a note on the calculation of adjusted profile likelihood summary an adjustment to the profile likelihood examined by cox and reid required orthogonalization of the nuisance parameters with respect to the parameter of interest an approximation to the adjusted profile likelihood not requiring explicit orthogonalization is derived and exemplified

moduli of fibered surface pairs from twisted stable maps in this paper we use the theory of twisted stable maps to construct compactifications of the moduli space of pairs x rightarrow c s f where x rightarrow c is a fibered surface s is a sum of sections f is a sum of marked fibers and x s f is a stable pair in the sense of the minimal model program this generalizes the work of abramovich vistoli who compactified the moduli space of fibered surfaces with no marked fibers furthermore we compare our compactification to alexeev s space of stable maps and the ksba compactification as an application we describe the boundary of a compactification of the moduli space of elliptic surfaces

error analysis of time delay estimation using a finite integration time correlator a methodology is presented for analyzing the error associated with time delay estimation using a finite integration time correlator processing waveforms received at two separate sensors the type of signal considered is a sinusoid whose amplitude is randomly modulated the signals are assumed to be imbedded in additive gaussian noise before they are correlated the received waveforms are converted to a lower center frequency by mixers whose local oscillators are assumed to contain phase noise by direct calculation in the time domain the variance of the error in the time delay estimate is shown to be a function of integration time signal to noise ratios signal and noise bandwidths and phase noise variance the phase noise is shown to limit the accuracy of the time delay estimate however without phase noise using the methodology the accuracy is shown to approach that obtained by the maximum likelihood estimator

configurations of kodaira fibers on rational elliptic surfaces i n t r o d u c t i o n l legend 6 the list 7 i d i s c r i m i n a n t cons ide ra t i ons 14 if po l a r s of p lane curves 16 ili in t r ins ic g e o m e t r y on g e n u s two curves 19 iv ell iptic qua r t i c s 25 v r a t i o n a l qua r t i c s 30 the reduc ib le case 43 to r s ion g r o u p s of ra t iona l elliptic surfaces 43 epi logue 46 references 46

adjusted estimates and wald statistics for the ar 1 model with constant abstract adjusted profile likelihood of cox and reid 1993 journal of the royal statistical society b 55 467 471 for the autoregressive coefficient of the ar 1 model with constant yields an estimator which can be more accurate than the maximum likelihood estimator in small samples whether the process is stationary or a random walk and in the latter case also asymptotically a related estimator is asymptotically distributed compactly and symmetrically around zero under a random walk but is not consistent in general the adjusted wald statistic can follow the asymptotic standard normal distribution much better than the non adjusted under stationarity or a drifted random walk

on conditional inference for a real parameter a differential approach on the sample space summary a one dimensional conditional procedure defines a partition of the sample space into curves which can be represented by means of a unit vector field a formula is given for the conditional distribution in terms of local properties of the vector field conditions are developed for reducing the first order effects of nuisance parameters and reproducing to higher order the likelihood change for the parameter of interest the emphasis is on extending exponential family methods after locally approximating the statistical model by an exponential family

saddlepoint expansions for directional test probabilities 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 n1 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

estimating functions and approximate conditional likelihood summary the approximate conditional likelihood method proposed by cox reid 1987 is applied to the estimation of a scalar parameter 0 in the presence of nuisance parameters the estimating function of 0 based on the approximate conditional likelihood is shown to be preferable to that based on the profile likelihood a sufficient condition for both approaches to be equivalent is given the role of parameter orthogonality is emphasized several examples including bivariate normal means with known coefficient of variation are presented

on expected volumes of multidimensional confidence sets associated with the usual and adjusted likelihoods we consider a general multiparameter set up where both the interest and the nuisance parameters are possibly vector valued we derive an explicit higher order asymptotic formula to compare the expected volumes of confidence sets given by likelihood ratio statistics arising from the usual profile likelihood and various adjustments thereof our general framework also allows us to include highest posterior density regions with approximate frequentist validity in the study the fact that our interest parameter is possibly vector valued complicates the derivation and warrants the development of special tools and techniques

log canonical models of elliptic surfaces abstract we classify the log canonical models of elliptic surface pairs f x c s f a where f x c is an elliptic fibration s is a section and f a is a weighted sum of marked fibers in particular we show how the log canonical models depend on the choice of the weights we describe a wall and chamber decomposition of the space of weights based on how the log canonical model changes in addition we give a generalized formula for the canonical bundle of an elliptic surface with section and marked fibers this is the first step in constructing compactifications of moduli spaces of elliptic surfaces using the minimal model program

on the choice of prior distribution for the box cox transformed linear model summary the noninformative prior distribution for the parameters of the linear model transformed following box cox 1964 has the non bayesian property of depending to some extent on the data an alternative choice of prior which is not outcome dependent was suggested by pericchi 1981 but it is argued here that this prior has some undesirable features an alternative family of non outcome dependent priors is suggested leading to a noninformative prior which is closer in spirit to that proposed by box cox the posterior consequences of adopting this prior are fully explored and an example discussed

properties of sufficiency and statistical tests 1 in a previous paper dealing with the importance of properties of sufficiency in the statistical theory of small samples attention was mainly confined to the theory of estimation in the present paper the structure of small sample tests whether these are related to problems of estimation and fiducial distributions or are of the nature of tests of goodness of fit is considered further

a simple method for the adjustment of profile likelihoods summary we propose a simple adjustment for profile likelihoods the aim of the adjustment is to alleviate some of the problems inherent in the use of profile likelihoods such as bias inconsistency and overoptimistic variance estimates the adjustment is applied to the profile loglikelihood score function at each parameter value so that its mean is zero and its variance is the negative expected derivative matrix of the adjusted score function for cases in which explicit calculation of the adjustments is difficult we give two methods to simplify their computation an automatic simulation method that requires as input only the profile loglikelihood and its first few derivatives first order asymptotic expressions some examples are provided and a comparison is made with the conditional profile log likelihood of cox and reid

a fast local maximum likelihood estimator for time delay estimation a fast algorithm for the local maximum likelihood determination of the difference of arrival time of a common signal at two spatially separated sensors with uncorrelated noise is given the fast algorithm consists of locally maximizing the cross correlation function from the two wide band signals by using newton s method for finding the root of an equation the probability density function of one iteration of newton s method is explicitly computed in terms of exponential and error functions using a theorem by rice on the probability density of local maxima of gaussian processes the probability density of the local maxima of the cross correlator is obtained these results are new when both the signal and the noises have flat power spectral densities the mean square error mse of two iterates of newton s method is practically equal to the mse computed from the probability density of the local maxima of the cross correlator via rice s theorem the above holds if the starting point used in newton s method is within a quarter signal resolution binwidth from the true delay and the signal to noise ratio snr at the cross correlator output is 15 db or higher the mse of the local maximum estimator obtained from rice s theorem is almost equal to the cramer rao bound even for low snr i e 5 db

moduli of weierstrass fibrations with marked section abstract we study the moduli space of stable pairs x s s a i f i consisting of a weierstrass fibration x its section s and some fibers f i we find a compactification which is a dm stack and we describe the objects on the boundary we show that the fibration in the definition of weierstrass fibration extends to the boundary and it is equidimensional when s 1 we prove that there are wall crossing morphisms when the weights s and a i change when s 1 this recovers the work of la nave 36 and a special case of the work of ascher bejleri 11

a simple predictive density function abstract an easily computable predictive density is considered although involving a plain maximum likelihood technique it coincides with the posterior predictive density with a noninformative prior for standard situations achievements are comparable with classical inference

approximate predictive likelihood summary a predictive likelihood is given which approximates both bayes and maximum likeli hood predictive inference by expansion of a posterior likelihood this synthesizes and extends previous results and is widely applicable the approximation usually differs from exact bayes posterior predictive density by op n 2 and from exact predictive likelihood by op n 1 but does not depend on the availability of prior information and is applicable when exact predictive likelihood cannot be found the results are applied to the prediction various treatments of the likelihood function form the basis of parametric inference but until recently few non bayesian attempts have been made to define versions of it suitable for prediction however lauritzen 1974 hinkley 1979 and more recently butler 1986 have proposed rather similar definitions of predictive likelihood based on the idea of conditioning on the value of a sufficient statistic unfortunately they apply only when sufficiency provides a genuine reduction of the data moreover even if for a given problem an exact predictive likelihood exists the calculations needed to derive it can be onerous this paper provides approximations to hinkley lauritzen predictive likelihood accur ate to op n 1 in many cases which are applicable in situations where maximum likelihood estimation is regular they also apply to problems which admit no exact predictive likelihood because sufficiency provides no real reduction of the data the approximations are derived from a posterior predictive density and may if prior informa tion is available be regarded as bayesian procedures their construction requires only the repeated maximization of a likelihood and evaluation of the observed information matrix at the maximum and is fast and accurate using usual numerical maximization methods there is a close connexion with the approximations to posterior moments and marginal densities given by tierney kadane 1986 who suggest the same approxima tion as used here but give no detailed results for prediction section 2 of this paper shows how laplace s method for integrals may be applied to a posterior predictive density to yield approximate predictive likelihood section 3 discusses a connection with conditional inference essentially the same approximation

a time series illustration of approximate conditional likelihood summary we consider the calculation of confidence intervals for the correlation parameter based on observations of a large number of short gaussian first order autoregressive processes with different means but common correlation and variance ordinary profile likelihood gives very misleading answers the approximate conditional likelihood of cox reid 1987 is calculated and shown to lead to consistent estimation of the correlation simulations verify that the usual asymptotic properties of the statistics derived from the conditional likelihood are quite satisfactory in this example the approximate conditional likelihood is the same as the marginal likelihood for the standardized residuals although this is not true in general

conditional score functions some optimality results summary the conditional score function has previously been shown to generate the optimal estimating equation for a parameter of interest when the conditioning statistic is complete and sufficient for the nuisance parameters godambe 1976 the present paper generalizes these results to partial likelihood factorizations and then examines the nature of the problem when the appropriate conditioning statistics depend on the parameter of interest in this case globally optimal estimating functions are impossible a weaker criterion of optimal weighting leads to a class of estimated conditional score functions which satisfy an information equality the purpose of this paper is to develop and justify an approach based on conditional likelihood to estimation in the presence of nuisance parameters the arguments focus on the case when there is a complete and sufficient statistic s 0 for the nuisance parameters when the parameter of interest 0 is fixed an extended notion of completeness will be presented in the context of partial likelihoods the problem takes on two separate natures depending on whether the statistic s 0 truly depends on 0 or not if s s 0 is free of 0 then the conditional likelihood of the data given s has been shown to generate optimal inferential procedures giving a best estimating equation godambe 1976 and a best asymptotically normal estimator in a mixture setting lindsay 1980 if s 0 depends on 0 then it will be shown that the best estimating equation also a conditional likelihood score will generally be only locally optimal leaving no single best strategy consideration of an optimal weighting strategy yields estimators which have second order properties similar to the conditional likelihoods of the first type

moduli spaces of weighted pointed stable curves abstract a weighted pointed curve consists of a nodal curve and a sequence of marked smooth points each assigned a number between zero and one a subset of the marked points may coincide if the sum of the corresponding weights is no greater than one we construct moduli spaces for these objects using methods of the log minimal model program and describe the induced birational morphisms between moduli spaces as the weights are varied in the genus zero case we explain the connection to geometric invariant theory quotients of points in the projective line and to compactifications of moduli spaces studied by kapranov keel and losev manin

bartlett adjustments to the likelihood ratio statistic and the distribution of the maximum likelihood estimator for rather general parametric models a simple connection is established between the bartlett adjustment factor of the log likelihood ratio statistic and the normalizing constant c of the formula c i i 1 2l for the conditional distribution of a maximum likelihood estimator as applied to the full model and the model of the hypothesis tested this leads to a relatively simple demonstration that division of the likelihood ratio statistic by a suitable constant or estimated factor improves the chi squared approximation to its distribution various expressions for these quantities are discussed in particular for the case of a one dimensional parameter an approximation to the constants involved is derived which does not require integration over the sample space

a bayesian approach to transformations to normality summary the analysis of transformation of observations in the linear model with normal errors proposed by box cox 1964 is considered a different choice of noninformative unnormed prior is advocated which is not outcome dependent this new selection of prior leads to a formal identity between likelihood and bayesian inference both for the estimation of the best transformation to normality and for the presence of homoscedasticity and additivity under this transformation extension to a related problem is mentioned

modeling of time delay and its application to estimation of nonstationary delays a method to model a time delay by a finite impulse response filter is presented it is useful in simulation work that involves time delays and transforms the time delay estimation problem into one of parameter estimation the benefits of this approach are the elimination of spectral estimation a choice of many parameter estimation algorithms and the capability to track time varying delays two examples of estimating nonstationary time delays are also given

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

information bias and adjusted profile likelihoods summary the bias and information bias of the ordinary profile score statistic are both typically of order 0 1 several additive adjustments to the profile score statistic that reduce its bias to order o n have been proposed in certain situations the information bias of these adjusted profile score statistics is also reduced to order o n q specifically the modified profile likelihood of barndorff nielsen yields an adjusted profile score statistic having both reduced bias and reduced information bias in general however a bias reducing adjustment to the profile score statistic will not automatically reduce the order of the information bias as well in this paper an analytical formula is obtained for the information bias of a general bias adjusted profile score statistic this formula is used to compare bias reducing adjustments to the profile score statistic as well as to construct further adjustments that reduce the information bias to o n 1 several examples are presented to illustrate use of the formula for information bias in particular the information bias formula may be utilized in a criterion for choosing between orthogonal parameterizations for the conditional profile likelihood of cox and reid

differential geometry of curved exponential families curvatures and information loss the differential geometrical framework is given for analyzing statistical problems related to multi parameter families of distributions the dualistic structures of the exponential families and curved exponential families are elucidated from the geometrical viewpoint the duality connected by the legendre transformation is thus extended to include two kinds of affine connections and two kinds of curvatures the second order information loss is calculated for fisher efficient estimators and is decomposed into the sum of two non negative terms one is related to the exponential curvature of the statistical model and the other is related to the mixture curvature of the estimator only the latter term depends on the estimator and vanishes for the maximum likelihood estimator a set of statistics which recover the second order information loss are given the second order efficiency also is obtained the differential geometry of the function space of distributions is discussed

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

a canonical bundle formula a higher dimensional analogue of kodaira s canonical bundle formula is obtained as applications we prove that the log canonical ring of a klt pair with 3is finitely generated and that there exists an effectively computable natural number m such that mk x induces the iitaka fibering for every algebraic threefold x with kodaira dimension 1

likelihood and observed geometries a partir du point de vue de la conditionnalite on etablit un parallele observe a la theorie de chentsov efron amari etc en utilisant l information observee et un tenseur d asymetrie observee

analysis of discrete implementation of generalized cross correlator a common discrete implementation of the cross correlator uses a parabolic fit to the peak when the delay is not an integral multiple of the sampling period this correspondence analyzes and assesses the pitfalls of this approach it is shown that this yields a biased estimate of the time delay with both the bias and variance of the estimate dependent on the location of the delay between samples snr signal and noise bandwidths and the prefilter or window used in the generalized correlator

variance stabilization for a scalar parameter summary we present a variance stabilizing transformation for inference about a scalar parameter that is estimated by a function of a multivariate m estimator the transformation proposed is automatic computationally simple and can be applied quite generally though it is based on an intuitive notion and entirely empirical the transformation is shown to have an appropriate justification in providing variance stabilization when viewed from both parametric and nonparametric perspectives further the transformation repairs deficiencies of existing methods for variance stabilization the transformation proposed is illustrated in a range of examples and its effectiveness to yield confidence limits having low coverage error is demonstrated in a numerical example