parameter transformations for improved approximate confidence regions in nonlinear least squares in a previous paper it was shown that parameter effects nonlinearities of a nonlinear regression model experimental design parameterization combination can be quantified by means of a parameter effects curvature array a based on second derivatives of the model function in this paper the individual terms of a are interpreted and local compensation methods are suggested a method of computing the parameter effects array corresponding to a transformed set of parameters is given and we discuss how this result could be used to determine reparameterizations which have zero local parameter effects nonlinearity

parameter plots in nonlinear regression summary we propose graphical methods for displaying relevant information on a selected parameter from a normal nonlinear regression model it is shown that the usual extension of added variable plots from linear to nonlinear regression can fail to reveal important diagnostic information and that this information can be recovered by using a parameter plot that depends on selected elements of the parameter effects curvature array

scaling of the discrete time algebraic riccati equation to enhance stability of the schur solution method a simple scaling procedure for discrete time riccati equations is introduced this procedure eliminates instabilities which can be associated with the linear equation solution step of the generalized schur method without changing the condition of the underlying problem a computable bound for the relative error of the solution of the riccati equation is also derived

computational experience with confidence regions and confidence intervals for nonlinear least squares we present the results of a monte carlo study of the leading methods for constructing approximate confidence regions and confidence intervals for parameters estimated by nonlinear least squares we examine three variants of the linearization method the likelihood method and the lack of fit method the linearization method is computationally inexpensive produces easily understandable results and is widely used in practice the likelihood and lack of fit methods are much more expensive and more difficult to report in our tests both the likelihood and lack of fit methods perform very reliably all three variants of the linearization method however often grossly underestimate confidence regions and sometimes significantly underestimate confidence intervals the linearization method variant based solely on the jacobian matrix appears preferable to the two variants that use the full hessian matrix because it is less expensive more numerically stable and at least as accurate the bates and watts curvat

confidence regions for parameter subsets in nonlinear regression on utilise la methode a deux niveaux proposee par hamilton watts et bates 1982 pour obtenir des approximations ameliorees des regions de confiance

confidence curves in nonlinear regression abstract standard wald confidence regions for parameters in a normal nonlinear regression model often fail to capture accurately the uncertainty of estimation as reflected by the corresponding profile log likelihood we present a graphical method along with a stable computational algorithm for inference on scalar parameters in a nonlinear regression model

approximate confidence limits for a parameter function in nonlinear regression on propose une methode a deux etapes la premiere utilisant une transformation des parametres simples pour definir de nouveaux parametres la seconde necessitant l utilisation d une region de confiance ellipsoidale pour obtenir un intervalle de confiance

better bootstrap confidence intervals abstract we consider the problem of setting approximate confidence intervals for a single parameter in a multiparameter family the standard approximate intervals based on maximum likelihood theory can be quite misleading in practice tricks based on transformations bias corrections and so forth are often used to improve their accuracy the bootstrap confidence intervals discussed in this article automatically incorporate such tricks without requiring the statistician to think them through for each new application at the price of a considerable increase in computational effort the new intervals incorporate an improvement over previously suggested methods which results in second order correctness in a wide variety of problems in addition to parametric families bootstrap intervals are also developed for nonparametric situations

marginal curvatures and their usefulness in the analysis of nonlinear regression models abstract interval estimation for individual parameters in nonlinear regression is examined in relation to curvature of the solution locus criteria are developed to define severe curvature and distinguish cases in which linear methods may be used from those in which parameter transformations are necessary a formula for improving the linear approximation in cases of relatively mild curvature is proposed for example suppose that interest is focused on one specific parameter j from among 1 2 p in a nonlinear regression model a crude approach to setting 95 confidence limits to j might proceed as follows first calculate the standard error of the estimate as the square root of its asymptotic variance then call the confidence limits in this article higher order correction terms and are proposed that enable us to develop a power series expansion for the confidence limits as and respectively here denotes some estimate of 2 the variance of a single observation

bootstrap confidence intervals for a class of parametric problems summary we consider the following class of problems having observed a multivariate normal data vector y with unknown mean vector q covariance matrix the identity find an approximate confidence interval for 0 t q a real valued function of q a simple geometric construction is given which leads to highly accurate solutions this construction shows that the standard approximation based on maximum likelihood theory 04 cz00 can be quite misleading when 0 is nonlinear in q we discuss bootstrap based confidence intervals which remove most of the error in the standard approximation at the expense of considerably more calculation the bootstrap intervals are invariant under transformation of both y and i and so they automatically produce accurate solutions in problems which can be transformed to multivariate normality without requiring knowledge of the normalizing transformation

numerical point of view on calculus for functions assuming finite infinite and infinitesimal values over finite infinite and infinitesimal domains abstract the goal of this paper consists of developing a new more physical and numerical in comparison with standard and non standard analysis approaches point of view on calculus with functions assuming infinite and infinitesimal values it uses recently introduced infinite and infinitesimal numbers being in accordance with the principle the part is less than the whole observed in the physical world around us these numbers have a strong practical advantage with respect to traditional approaches they are representable at a new kind of a computer the infinity computer able to work numerically with all of them an introduction to the theory of physical and mathematical continuity and differentiation including subdifferentials for functions assuming finite infinite and infinitesimal values over finite infinite and infinitesimal domains is developed in the paper this theory allows one to work with derivatives that can assume not only finite but infinite and infinitesimal values as well it is emphasized that the newly introduced notion of the physical continuity allows one to see the same mathematical object as a continuous or a discrete one in dependence on the wish of the researcher i e as it happens in the physical world where the same object can be viewed as a continuous or a discrete in dependence on the instrument of the observation used by the researcher connections between pure mathematical concepts and their computational realizations are continuously emphasized through the text numerous examples are given

bent cable regression theory and applications we use the so called bent cable model to describe natural phenomena that exhibit a potentially sharp change in slope the model comprises two linear segments joined smoothly by a quadratic bend the class of bent cables includes as a limiting case the popular piecewise linear model with a sharp kink otherwise known as the broken stick associated with bent cable regression is the estimation of the bend width parameter through which the abruptness of the underlying transition may be assessed we present worked examples and simulations to demonstrate the regularity and irregularity of bent cable regression encountered in finite sample settings we also extend existing bent cable asymptotics that previously were limited to the basic model with known linear slopes of 0 and 1 practical conditions on the design are given to ensure regularity of the full bent cable estimation problem if the underlying bend segment has nonzero width under such conditions the least squares estimators are shown to be co

confidence bounds for nonlinear dose response relationships an important aim of drug trials is to characterize the dose response relationship of a new compound such a relationship can often be described by a parametric nonlinear function that is monotone in dose if such a model is fitted it is useful to know the uncertainty of the fitted curve it is well known that wald confidence intervals are based on linear approximations and are often unsatisfactory in nonlinear models apart from incorrect coverage rates they can be unreasonable in the sense that the lower confidence limit of the difference to placebo can be negative even when an overall test shows a significant positive effect bootstrap confidence intervals solve many of the problems of the wald confidence intervals but are computationally intensive and prone to undercoverage for small sample sizes in this work we propose a profile likelihood approach to compute confidence intervals for the dose response curve these confidence bounds have better coverage than wald intervals and are more precise and generally faster than bootstrap methods moreover if monotonicity is assumed the profile likelihood approach takes this automatically into account the approach is illustrated using a public dataset and simulations based on the emax and sigmoid emax models

residuals in nonlinear regression abstract the authors employ a quadratic expansion to investigate the behavior of the ordinary residuals in nonlinear regression in particular they derive quadratic approximations for the mean and variance of the ordinary residuals and the covariances between the ordinary residuals and the fitted values this investigation leads to the conclusion that the ordinary residuals can produce misleading results when used in diagnostic methods analogous to those for linear regression consequently a new type of residual that overcomes many of the potential shortcomings of the ordinary residuals is suggested author

inference on segmented polynomial models segmented polynomial regression models with unknown change points are used in a wide variety of biological settings the application that stimulated this work uses a segmented polynomial model to examine the optimal hematocrit hypothesis we discuss problems in the fitting of these models and compare by simulation two methods of inference in these models that based on the chi squared approximation to the distribution of the likelihood ratio statistic and that based on the asymptotic normality of the least squares estimates the results show that of the two only the likelihood ratio statistic produces reliable inference concerning the change point

comments on scaling of the discrete time algebraic riccati equation to enhance stability of the schur solution method with reply it is pointed out that a relative error bound of the solution of the discrete riccati equation derived in the above titled work by t gudmundson et al ibid vol 37 no 4 pp 513 518 apr 1992 is already available in previously published literature in reply gudmundson et al point out that the main point in their paper was simply to note that apparent instability of the schur method for discrete time riccati equations wa due to improper scaling and the proposed a scaling procedure for dealing with the problem

the signed root deviance profile and confidence intervals in maximum likelihood analysis abstract we investigate a natural extension of the profile t plot of bates and watts to a general parametric function g of the parameters in a general maximum likelihood analysis although the basic purpose of the extension called the signed root deviance profile srdp is to construct likelihood ratio lr confidence intervals for g it has various other applications that significantly extend its usefulness the tangent to the plot of the srdp at the maximum likelihood estimate of g gives the linear approximation la interval based on the observed information matrix the plot may be used as a diagnostic tool to compare la and lr intervals and to suggest transformations of g whose la intervals when inverted are close to the lr intervals for g the standard way to construct any profile is through repeated optimizations but problems associated with nonlinear constraints can make this difficult an alternative method based on integration is presented that avoids these problems an exam

semiparametric regression in likelihood based models abstract a weighted likelihood is used to estimate the parameters in a semiparametric model involving two covariates and allowing an association between the covariates the development is for arbitrary but specified densities of the observations the estimators are consistent and asymptotically normal hypothesis testing of the parametric component can be performed using a wald test simulations and analysis of data with bernoulli observations demonstrate the estimators application speckman developed kernel estimators where the conditional density of the observations is normal with p parametric covariates speckman s estimators and the new estimators are asymptotically equivalent with the bias of speckman s estimators being smaller as an example we study the relationship between a binary response indicating the occurrence of an intraoperative cardiac complication icc in vascular surgery patients and two risk factors duration of the operation or and asa score which is an evaluation of the patien

diagnostics for assessing the accuracy of normal approximations in exponential family nonlinear models abstract diagnostics are investigated for assessing the agreement between likelihood and standard large sample confidence regions for parameters from an exponential family nonlinear model cordeiro and paula 1989 the development connects the contour methods proposed by hodges 1985 1987 with the curvature measures for normal nonlinear regression proposed by bates and watts 1980 and jennings s 1982 1986 diagnostics for logistic regression the proposed methodology is illustrated with several examples

inverting the indirect the ellipse and the boomerang visualizing the confidence intervals of the structural coefficient from two stage least squares abstract in the just identified model the exact distribution of the two stage least squares 2sls estimator of the coefficient of the endogenous regressor is a ratio of two normally distributed random variables basmann 1960 basmann 1974 used fieller s 1932 result to derive the density function of the estimator in this paper we present a novel graphical exposition of fieller s 1954 technique to approximate the confidence interval for the 2sls estimator we use this approach to examine how the degree of endogeneity and instrument relevance influences the correspondence between the fieller and traditional asymptotic confidence intervals for the estimator

bias in nonlinear regression abstract this document investigates the biases of the residuals and the maximum likelihood parameters estimates from standard normal theory nonlinear regression models emphasis is placed on determining the influence of individual cases on the biases and on understanding how the residual biases can affect the usefulness of standard diagnostic methods it is shown that the various bias expressions in the literature are equivalent that the biases in nonlinear regression can be studied usefully in the context of linear regression and that diagnostic plots using residuals can be misleading because of substantial residual biases for a class of partially nonlinear models it is shown that the maximum intrinsic curvature is closely related to the residual expectations finally the model associated with power transformations of single explanatory variables in linear regression is investigated in further detail and several numerical illustrations are presented author

a profile based approach to parametric sensitivity analysis of nonlinear regression models predictions from a nonlinear regression model are subject to uncertainties propagated from the estimated parameters in the model parameters exerting the strongest influence on model predictions can be identified by a sensitivity analysis in this article a new parametric sensitivity measure is introduced based on the profiling algorithm developed by bates and watts for constructing likelihood intervals for the individual parameters in nonlinear regression models in contrast with traditional sensitivity coefficients this profile based sensitivity measure accounts for both correlation structure among the parameters and model nonlinearity it also provides sensitivity information over wide ranges of parameter uncertainties application of the proposed approach is illustrated with three examples

inverse estimates from a multiresponse model summary in a multiresponse model a multivariate y is modelled as a nonlinear function of the independent variable x the inverse estimate of x for a given observation on y is the object of study in this paper a straightforward method is proposed for estimating x and setting confidence limits to the estimate an example of determining age from the lengths of horns of a rhinoceros is given animal based on horn lengths only we specifically deal with the situation in which the response functions are nonlinear in the unknown parameters the method proposed is to augment the original data set with the additional measure ments lag and vpg of the extra animal whose age is needed and a guessed age xi varying values of xg will generate varying values of the maximized likelihood conditional on xg and these define the profile likelihood kalbfleisch and sprott 1970 for xg this profile likelihood can be used and interpreted in a number of ways all basically intended to illustrate the reliability of the age estimate a cautionary word is required the step of associating the profile likelihood with classical confidence limits for the true value of xg is difficult and only approximate solutions are available to that extent this paper should not be regarded as proposing a definitive answer there are open questions raised in particular it would be valuable to quantify more precisely the sampling distribution of the profile likelihood function the problem posed here is a version of the multivariate calibration problem the literature although sometimes couched in more general terms invariably deals in detail only with response models that are linear in the unknown parameters 0 brown 1982 gives a broad review of the linear multivariate calibration problem and outlines both bayesian and non bayesian solutions hunter and lamboy 1981 review the linear problem from a bayesian viewpoint brown and sundberg 1987 1989 propose using the profile likelihood of x minder and whitney 1975 derived a marginal likelihood approach the classic paper on univariate linear calibration is that of fieller 1940 and many modern ideas can be related back to the fieller fiducial limits oman and wax 1984 give an interesting application of linear multivariate calibration in estimating fetal age given two

curvatures for parameter subsets in nonlinear regression on etend les mesures de nonlinearite proposees par bates et watts 1980 a un sous ensemble arbitraire des parametres dans un modele de regression non lineaire normal

correction factors for f ratios in nonlinear regression summary multiplicative correction factors are derived for the limiting f distributions of two test statistics for parameter subsets in nonlinear regression the factors depend on the first and second derivatives of the model and are related to measures of intrinsic nonlinearity an example is given for which the correction is substantial a similar factor is obtained for the lack of fit test in nonlinear regression