THE FUNDAMENTAL FORMULA OF BLASCHKE IN INTEGRAL GEOMETRY AND GEOMETRICAL PROBABILITY, AND ITS ITERATION, FOR DOMAINS WITH FIXED ORIENTATIONS

The so-called “principal formulae” of planar integral geometry are conventionally couched in terms of the “kinematic density”dxdydθ. Here a corresponding theory with respect to the “Lebesgue density”dxdy, that is with rotations suppressed, is developed. The only real difference is that the new “fundamental formula of Blaschke”contains a term depending upon the relative orientations of the two domains involved. In particular, the remarkable iteration property of these formulae carries over. The usual principal formulae follow as a corollary of the formulae given here, upon, averaging over orientations.     

Defining the linear portion of a sigmoid‐shaped curve: bend points

Formulae are provided that define the ‘bend points’, the beginning and end of the essentially linear dose–response region, for the four‐parameter logistic model. The formulae are expressed in both response and dose units. The derivation of the formulae is shown in order to illustrate the general nature of the methodology. Examples are given that describe how the formulae may be used while planning and conducting bioassays. Copyright © 2003 John Wiley & Sons, Ltd.     

Information loss on the mean for spatial processes when some values are missing

The loss of information on the mean due to the presence of missing values is discussed for a Gaussian univariate process on a rectangular lattice. The exact as well as the approximate formulae for this loss are given for general conditional autoregressive (CAR) and simultaneous autoregressive (SAR) processes. The formulae are evaluated for some low order CAR and SAR processes. The approximate formula is shown to give a good insight into how the loss varies over the different configurations of missing sites.     

Choosing Sampling Effort for Recapture Experiments with Behavioural Response

One of the main aims of a recapture experiment is to estimate the unknown size, N of a closed population. Under the so-called behavioural model, individual capture probabilities change after the first capture. Unfortunately, the maximum likelihood estimator given by Zippin (1956) may give an infinite result and often has poor precision. Chaiyapong & Lloyd (1997) have given formulae for the asymptotic bias and variance as well as for the probability that the estimate is infinite.
The purpose of this article is to tabulate the inversions of the above cited formulae so that practitioners can plan the required capture effort. This paper develops simple approximations for the minimum capture effort required to achieve (i) no more than a certain probability of breakdown, (ii) a given relative standard error.     

A Family Which Integrates the Generalized Charlier Series and Extended Noncentral Negative Binomial Distributions

The generalized Charlier series distribution includes the binomial distribution, and the noncentral negative binomial distribution extends the negative binomial distribution. The present article proposes a family of counting distributions, which contains both the generalized Charlier series and extended noncentral negative binomial distributions. Compound and mixture formulations of the proposed distribution are given. The probability mass function is expressible in terms of the confluent hypergeometric function as well as the Gauss hypergeometric function. Recursive formulae for probability mass function have been studied by Panjer, Sundt and Jewell, Schröter, Sundt, and Kitano et al. in the context of insurance risk. This article explores horizontal, vertical, triangular, and diagonal recursions. Recursive formulae as well as exact expressions for descending factorial moments are studied. The proposed distribution allows overdispersion or underdispersion relative to a Poisson distribution. An illustrative example of data fitting is given.     

BARTLETT CORRECTED LIKELIHOOD RATIO TESTS IN LOCATION-SCALE NONLINEAR MODELS

The paper derives Bartlett corrections for improving the chisquare approximation to the likelihood ratio statistics in a class of location-scale family of distributions, which encompasses the elliptical family of distributions and also asymmetric distributions such as the extreme value distributions. We present, in matrix notation, a Bartlett corrected likelihood ratio statistic for testing that a subset of the nonlinear regression coefficients in this class of models equals a given vector of constants. The formulae derived are simple enough to be used analytically to obtain several Bartlett corrections in a variety of important models. We show that these formulae generalize a number of previously published results. We also present simulation results comparing the sizes and powers of the usual likelihood ratio tests and their Bartlett corrected versions when the scale parameter is considered known and when this parameter is uncorrectly specified.     

Fourth-order moments of augmented arch processes

This paper is concerned with establishing the correspondence between the fourth- order moments structure and the parametric representation for augmented ARCH processes. In an introduction, the possible types of fourth-order moments are classified and some tentative interpretation for each class is given. Then, algebraic formulae are developed that permit the calculation of all fourth- order cross moments that are possibly non-zero within the AARCH class on the basis of known AARCH parameters. These formulae are useful for evaluating properties of estimated AARCH models, if estimation is conducted via maximum likelihood methods or approximations, and also for possible use in method-of-moments estimation.     

Cusum procedure for monitoring variability

In this paper, a CUSUM procedure is given for monitoring for a decrease in the variance (process improvement) as well as a two-sided CUSUM which monitors for both increases and decreases in the variance. The observations are assumed to be independent and normally distributed. The procedure is based on the log¬arithm of the likelihood ratio of the probability density functions under the two competing hypotheses. Formulae that approximate the average run length of the CUSUM procedure for detecting an increase (or decrease) in the variance of a normal distribution are given. These formulae, when corrected for the overshoot from the boundary, provide a very accurate approximation     

Further results on the trace of a noncentral wishart matrix

In an earlier article Mathai (1980) has given compact representations for the moments and cumulants of the trace of a noncentral Wishart matrix. He has also shown that (trA-ntr;∑)/(2ntri∑²)172. is asymptotically standard normal where A is a noncentral wishart matrix with n degrees of freedom and covariance matrix [0, In the present article explicit expressions for the exact density of the trace are given in terms of confluent hypergeometric functions and in terms of zonal polynomials for the general case and as finite sums when the sample size is odd. As a consequence of some of these representations some summation formulae for zonal polynomials are also given     

THE AUTOREGRESSIVE MOVING AVERAGE MODEL FOR SPATIAL ANALYSIS

A two dimensional autoregressive moving average spatial model is used to analyse spatial interaction. Maximum likelihood estimates of the unknown parameters are derived as the solution of a system of nonlinear equations, and are shown to be best asymptotic normal. One important computational procedure is discussed. The argument is extended to the general regression model with autoregressive moving average residuals. Explicit computational formulae are given.     

Population moments and moments of the sample mean

This paper presents new formulae which simultaneously express and estimate moments of the sample mean and estimate population moments, from a simple random sample drawn without replcement from a finite population. By avoiding the generality of the multivariate case, these two problems are not only unified but are made significantly more tractable. Explicit solution are given up to eighth moments. Asymptotic results for infinite populations are also given.     

Perfect cells,direct models and contingency table outliers

A peifect cell frequency is one exactly satisfying a given model General formulae foi peifect cells are provided for direct models having two Or hiee components. Foimulae are piovided for the on-diagonal cells of the corresponding hat matrices Methods for identifying outliers are discussed.     

Information loss due to incomplete data from a spatial gaussian one-parameter first-order conditional process

Formulae are given for the Fisher information loss on parameters for the mean and the variance when some values of a Gaussian process are not observed. The special case of a one-parameter first-order conditional process on a rectangular lattice is considered in detail, and formulae are compared with numerical results.     

A confidence set for that <Emphasis Type="Italic">x</Emphasis>-coordinate where a quadratic regression model has a given gradient

An exact confidence set for that x-coordinate where a quadratic regression model has a given gradient is derived. The limits of the confidence set are given by mathematical formulae. They are implemented in Fortran programs that can be downloaded from the web. The confidence set need not be an interval. Its increase and its changing shape for increasing confidence level is extensively described and visualized in a figure that relates to data from nitrogen-rate trials in Germany. The wheat yields in this example are modeled as quadratic functions of the nitrogen input in order to determine a confidence set for the economically optimum nitrogen fertilization. The disadvantage that the confidence set does not distinguish between concave and convex parabolae, between profit maxima and minima, is also discussed.     

COVARIANCES FOR FIXED INTERVAL SMOOTHED KALMAN FILTER PARAMETER ESTIMATES

Given time series data for fixed interval t= 1,2,…, M with non-autocorrelated innovations, the regression formulae for the best linear unbiased parameter estimates at each time t are given by the Kalman filter fixed interval smoothing equations. Formulae for the variance of such parameter estimates are well documented. However, formulae for covariance between these fixed interval best linear parameter estimates have previously been derived only for lag one. In this paper more general formulae for covariance between fixed interval best linear unbiased estimates at times t and t - l are derived for t= 1,2,…, M and l= 0,1,…, t - 1. Under Gaussian assumptions, these formulae are also those for the corresponding conditional covariances between the fixed interval best linear unbiased parameter estimates given the data to time M. They have application, for example, in determination via the expectation-maximisation (EM) algorithm of exact maximum likelihood parameter estimates for ARMA processes expressed in statespace form when multiple observations are available at each time point.     

Test and confidence set for the difference of the x-coordinates of the vertices of two quadratic regression models

In this article, an exact test and a confidence set for the difference of the x-coordinates of the vertices of quadratic regression models is derived. It is assumed that the quadratic coefficients of both parabolae are equal, so the mentioned difference can equally be related to those x-coordinates where the parabolae have definite given gradient. The limits of the confidence set are given by mathematical formulae. They are implemented in Fortran programs that can be downloaded from the web. The confidence set need not be an interval. This is shown by an example based on nitrogen-rate trials in Germany, where the wheat yields are modeled as quadratic functions of the nitrogen input.     

Influence curve of the monotonic dependence function for binormal distributions

Explicit formulae are derived for the influence curve of the monotonic dependence function in the case of binormal distributions, and are compared with those for the correlation coefficient. The interpretation of the comparison is given.     

Estimation of quantiles of exponential and double exponential distributions based on two order statistics

Asymptotically best linear unbiased estimators (ABLUE) of quantiles, x^., in the two-parameter (location-scale) exponential and double exponential families are obtained as linear combinations of two suitably chosen order statistics. Exact formulae for the linear combinations are given as functions of £. The derived estimators in both cases compare favorably with the usual nonparametric estimator. Also, in the exponential case the derived estimator compares favorably with the Sarhan-Greenberg BLUE based on a complete sample     

Computing inclusion probabilities for order sampling

Rosèn [1997. J. Statist. Plann. Inference 62, 159–191] introduced order sampling schemes of fixed shape which have inclusion probabilities roughly proportional to given size measures ($\pi$ps schemes). Three particular cases where the fixed shape distributions are Pareto, exponential and uniform, respectively, are specially treated. In this paper, we give general algorithms for computing the first- and second-order inclusion probabilities for a general fixed shape order sampling scheme and explicit formulae for the three special cases. Identities are given that can be used to check the accuracy of the numerical results. Examples are included as well as some comments on improving the computational efficiency and accuracy of the algorithms.     

A second order approximation to taguchps online control procedure

In this paper, a second order approximation of the average loss function for on-line control procedure is developed under a Brownian motion model. The approximation depends on whether the ratio of the adjustment cost to the inspection cost is small or large. Both cases are dealt with. Simple formulae for the optimal parameters are given for both cases. These approximations are shown to be better than the ones given by Taguchi et al.(1989) and Adams and WoodaU(1989).