistent bias and variance estimators, and for linear combinations of the pseudovalues to be asymptotically normal. Regularity conditions are stated in terms of the existence of strong Fréchet differentials of the functional version of the statistic being jackknifed.
Kernels for Nonparametric Curve Estimation
Authors: Gasser T. , MÜller H-G. , Mammitzsch V.
Published: December 2018

The choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated. Explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error, IMSE (the present proof of the optimality of the latter kernels is restricted up to order k = 5). These kernels are also of interest for the nonparametric estimation of probability densities and spectral densities. A finite sample study indicates that higher order kernels – asymptotically improving the rate of convergence – may become attractive for realistic finite sample size. Suitably modified kernels are considered for estimating at the extremities of the data, in a way which allows to retain the order of the bias found for interior points.
Using Bricks to Build Block Designs
Authors: Jones Byron
Published: December 2018

A method is suggested for constructing incomplete block designs in which uncorrelated estimates of contrasts of interest have variances in a specified ratio. The designs are built up by using sets of blocks called bricks. Many of the constructed designs have blocks of different sizes. The choice between competing designs is made by choosing the one that maximizes the weighted sum of the effective replications. An example is given to illustrate the method.
Stochastic Models of Computer Communication Systems
Authors: Kelly F. P.
Published: December 2018

This paper describes some examples of the stochastic models found useful in the design and analysis of advanced computer and communication systems. Our major theme might be termed the control of contention. As illustrations of this theme we discuss concurrency control procedures for databases, dynamic channel assignment for cellular radio, and random access schemes for the control of a broadcast channel. We emphasize asymptotic properties of product-form distributions and we present some new results on the stability of acknowledgement based random access schemes.
Response Time Problems in Communication Networks
Authors: Mitrani I.
Published: December 2018

Two problem areas are discussed. The first concerns the distribution of sojourn, or response times in networks which allow overtaking. Exact results for a few special cases are presented. The second problem area has to do with determining the average response times in local area networks under particular communication protocols. Models which do not lend themselves to exact solution are studied approximately.
Scheduling and Characterization Problems for Stochastic Networks
Authors: Whittle P.
Published: December 2018

Optimal design rules are developed for a Jackson network; these suggest an adaptive routing rule for the network. A concept of weak coupling is developed, which characterizes and greatly generalizes what seems to be the essential property of a Jackson network.
Asymptotic Normality of Goodness-Of-Fit Statistics for Sparse Product Multinomials
Authors: Dale Jocelyn R.
Published: December 2018

The asymptotic distribution of the Pearson X  2 and likelihood ratio G  2 goodness-of-fit statistics is derived for k independent, non-identically distributed multinomials, as k approaches infinity, with expected cell frequencies bounded. The application is to large datasets of counts with many zero cells. Conditions are given under which the limiting distributions are normal for originating probabilities (i) known and (ii) dependent on a common, finite-dimensional parameter. A logistic regression model is given which satisfies (ii). Finally, skewness is discussed as a measure of closeness of moderate upper percentage points of X  2 to normal percentiles.
On the Rate of Convergence of Orthogonal Series Density Estimators
Authors: Hall Peter
Published: December 2018

We derive a lower bound to the rate of convergence of nonparametric density estimators on compact sets, and show that it is achieved by estimators based on orthogonal series. Unlike earlier work on this type of problem, the bound depends on the size of Fourier coefficients rather than explicitly on smoothness assumptions.
Orientation Statistics Without Parametric Assumptions
Authors: Prentice Michael J.
Published: December 2018

Maximum likelihood estimation using the matrix von Mises-Fisher distribution in orientation statistics leads to unacceptably complicated likelihood equations, essentially because of the inconvenient form of the normalizing constant in the probability distribution. For the case of 3 × 2 or 3 × 3 orientations, the main cases of practical importance, an alternative approach is developed here by transforming the data into unsigned four-dimensional directions and using known results on the sampling properties of the spectral decomposition of the resulting sample moment of inertia matrix. It is demonstrated that the necessary computations are relatively simple by applying some of the techniques to a set of vectorcardiogram data.
Recovery of Row and Column Information in Row-Column Designs with Adjusted Orthogonality
Authors: Eccleston J. A. , John J. A.
Published: December 2018

The analysis of row-column designs which satisfy the property of adjusted orthogonality is considered. For such designs a series of results are derived which relate the information matrix, its inverse, the estimator of the treatment effects and the adjusted treatment sum of squares to those of its component designs. It is also shown that estimators of the stratum variances required for the recovery of information from between rows and between columns can be obtained by a relatively simple method.
Consistency of Deviance-Based M Estimators
Authors: Lenth Russell V. , Green Peter J.
Published: December 2018

In a general estimation problem, the deviance function generates statistics that are similar to squared standardized residuals. A deviance-based M estimator (DBME) is defined as an adaptively weighted maximum-likelihood estimator, where the weights depend upon the deviances. In both a single-parameter and a regression setting, we give some general conditions under which a DMBE is consistent. For a suitable weighting scheme, these conditions are satisfied in many continuous Cramer-Rao-regular families and in related linear or nonlinear regression cases. The conditions fail (and the estimator is inconsistent) in most discrete families.
Multiplicative Errors: Log-Normal or Gamma?
Authors: Firth David
Published: December 2018

Estimation in regression models with multiplicative error is commonly based on either the log-normal or the gamma likelihood. When the regression effects are also multiplicative, the gamma maximum likelihood estimate is found to be the more efficient of the two, but only slightly so, under reciprocal misspecification.
Saddlepoint Expansions for Directional Test Probabilities
Authors: Skovgaard Ib M.
Published: December 2018

An expansion is derived for the conditional tail probability of a multivariate random variable given its direction from the expected value. In particular, if the score function of a statistical model is chosen as this variable, such a conditional probability gives a test for a simple hypothesis, which is asymptotically equivalent to the likelihood ratio test. The expansion is of the large deviation type and is derived through the use of saddlepoint methods. Thus the relative error of the test probability is O(n  –1) uniformly in a bounded set for an average of n independent replications. The approximation is based on the cumulant transform for the random variable, but is otherwise a simple expression in terms of chi-squared distributions. In the one-dimensional case it reduces to the expansion obtained by Lugannani and Rice. Numerical examples show an excellent fit, comparable with other saddlepoint expansions, when the dimension is low, even for very small sample sizes, but for higher dimensions more replications are required to give a similar approximation.
A Note on the Existence of Maximum Likelihood Estimates in Linear Regression Models Using Interval-Censored Data
Authors: Hamada M. , Tse S. K.
Published: December 2018

For a class of linear regression models with interval-censored data, including the exponential regression model as a special case, a necessary and sufficient condition is given for the existence of the maximum likelihood estimator (MLE). The condition is especially simple to check for simple linear regression. This result is useful for comparing the small sample properties of MLEs from different special cases of interval-censored data.
A Review of Bootstrap Confidence Intervals
Authors: Diciccio Thomas J. , Romano Joseph P.
Published: December 2018

A survey of bootstrap procedures for constructing confidence regions is given. In particular, several distinct bootstrap methods are considered, with emphasis on the mathematical correctness of these procedures. The percentile, bias-corrected percentile and accelerated bias-corrected percentile methods, developed by Efron, are reviewed in both parametric and nonparametric situations. A procedure related to the accelerated bias-corrected method, which avoids explicit calculation of the analytical corrections required in Efron's method, is also introduced. In the context of a functional approach for the construction of confidence regions, the bootstrap is motivated as a method to estimate the distributions of approximate pivots. Finally, iterative bootstrap methods are discussed as means to improve coverage accuracy.
Bayes Methods for Outliers in Exponential Samples
Authors: Pettit L. I.
Published: December 2018

A Bayesian approach to the modelling of outliers is presented and examined in relation to members of the exponential family in general, and the exponential distribution in particular. The approach is appropriate when the analyst's prior knowledge is such that any aberrant observations will be surprising with high probability. It is shown that the most surprising observation, as measured by the predictive ordinate, must lie on the ‘outside’ of the sample. Bayes factors for models with and without outliers are given for exponential samples. It is shown that masking and swamping cannot occur. An exponential regression problem is considered.
On the Distribution of a Studentized Quantile
Authors: Hall Peter , Sheather Simon J.
Published: December 2018

The asymptotic variance of a sample quantile depends on the value of the population density at the population quantile. Therefore Studentizing a sample quantile involves density estimation, either explicitly or implicitly. One popular way of Studentizing is to use the Siddiqui–Bloch–Gastwirth estimator, whose construction depends crucially on the choice of a smoothing parameter m. We examine the effect which the selection of m has on the level error of tests or confidence intervals based on Studentized quantiles and show that, if we wish to minimize this error, m should be of a smaller order of magnitude than is recommended by squared error theory. The cases of one- and two-sided procedures are distinctly different, the former being less sensitive to the choice of m.
Bahadur Efficiency of Two-Sample Tests Based on One-Sample Sign Statistics
Authors: Tableman Mara , Hettmansperger Thomas P.
Published: December 2018

This paper discusses the two-sample test of location based on the comparison of two distribution-free one-sample confidence intervals derived from sign statistics. This test procedure, first introduced by Hettmansperger (1984), rejects the null hypothesis of equal population medians when the two intervals are disjoint. He presents three different ways to select the two one-sample intervals and one choice leads to Mood's test. All solutions have the same Pitman efficiency. This paper shows that the choices can be distinguished on the basis of Bahadur's efficiency. We formulate the problem in terms of (asymptotically) fixed width confidence intervals. In this context various median tests (including Mood's test) arise as special cases and they yield different performance. The solution that specifies equal asymptotic lengths for the one-sample intervals (which is different from Mood's test) is recommended.
Estimating Weighted Integrals of the Second-Order Intensity of a Spatial Point Process
Authors: Berman Mark , Diggle Peter
Published: December 2018

This paper considers the nonparametric estimation of the integral J=∫0Ttλ2(t)φ(t)dt

where λ2(t) is the unknown second-order intensity function of a two-dimensional stationary isotropic point process observed in some region and φ(t) is known for t ϵ [0, T]. An unbiased estimator of J is derived, and a computationally fast approximation to it is proposed. The estimator is then used to obtain a kernel method for smoothing point process data, a new estimator of the Fourier transform of the second-order intensity and some tests for spatial association between a point process and another stochastic process.
An Affine Invariant Bivariate Version of the Sign Test
Authors: Brown B. M. , Hettmansperger T. P.
Published: December 2018

The generalized median of Oja yields a notion of bivariate quantile and, in turn, an affine invariant bivariate analogue of the sign test. its properties include a simple null covariance formula, facilitating a permutation or sign change test in the case of bivariate symmetry, normal efficiency coinciding with that of the Oja median and bounded influence, and hence strong robustness.
An Exact Decomposition Theorem for a Sample from the Three-Dimensional Hyperboloid Distribution
Authors: Massam Hélène
Published: December 2018

Consider a sample (v  1, ..., vn) from the three-dimensional hyperboloid distribution. The joint probability distribution function (PDF) of (v  1 ..., vn) can be expressed as the product of the PDF of n independent quantities. Each of these quantities has an exponential κ distribution. The splitting is done in such a way that the data points vi are incorporated into the successive quantities one at a time. A similar result has been proved in the literature for the normal and the inverse Gaussian distributions. Since the normal, gamma and inverse Gaussian distributions are the only one-dimensional distributions which are reproductive and are equal to their saddlepoint approximation, a parallel result was to be expected for the gamma distribution. We also prove this result and an application to recursive testing is presented.
Optimal Screening Methods
Authors: Turkman K. F. , Turkman M. A. Amaral
Published: December 2018

Optimal screening methods from a predictive point of view and a decision theoretical approach are devised and are shown to have a similar form. In the multivariate normal case the optimal screening region is found to be a quadratic function of the screening variables. The results are illustrated by an example.
Design of Experiments with Both Mixture and Qualitative Factors
Authors: Donev A. N.
Published: December 2018

The paper establishes the conditions for D optimality of designs with mixture and qualitative factors. The solutions of the optimization problem for exact designs consist of several D-optimum designs: the one to be chosen can therefore also satisfy other optimality conditions. The problem of blocking of mixture experiments is also discussed.
Quasisymmetric Models for the Analysis of Square Contingency Tables
Authors: Becker Mark P.
Published: December 2018

A parametric model is presented for the analysis of square contingency tables where there is a one-to-one correspondence between the categories of the row variable and the categories of the column variable. The model is applicable to cross-classifications where the categories are unordered, partially ordered or completely ordered, but the interpretation of the model and associated parameters is most straightforward when the categories are at least partially ordered. Connections with the mover-stayer model and the usual model of quasisymmetry are described. Two data sets are used to illustrate the utility of the model.
Approximate Interval Probabilities
Authors: Barndorff-Nielsen O. E.
Published: December 2018

For models of parametric dimension 1 two parameterization invariant approximations to the conditional distribution function of the maximum likelihood estimator are considered. The first is derived by partial integration of a formula for the conditional density given an ancillary statistic (the p*-formula), a technique which in addition yields a useful expression for the error term. For exponential models of order 1 the approximation specializes to the Lugannani–Rice formula. The second approximation is based on an adjusted version (denoted r*) of the signed log-likelihood ratio statistic, which was also derived from p*. These two approximations generally agree closely, even in the far tails of the distribution. A third approximation is a modification of the first that avoids the specification of an ancillary. Some numerical comparisons are given.
A Graphical Procedure for Comparing Goodness-Of-Fit Tests
Authors: Mudholkar Govind S. , Kollia Georgia D. , Lin C. Thomas , Patel Kartik R.
Published: December 2018

A method for comparing tests of normality using their ‘isotones‘, i.e. contours on the surfaces representing p-values, is proposed. Select a family of distributions labelled by two parameters λ1 and λ2 to represent the null and alternative hypotheses. For each (λ1, λ2) take an ‘ideal sample’ from that distribution, calculate the value of the goodness-of-fit statistic and determine the corresponding P-value under the normality assumption. This generates a surface of the p-values of the ‘values of the test statistic' in the (λ1, λ2) plane. In this paper the construction and interpretation of the isotones, the contours on the above surfaces, for competing tests, is illustrated using the Shapiro–Wilk W, Vasicek's entropy test and Lin and Mudholkar's Zp-test of the composite hypothesis of normality.
Asymptotic Properties of Some Estimators for the Infection Rate in the General Stochastic Epidemic Model
Authors: Rida Wasima N.
Published: December 2018

Some estimators for the infection rate in the general stochastic epidemic model are presented. The first estimator follows the approach of maximum likelihood. However, this approach requires us to observe most if not all of the epidemic process. As an alternative, an estimator which uses less detailed data is derived from a suitable martingale. Both estimators are shown to be consistent only when a major outbreak of disease is observed. Minor out-breaks do not provide enough information on the infection rate. The asymptotic distributions of these estimators are also obtained. Asymptotic normality holds only for a major outbreak. Finally, the estimators are applied to data on a smallpox outbreak.
Weighted Kaplan-Meier Statistics: Large Sample and Optimality Considerations
Authors: Pepe Margaret Sullivan , Fleming Thomas R.
Published: December 2018

We propose a cumulative weighted difference in the Kaplan-Meier estimates as a test statistic for equality of distributions in the two-sample censored data survival analysis problem. For stability of such a statistic, the absolute value of the possibly random weight function must be bounded above by a multiple of (C  −)1/2 + δ where 1 – C  − is the left continuous censoring distribution function and δ > 0. For these weighted Kaplan–Meier (WKM) statistics, asymptotic distribution theory is presented along with expressions for the efficacy under a sequence of local alternatives. A simple censored data generalization of the two-sample difference in means test (z-test) is a member of this class and in large samples is seen to be quite efficient relative to the popular log-rank test under a range of alternatives including the proportional hazards alternative. Optimal weight functions are also calculated. The optimal WKM statistic is as efficient as the optimal weighted log-rank statistic for any particular sequence of local alternatives. Stratified statistics and trend statistics are also presented.
Error Model Diagnostics in the General Linear Model Relevant to the Analysis of Repeated Measurements and Field Experiments
Authors: Cullis B. R. , McGilchrist C. A. , Gleeson A. C.
Published: December 2018

The large sample distribution of the recursive residual maximum likelihood residuals from the general linear model with dependent errors is derived. Further results are obtained from this distribution to aid the identification of an appropriate error model in the general linear model.
Sound Confidence Intervals in the Heteroscedastic Linear Model Through Releveraging
Authors: Dorfman Alan H.
Published: December 2018

Under heteroscedasticity, ordinary least squares regression can fail to yield adequate inference on parameter coefficients with respect to hypothesis testing or confidence intervals. For example, confidence intervals can have coverage well below that claimed. This is especially the case in small or moderate-sized imbalanced samples and holds even if the ordinary least squares variance estimator is replaced by a robust-against-heteroscedasticity variance estimator. A new simple weighting scheme corrects this problem, although at possibly serious cost in efficiency. Inefficient methods have been rated as useful at the data exploration stage of analysis. The present method is also useful as an adjunct to least squares regression, at the stage of regression diagnostics. As such, it can sometimes replace the need for them altogether.
Minimum Distance Methods Based on Quadratic Distances for Transforms in Simple Linear Regression Models
Authors: Luong Andrew
Published: December 2018

Minimum distance estimators and test statistics based on quadratic distances for transforms are introduced in simple linear regression models. Asymptotic properties of quadratic distance estimators and tests for the error distributions are derived. The modified quadratic distance estimator is shown to be efficient provided that the error distribution is symmetric and the derivative of the logarithm of the error density is a linear combination of the transforms.
Bias Correction in Generalized Linear Models
Authors: Cordeiro Gauss M. , McCullagh Peter
Published: December 2018

In this paper we derive general formulae for first-order biases of maximum likelihood estimates of the linear parameters, linear predictors, the dispersion parameter and fitted values in generalized linear models. These formulae may be implemented in the GLIM program to compute bias-corrected maximum likelihood estimates to order n  −1, where n is the sample size, with minimal effort by means of a supplementary weighted regression. For linear logistic models it is shown that the asymptotic bias vector of β^ is almost collinear with β. The approximate formula βp/m+ for the bias of β^ in logistic models, where p = dim(β) and m+ = ∑ mi is the sum of the binomial indices, is derived and checked numerically.
Multivariate Mean Parameter Estimation by Using a Partly Exponential Model
Authors: Zhao Lue Ping , Prentice Ross L. , Self Steven G.
Published: December 2018

A class of partly exponential models is proposed for the regression analysis of multivariate response data. The class is parameterized in terms of the response mean and a general shape parameter. It includes the generalized linear error model and exponential dispersion models as special cases. Maximum likelihood equations for mean parameters are shown to be of the same form as certain generalized estimating equations, and maximum likelihood estimates of mean and shape parameters are asymptotically independent. Some results are given on the efficiency of the estimating equation procedure under misspecification of the response covariance matrix.
Semiparametric Additive Regression
Authors: Cuzick Jack
Published: December 2018

A simple estimator for β is proposed for the model y = x'β + g(t)+ error, g smooth but unknown. The approach is to approximate the estimating equation obtained from a ***semiparametric likelihood and in the simplest case reduces to minimizing the distance between the 'pseudoresiduals' y - x'β and a local linear cross-validated estimate of them. When the errors are independent with finite variance, the bias and variance of the estimate are computed and compared against the least squares estimate with g known. It is shown that n  1/2-consistent estimates are possible when the bandwidth contains only a finite number of points and their asymptotic relative efficiency is computed in special cases. When the error distribution is known up to a scale factor and the design points (X, T) are assumed to be drawn as independent samples from a bivariate distribution, the Hajek-LeCam lower bound for the variance of any regular estimator of β is computed and a sequence of estimators is exhibited which achieves this bound. When X and T are independent these estimators do as well asymptotically as the maximum likelihood estimator with g known.
Confidence Sets Having the Shape of a Half-Space
Authors: Perron FranÇois
Published: December 2018

For the problem of estimating the mean of a p-dimensional normal distribution, p > 1, confidence regions based on half-spaces bounded by a hyperplane having the vector of observations as normal are proposed. Confidence regions with exact probability of coverage are constructed. Tables are provided.
Maximum Entropy and the Nearly Black Object
Authors: Donoho David L. , Johnstone Iain M. , Hoch Jeffrey C. , Stern Alan S.
Published: December 2018

Maximum entropy (ME) inversion is a non-linear inversion technique for inverse problems where the object to be recovered is known to be positive. It has been applied in areas ranging from radio astronomy to various forms of spectroscopy, sometimes with dramatic success. In some cases, ME has attained an order of magnitude finer resolution and/or an order of magnitude smaller noise level than that obtainable by standard linear methods. The dramatic successes all seem to occur in cases where the object to be recovered is ‘nearly black‘: essentially zero in the vast majority of samples. We show that near-blackness is required, both for signal-to-noise enhancements and for superresolution. However, other methods—in particular, minimum l  1-norm reconstruction—may exploit near-blackness to an even greater extent.
Jackknife-After-Bootstrap Standard Errors and Influence Functions
Authors: Efron Bradley
Published: December 2018

This paper shows how to derive more information from a bootstrap analysis, information about the accuracy of the usual bootstrap estimates. Suppose that we observe data x = (x  1  x  2, . . ., xn), compute a statistic of interest s(x) and further compute B bootstrap replications of s, say s(x*1) s(x*2), . . ., s(x*B), where B is some large number like 1000. Various accuracy measures for s(x) can be obtained from the bootstrap values, e.g. the bootstrap estimates of standard error and bias, or the length and shape of bootstrap confidence intervals. We might wonder how accurate these accuracy measures themselves are, or how sensitive they are to small changes in the individual data points xi. It turns out that these questions can be answered from the information in the original bootstrap sample s*  1  s*  2, . . ., s*B, with no further resampling required. The answers, which make use of the jackknife and delta method influence functions, are easy to apply and can give informative results, as shown by several examples.
Approximate Bayes Factors and Orthogonal Parameters, with Application to Testing Equality of Two Binomial Proportions
Authors: Kass Robert E. , Vaidyanathan Suresh K.
Published: December 2018

We use asymptotic expansions to approximate Bayes factors, improving on a method used by Jeffreys. Suppose that the hypothesis H  0: ψ = ψ0 is to be tested against H  A: ψ ≠ ψ0 in the presence of a nuisance parameter β, and initially priors π0(β) under H  0 and π(β, ψ) under H  A are used. We consider the problem of assessing sensitivity of the Bayes factor to small changes in π0 and π. We show that for local alternatives (which, for moderate sample sizes, are consistent with small or moderately large values of the Bayes factor in favour of the alternative), if β and ψ are what we call ‘null orthogonal’ parameters, then alterations in π0 have no effect on the Bayes factor up to order O(n  –1). Under similar conditions we also derive an order O(n  –1) approximation to the minimum Bayes factor over all priors π under H  A such that the marginal prior on ψ is normal with mean ψ0. We then go on to consider sensitivity to specific changes in the marginal prior on ψ and show how asymptotics may be used for this, applying a second-order approximation due to Tierney and Kadane. We illustrate the results with a test of equality of two binomial proportions and briefly investigate the accuracy of the approximations is this context.
Testing for Parameter Variation in Non-Linear Regression Models
Authors: McCabe B. P. M. , Leybourne S. J.
Published: December 2018

This paper addresses the problem of testing for purely random parameter variation in nonlinear regression models. Based on different approximations to the true density of the data, score-type tests are constructed and their asymptotic distributions are derived. The local power of the tests is investigated both theoretically and via Monte Carlo simulation. An empirical testing example, involving a well-known non-linear aggregate demand for money function, is also given.
Convergence Rates of the Gibbs Sampler, the Metropolis Algorithm and Other Single-Site Updating Dynamics
Authors: Frigessi Arnoldo , Stefano Patrizia , Hwang Chii-Ruey , Sheu Shuenn-Jyi
Published: December 2018

Sampling from a Markov random field II can be performed efficiently via Monta Carlo methods by simulating a Markov chain that converges weakly to II. We consider a class of local updating dynamics that are reversible with respect to II. It includes the Metropolis algorithm (MA) and the Gibbs sampler (GS). We investigate the speed of weak convergence of these Markov chains in terms of their second-largest eigenvalues in absolute value. We study the general algebraic structure and then the stochastic Ising model in detail. We conclude the following: the GS is faster than locally updating twice by the MA; in the Ising case, the MA is the best at low temperature but the worst at high temperature; there are dynamics faster than the GS at high temperature. The results clear up some intuitive misconceptions.
Nearly Orthogonal Randomized Designs
Authors: Hooper Peter M.
Published: December 2018

A new method of experimental randomization is presented that is suitable under additive general linear models for treatment and unit effects. The method is designed for a standard normal theory analysis of the data. The method is motivated by thinking of the randomization strategy, which may be represented as a distribution on the group of n × n permutation matrices, as a discrete approximation to a fictitious, ideal randomization strategy corresponding to a continuous distribution on the group of n ×  n orthogonal matrices. Designs are generated by using a simulated annealing algorithm. The method requires the selection of a tuning constant that determines a trade-off between validity and efficiency. The tuning constant is chosen with the aid of a new diagnostic measure of validity. The ideas are illustrated by using continuous, piecewise linear models of potential use in field experiments.
On the Empirical Validity of the Bayesian Method
Authors: Vovk V. G. , V'Yugin V. V.
Published: December 2018

The ideas in Kolmogorov's programme for algorithmic substantiation of applications of probability make it possible to define a measure of disagreement between the probability distribution representing the attitude of a coherent individual towards a random experiment and the outcome of the experiment. When there is agreement we say that the probability distribution is empirically valid. We prove quantitatively that formulae of Bayesian statistics transform empirically valid probability distributions into other empirically valid distributions.
Computing Theoretical Autocovariances of Multivariate Autoregressive Moving Average Models by Using a Block Levinson Method
Authors: Mittnik Stefan
Published: December 2018

A closed form expression relating the theoretical autocovariances of a multivariate autoregressive moving average (ARMA) model to the ARMA parameters in terms of a system of linear equations has been presented recently. Using this result and the special Hankel plus Toeplitz structure of the coefficient matrix obtained after suitable partitioning, this paper presents a method for computing multivariate autocovariance functions that is based on a block Levinson procedure. The method is particularly advantageous for models with high order autoregressive components and/or a large number of variables.
A Fourier Approach to Nonparametric Deconvolution of a Density Estimate
Authors: Diggle Peter J. , Hall Peter
Published: December 2018

We consider the problem of constructing a nonparametric estimate of a probability density function h from independent random samples of observations from densities a and f, when a represents the convolution of h and f. Our approach is based on truncated Fourier inversion, in which the truncation point plays the role of a smoothing parameter. We derive the asymptotic mean integrated squared error of the estimate and use this formula to suggest a simple practical method for choosing the truncation point from the data. Strikingly, when the smoothing parameter is chosen in this way then in many circumstances the estimator behaves, to first order, as though the true f were known. Simulated data sets are used to demonstrate that the method can give good results in practice.
Interpolated Nonparametric Prediction Intervals and Confidence Intervals
Authors: Beran Rudolf , Hall Peter
Published: December 2018

In several important statistical problems, prediction intervals and confidence intervals can be constructed with coverage levels which are known precisely but cannot be rendered equal to predetermined levels such as 0.95. One solution to this difficulty is to interpolate between such intervals. We show that simple linear interpolation reduces the order of coverage error, but that higher orders of interpolation produce no further improvement. The error is reduced by a factor n  −1 for prediction intervals and n  −1/2 for confidence intervals, where n denotes sample size. In the case of confidence intervals for quantiles, linear interpolation provides particularly accurate intervals which err on the side of conservatism.
Shape Analysis for the Anisotropic Corpuscle Problem
Authors: Girling A. J.
Published: December 2018

The problem of inferring the shape distribution of anisotropically oriented spheroidal particles from their elliptical traces in planar samples is addressed by generalizing a size-shape independence condition suggested by Wicksell in his original discussion of the corpuscle problem. Shape statistics are introduced through which the eccentricities of the sampled ellipses may be related to some suitably defined moments of the underlying particle shape distribution, taking account of particle orientation. Practical methods which employ bootstrap resampling of shape statistics are developed with reference to data from a recent neurological study.
Systematic Row-And-Column Designs Balanced for Low Order Polynomial Interactions between Rows and Columns
Authors: Edmondson R. N.
Published: December 2018

Systematic row-and-column designs intended to accommodate low order polynomial rows × columns interaction effects are discussed. Latin square, Graeco-Latin square and quasi-Latin square designs of size 2N × 2M and size 9 × 9 are examined by using plot pseudofactor methods. Latin square and Graeco-Latin square designs of size 5 × 5 and 7 × 7 are examined by using a published technique for the design of quantitative factorial experiments with a prime number of factor levels. A 6 × 6 Latin square design is constructed by trial and error. Finally, the control of positional effects by fitted quadratic response surfaces is discussed.
A Contamination Model and Resistant Estimation Within the Sampling Paradigm
Authors: Titterington D. M.
Published: December 2018

Copas's model for contamination in binary regression structures is translated into the sampling paradigm approach that is familiar in discriminant analysis. A parallel treatment of the two models is afforded by a common expression of the log-likelihood. Resistant parameter estimation is discussed for prevalence rates and for parameters, including discriminant functions, within component distributions. The methods are illustrated on a breast cancer example.
High Breakdown Methods of Time Series Analysis
Authors: Tatum Lawrence G. , Hurvich Clifford M.
Published: December 2018

A robust form of the discrete Fourier transform is developed that can handle large amounts of contamination and patchy outliers. We use robust regression to fit a sine and cosine coefficient at each Fourier frequency, and these coefficients are then inverse Fourier transformed to give a filtered version of the data. The filtered series can then be analysed with conventional methods. The limiting breakdown bound of the filter is 50%. Other properties of the filter are also given. The performance of our method is compared by a Monte Carlo study with that of a data cleaner of Martin and Thomson. A comparison of the methods, including an outlier detection procedure, is also done by using a real data set with patchy outliers.
Approximate Non-Gaussian Bayesian Estimation and Modal Consistency
Authors: Gordon N. J. , Smith A. F. M.
Published: December 2018

A new recursive estimation procedure is proposed for the location of a dynamic linear model with non-normal errors. The procedure is a modification of a modal approximation algorithm, which is shown to be prone to instabilities. The modification is motivated by a notion of posterior modal consistency.
Bounded Influence Rank Regression
Authors: Naranjo J. D. , Hettmansperger T. P.
Published: December 2018

When εi = yi – x′i β, it is known that minimizing ΣΣ|εi – εj| yields an estimate of regression that attains a bounded influence of the residual with 95% efficiency for the normal distribution. We show that introducing weights ΣΣbij|εi – εj| achieves bounded total influence with positive breakdown. Mallows weights in particular are optimally efficient under a predefined bound on the gross error sensitivity. A generalization of Mallows weights allows additional local stability against high leverage points. Two numerical examples illustrate the behaviour of the estimate.