modified signed log likelihood ratio summary the signed log likelihood ratio r for a one dimensional interest parameter can be modified as r r r log u r so that r is asymptotically standard normally distributed with error of order o n 3 2 u being interpretable as a test statistic a new more direct derivation of this result is given which leads to a more explicit expression for u and provides a better handle on the error term the relation of the tail area approximation determined by r to a generalization of the lugannani rice approximation is discussed and the applicability of r to the analysis of residual variation is indicated

properly posed sets of nodes for multivariate lagrange interpolation in c s in this paper we apply techniques from the theory of ideals and varieties in algebraic geometry to study the geometric structure of a properly posed set of nodes or ppsn for short for multivariate lagrange interpolation along an algebraic hypersurface we provide a hyperplane superposition process to construct the ppsn for interpolation along an algebraic hypersurface and as a result we offer a clear understanding of the geometric structure of the ppsn for multivariate lagrange interpolation in cs

approximations of marginal tail probabilities and inference for scalar parameters summary in many situations inference for a scalar parameter in the presence of nuisance parameters requires integration of either a joint density of pivotal quantities or a joint posterior density for such inference accurate approximations of marginal tail probabilities are useful to avoid high dimensional integrals two tail probability approximations are developed in this paper numerical results given for conditional inference in location scale and linear regression models show the approximations to be generally accurate even for small sample sizes

on the multivariate horner scheme we present an efficient version of the horner scheme for the evaluation of multivariate polynomials and study its stability properties in particular we show that the backward error of evaluation is bounded by a quantity that is linear in the total degree of the polynomial which itself is usually significantly smaller than the number of operations involved in the evaluation process moreover we show that the same type of estimates holds also for the forward error provided that the polynomial basis is chosen suitably

asymptotic normality with small relative errors of posterior probabilities of half spaces some examples of half spaces of interest in parameter spaces are in a clinical trial of a treatment versus placebo the half spaces where the treatment is a helpful or b harmful thus one may not only want to test the hypothesis that the treatment c makes no difference but to assign posterior probabilities to a b and c under conditions as unrestrictive as possible on the choice of prior probabilities e g dudley and haughton 2001 more generally we have in mind applications to model selections as in the bic criterion of schwarz 1978 and its extensions poskitt 1987 haughton 1988 specifically to one sided models and multiple data sets dudley and haughton 1997

practical use of higher order asymptotics for multiparameter exponential families recently developed asymptotics based on saddlepoint methods provide important practical methods for multiparameter exponential families especially in generalized linear models the aim here is to clarify and explore these attention is restricted to tests and confidence intervals regarding a single parametric function which can be represented as a natural parameter of a full rank exponential family excellent approximations to exact conditional inferences are often available in terms of simple adjustments to the signed square root of the likelihood ratio statistic

stable and invariant adjusted directed likelihoods summary for the purpose of making inferences for a one dimensional interest parameter or constructing approximate complementary ancillaries or residuals the directed likelihood or signed square root of the likelihood ratio statistic can be adjusted so that the resulting modified directed likelihood is under ordinary repeated sampling approximately standard normal with error of o n 312 conditional on a suitable ancillary statistic and hence unconditionally in general suitable specification of the ancillary statistic may be difficult we introduce two adjusted directed likelihoods which are similar to the modified directed likelihood but do not require the specification of the ancillary statistic the error of the standard normal approximation to the distribution of these new adjusted directed likelihoods is o n 1 conditional on any reasonable ancillary statistic which is still an improvement over the unadjusted directed likelihoods

modified profile likelihoods in models with stratum nuisance parameters it is well known at least through many examples that when there are many nuisance parameters modified profile likelihoods often perform much better than the profile likelihood ordinary asymptotics almost totally fail to deal with this issue for this reason we study asymptotic properties of the profile and modified profile likelihoods in models for stratified data in a two index asymptotics setting this means that both the sample size of the strata m and the dimension of the nuisance parameter q may increase to infinity it is shown that in this asymptotic setting modified profile likelihoods give improvements with respect to the profile likelihood in terms of consistency of estimators and of asymptotic distributional properties in particular the modified profile likelihood based statistics have the usual asymptotic distribution provided that l m o q 1 3 while the analogous condition for the profile likelihood is l m o q 1

studentization and deriving accurate p values the original studentization was the conversion of a sample mean departure into the familiar t statistic plus the derivation of the corresponding student distribution function the observed value of the distribution function is the observed p value as presented in an elemental form we examine this process in a broadly general context a null statistical model is available together with observed data a statistic t y has been proposed as a plausible measure of the location of the data relative to what is expected under the null a modified statistic say t y is developed that is ancillary the corresponding distribution function is determined exactly or approximately and the observed value of the distribution function is the p value or percentage position of the data with respect to the model such p values have had extensive coverage in the recent bayesian literature with many variations and some preference for two versions labelled pppost and pcpred the bootstrap method also directly addresses this studentization process we use recent likelihood theory that gives a third order factorization of a regular statistical model into a marginal density for a full dimensional ancillary and a conditional density for the maximum likelihood variable the full dimensional ancillary is shown to lead to an explicit determination of the studentized version t y together with a highly accurate approximation to its distribution function the observed value of the distribution function is the p value and is available numerically by direct calculation or by markov chain monte carlo or by other simulations in this paper for any given initial or trial test statistic proposed as a location indicator for a data point we develop an ancillary based p value designated panc a special version of the bayesian pcpred and a bootstrap based p value designated pbs we then show under moderate regularity that these are equivalent to the third order and have uniqueness as a determination of the statistical location of the data point as of course derived from the initial location measure we also show that these p values have a uniform distribution to third order as based on calculations in the moderate deviations region for implementation the bayesian and likelihood procedures would perhaps require the same numerical computations while the bootstrap would require a magnitude more in computation and would perhaps not be accessible examples are given to indicate the ease and exibility of the approach

multivariate scattered data fitting during the week of 2 8 march 1995 a meeting entitled international conference on scattered data fitting was held in cancun mexico the meeting was attented by 42 researchers from europe and the americas and 34 lectures were presented this special issue of journal of computational and applied mathematics includes 14 papers based on presentations at the conference

exponential tilting in bayesian asymptotics we use exponential tilting to obtain versions of asymptotic formulae for bayesian computation that do not involve conditional maxima of the likelihood function yielding a more stable computational procedure and significantly reducing computational time in particular we present an alternative version of the laplace approximation for a marginal posterior density implementation of the asymptotic formulae and a modified signed root based importance sampler are illustrated with an example

accurate directional inference for vector parameters we consider a vector valued parameter of interest in the presence of a nitedimensional nuisance parameter based on higher order asymptotic theory for likelihood inference we propose a directional test for the vector parameter of interest that is computed using one dimensional integration for discrete responses this extends the development of davison et al 2006 and several examples below concern testing hypotheses in contingency tables for continuous responses the work extends the directional test of cheah et al 1994 exponential family examples and simulations illustrate the high accuracy of the method which we compare with an adjusted likelihood ratio test of skovgaard 2001 in a high dimensional covariance selection example the approach works essentially

on lagrange and hermite interpolation in rk a method for the construction of a set of data of interpolation in several variables is given the resulting data which are either function values or directional derivatives values give rise to a space of polynomials in such a way that unisolvence is guaranteed the interpolating polynomial is calculated using a procedure which generalizes the newton divided differences formula for a single variable

higher order asymptotics and the likelihood principle one parameter models summary inferences based on higher order asymptotics in contrast to those based on standard first order methods approximate exact inferences closely enough to depend on the reference set we use this feature to quantify the effect of the reference set on exact inferences in particular the difference in p values for two reference sets can be at most o n and it is seen that one can often identify the coefficient of n although this is useful from a foundational viewpoint the question arises of whether higher order adjustments to firstorder p values reflect only the choice of reference set it is indicated that for one parameter problems this may largely be the case but that when there are nuisance parameters the situation is likely to be quite different the nuisance parameter setting is not investigated thoroughly in this paper but we believe that in that case the most important part of the higher order corrections largely conforms to the likelihood principle

general order newton pade approximants for multivariate functions pade approximants are a frequently used tool for the solution of mathematical problems one of the main drawbacks of their use for multivariate functions is the calculation of the derivatives off x 1 x p therefore multivariate newton pade approximants are introduced their computation will only use the value off at some points in sect 1 we shall repeat the univariate newton pade approximation problem which is a rational hermite interpolation problem in sect 2 we sketch some problems that can arise when dealing with multivariate interpolation in sect 3 we define multivariate divided differences and prove some lemmas that will be useful tools for the introduction of multivariate newton pade approximants in sect 4 a numerical example is given in sect 5 together with the proof that forp 1 the classical newton pade approximants for a univariate function are obtained

uniform bivariate hermite interpolation i coordinate degree on considere des schemas d interpolation d hermite uniforme bivariable a partir d espaces de polynomes de degre coordonne m n on se pose la question de la presque regularite de tels schemas c est a dire celle de l existence d ensembles de n uds pour lesquels avec une constellation de derivees donnees l interpolation est possible on montre que dans le cas affirmatif le schema en question peut etre utilise pour presque tous les ensembles de n uds l etude est basee sur des techniques de reduction

bivariate hermite interpolation and applications to algebraic geometry we try to solve the bivariate interpolation problem 1 3 for polynomials 1 1 wheres is a lower set of lattice points and for theq th interpolation knot a q is the set of orders of derivatives that appear in 1 3 the number of coefficients s is equal to the number of equations a q if this is possible for all knots in general position the problem is almost always solvable a a s we seek to determine whether 1 3 is a a s an algorithm is given which often gives a positive answer to this it can be applied to the solution of a problem of hirschowitz in algebraic geometry we prove that for hermite conditions 1 3 when alla q are lower triangles of orderp andp is of total degreen 1 3 is a a s for allp 1 2 3 and alln except for the two casesp 1 n 2 andp 1 n 4

adjusted versions of profile likelihood and directed likelihood and extended likelihood several adjusted and invariant versions of profile likelihood and of directed likelihood short for signed log likelihood ratio are discussed these adjusted versions which fall into two groups are closely related to modified profile likelihood and modified directed likelihood the first group has the advantage of not requiring an explicit specification of an ancillary but partly at a cost of a loss of asymptotic accuracy in comparison with the modified versions the second group consists of some asymptotically sharpened versions of modified profile likelihood furthermore a concept of extended likelihood is proposed which associates a likelihood like function to certain combinants

on the solvability of bivariate hermite birkhoff interpolation problems in a recent paper by hack 1987 certain bivariate polynomial hermite birkhoff interpolation problems are considered and sufficient conditions for unique solvability are obtained by applying a general result on interpolation with tensor functionals in this paper using a matricial approach and with the guideline of a result due to stenger 1968 these conditions are shown to be also necessary and expressed in terms of the unisolvence of certain univariate interpolation problems the results are valid not only for the polynomial case but also for interpolation with bivariate functions built from univariate extended complete tchebycheff ect systems

on bivariate hermite interpolation with minimal degree polynomials a newton type approach is used to deal with bivariate polynomial hermite interpolation problems when the data are distributed in the intersections of two families of straight lines as a generalization of regular grids the interpolation operator is degree reducing and the interpolation space is a minimal degree space integral remainder formulas are given for the lagrange case then extended to the hermite case and finally used to obtain error estimates

a new iterative scheme for obtaining eigenvectors of large real symmetric matrices a rapidly convergent block iterative scheme for the computation of a few of the lowest eigenvalues and eigenvectors of a large dense real symmetric matrix is proposed the method is especially applicable to matrix eigenvalue problems that arise from the discretization of self adjoint partial differential equations one such application to certain symmetric matrices that arise in solid state band structure calculations is considered in detail the most timeconsuming parts of the present algorithm are a matrix multiplication and a gauss siedel relaxation step which are performed on each iteration these two parts can however be very efficiently implemented on a vector or parallel processing computer

a new method for diagonalising large matrices the structure and implementation of a new general iterative method for diagonalising large matrices the residual minimisation direct inversion in the iterative subspace method of bendt and zunger are described and contrasted with other more commonly used iterative techniques the method requires the direct diagonalisation of only a small submatrix does not require the storage of the large matrix and provides eigensolutions to within a prescribed precision in a rapidly convergent iterative procedure numerical results for two rather different matrices a real 50 50 non diagonally dominant matrix and a complex hermitian 181 181 matrix corresponding to the pseudopotential band structure of a semiconductor in a plane wave basis set are used to compare the new method with the competing methods the new method converges quickly and should be the most efficient for very large matrices in terms both of computation time and central storage requirements it is quite insensitive to the properties of the matrices used this technique makes possible efficient solution of a variety of quantum mechanical matrix problems where large basis set expansions are required

confidence intervals of effect size in randomized comparative parallel group studies we have presented a new likelihood based approach for constructing confidence intervals of effect size that are applicable to small samples we also conduct a simulation study to compare the coverage probability of the new likelihood based method with other three methods proposed by hedges and olkin and kraemer and paik simulation studies show that the confidence interval generated by the modified signed log likelihood ratio method possesses essentially exact coverage probabilities even for small samples although the coverage probabilities are consistently but slightly less than the nominal level the methods are also applied to two examples

mean loglikelihood and higher order approximations summary higher order approximations to p values can be obtained from the log likelihood function and a reparameterization that can be viewed as a canonical parameter in an exponential family approximation to the model this approach clarifies the connection between skovgaard 1996 and fraser et al 1999a and shows that the skovgaard approximation can be obtained directly using the mean log likelihood function

asymptotics and the theory of inference asymptotic analysis has always been very useful for deriving distributions in statistics in cases where the exact distribution is unavailable more importantly asymptotic analysis can also provide insight into the inference process itself suggesting what information is available and how this information may be extracted the development of likelihood inference over the past twenty some years provides an illustration of the interplay between techniques of approximation and statistical theory

likelihood for component parameters for a statistical model with data likelihood for the scalar or vector full parameter 0 of dimension p say is typically well defined and easily computed in this paper we investigate likelihood for a component parameter 0 of dimension d p and make use of the recent likelihood theory that has been successful in producing highly accurate third order p values for scalar parameters of continuous models the theory leads under moderate regularity to a definitive third order determination of likelihood for a component parameter 0 of dimension d where 1 d p we use the simple location model on the plane with standard normal errors to motivate the development the example exhibits most of the key characteristics of the general case and the recent theory then extends the determination of likelihood to the general context for the scalar interest parameter case with d 1 the usual determinations are typically of second order accuracy the example indicates how the new determination achieves third order accuracy the implementation is straightforward and uses familiar ingredients to other determinations such as the full maximum likelihood value o the constrained value o given 0 and the observed information j o for a complementing nuisance parameter 0 it does however require a special version of the nuisance information j o a version calibrated relative to a symmetric choice of the exponential type reparameterisation 0 underlying the recent theory but this is easily computed various examples are given and the motivating example is discussed in detail

polynomial interpolation of minimal degree minimal degree interpolation spaces with respect to a finite set of points are subspaces of multivariate polynomials of least possible degree for which lagrange interpolation with respect to the given points is uniquely solvable and degree reducing this is a generalization of the concept of least interpolation introduced by de boor and ron this paper investigates the behavior of lagrange interpolation with respect to these spaces giving a newton interpolation method and a remainder formula for the error of interpolation moreover a special minimal degree interpolation space will be introduced which is particularly beneficial from the numerical point of view

generalized hermite interpolation via matrix valued conditionally positive definite functions in this paper we consider a broad class of interpolation problems for both scalarand vector valued multivariate functions subject to linear side conditions such as being divergence free where the data are generated via integration against compactly supported distributions we show that by using certain families of matrix valued conditionally positive definite functions such interpolation problems are well poised that is the interpolation matrices are invertible as a sample result we show that a divergence free vector field can be interpolated by a linear combination of convolutions of the data generating distributions with a divergence free 3 x 3 matrix valued conditionally positive definite function in addition we obtain norm estimates for inverses of interpolation matrices that arise in a class of multivariate hermite interpolation problems

effects of the reference set on frequentist inferences summary we employ second order likelihood asymptotics to investigate how ideal frequentist inferences depend on the probability model for the data through more than the likelihood function referring to this as the effect of the reference set there are two aspects of higher order corrections to first order likelihood methods namely i that involving effects of fitting nuisance parameters and leading to the modified profile likelihood and ii another part pertaining to limitation in adjusted information generally each of these involves a first order adjustment depending on the reference set however we show that for some important settings likelihood irrelevant model specifications have a second order effect on both of these adjustments this result includes specification of the censoring model for survival data on the other hand for sequential experiments the likelihood irrelevant specification of the stopping rule has a second order effect on adjustment i but a first order effect on adjustment ii these matters raise the issue of what are ideal frequentist inferences since consideration of exact frequentist inferences will not suffice we indicate that to second order ideal frequentist inferences may be based on the distribution of the ordinary likelihood ratio statistic without commonly considered adjustments thereto

on birkhoff interpolation by polynomials in several variables chui and lai 1987 have discussed a kind of multivariate polynomial interpolation problem defined on the straight line type node configuration c sltncc in this paper we define general birkhoff interpolation problems for the sltncc and show under some restrictions that these interpolation problems are unisolvent also we give some generalizations of the sltncc

inference for the lognormal mean and quantiles based on samples with left and right type i censoring interval estimation of the mean and quantiles of a lognormal distribution is addressed based on a type i singly censored sample a special case of interest is that of a sample containing values below a single detection limit generalized inferential procedures which use maximum likelihood estimation based approximate pivotal quantities and some likelihood based methods are proposed the latter include methodology based on the signed log likelihood ratio test slrt statistic and the modified signed log likelihood ratio test mslrt statistic the merits of the methods are evaluated for a left censored sample using monte carlo simulation for inference concerning the lognormal mean the slrt is to be preferred for left tailed testing generalized inference for right tailed testing and all three approaches provide nearly the same performance for two tailed testing these conclusions hold even when the proportion of censored values is as large as 0 70 for inference concerning quantiles both the generali

simple modifications for signed roots of likelihood ratio statistics summary in the context of inference about a scalar parameter in the presence of nuisance parameters some simple modifications for the signed root of the log likelihood ratio statistic r are developed that reduce the order of error in the standard normal approximation to the distribution of r from o n 1 2 to o n 1 barndorff nielsen has introduced a variable u such that the error in the standard normal approximation to the distribution of r r r log u r is of order o n 3 2 but calculation of urequires the specification of an exact or approximate ancillary statistic a this paper proposes an alternative variable to u denoted by t that is available without knowledge of a and satisfies t u op n 1 in general thus the standard normal approximation to the distribution of r r log t r has error of order o n 1 and it can be used to construct approximate confidence limits having coverage error of order o n 1 in certain cases however tand uare identical the derivation of tinvolves the bayesian approach to constructing confidence limits considered by welch and peers and peers similar modifications for the signed root of the conditional likelihood ratio statistic are also developed and these modifications are seen to be useful when a large number of nuisance parameters are present several examples are presented including inference for natural parameters in exponential models and inference about location scale models with type ii censoring in each case the relationship between tand u is discussed numerical examples are also given including inference for regression models inference about the means of log normal distributions and inference for exponential lifetime models with type i censoring where barndorff nielsen s variable u is not available

modification of the liu davidson method for obtaining one or simultaneously several eigensolutions of a large real symmetric matrix a new and efficient iteration method for obtaining simultaneously several eigensolutions and even for obtaining only one solution of a large real symmetric matrix is presented by modifying the simultaneous expansion method by davidson and liu the method is basically the ritz iteration method to correct trial vector s simultaneously using correction vectors however the number of the correction vectors determined in each iteration need not be the same as the number of the desired solutions it is advantageous for the former number to be smaller than the latter when many eigensolutions are sought in addition trial vectors which need not be obtained exactly nor be corrected are included so as to bound the desired solutions which are to be obtained exactly from above and or below when interior solutions are sought the correction vector space may be kept constant throughout the iterations when the correction vector space has room enough to include the trial vectors of the previous iterations the old trial vectors can serve as expansion vectors the performance of the algorithm is tested for five matrices and is compared with the original liu davidson algorithm the results demonstrate that the present algorithm attains efficient use of memory space and reduction of iteration cycles arithmetic operations and i o processings

the proper interpolation space for multivariate birkhoff interpolation multivariate birkhoff interpolation problem has many important applications such as in finite element method in this paper two algorithms are given to compute the basis of the minimal interpolation space and the lower interpolation space respectively for an arbitrary given node set and the corresponding interpolation conditions on each node we can get the monomial basis newton type basis as well as lagrange type basis the interpolation polynomial can be derived from the basis directly

a simple general formula for tail probabilities for frequentist and bayesian inference summary we describe a simple general formula for approximating the p value for testing a scalar parameter in the presence of nuisance parameters the formula covers both frequentist and bayesian contexts and does not require explicit nuisance parameterisation implementation is discussed in terms of computer algebra packages examples are given and the relationship to barndorff nielsen s approximation is discussed

successive improvement of the order of ancillarity summary it is shown by an argument similar to one of welch 1947 that it is under usual regularity conditions in general possible to improve the order of ancillarity successively this means that when an asymptotic ancillary has been constructed such that its distribution is independent of the parameter apart from terms of order o n k then a new ancillary may be constructed with a distribution which depends on the parameter only through terms of order o nk22 it is shown how the new ancillary may be constructed explicitly when the expansion is of the edgeworth type more specifically when it is a normal density multiplied by linear combinations of hermite polynomials

grobner bases h bases and interpolation the paper is concerned with a construction for h bases of polynomial ideals without relying on term orders the main ingredient is a homogeneous reduction algorithm which orthogonalizes leading terms instead of completely canceling them this allows for an extension of buchberger s algorithm to construct these h bases algorithmically in addition the close connection of this approach to minimal degree interpolation and in particular to the least interpolation scheme due to de boor and ron is pointed out

on the history of multivariate polynomial interpolation multivariate polynomial interpolation is a basic and fundamental subject in approximation theory and numerical analysis which has received and continues receiving not deep but constant attention in this short survey we review its development in the first 75 years of this century including a pioneering paper by kronecker in the 19th century

on the singularity of multivariate hermite interpolation in this paper we study the singularity of multivariate hermite interpolation of type total degree we present two methods to judge the singularity of the interpolation schemes considered and by methods to be developed we show that all hermite interpolation of type total degree on m d k points in r d is singular if d 2k and then we solve the hermite interpolation problem on email protected d 3 nodes completely precisely all hermite interpolations of type total degree on email protected d 1 points with d 2 are singular only three cases for m d 2 and one case for m d 3 can produce regular hermite interpolation schemes respectively besides we also present a method to compute the interpolation space for hermite interpolation of type total degree

newton basis for multivariate birkhoff interpolation multivariate birkhoff interpolation is the most complex polynomial interpolation problem and people know little about it so far in this paper we introduce a special new type of multivariate birkhoff interpolation and present a newton paradigm for it using the algorithms proposed in this paper we can construct a hermite system for any interpolation problem of this type and then obtain a newton basis for the problem w r t the hermite system

a new splitting to solve a large hermitian eigenproblem abstract a particular case of rohe s convergent splitting is presented which gives a rapidly converging method of estimating the lowest eigenvalues and corresponding vectors of a large hermitian matrix when a leading principal minor of that matrix provides a good approximation to the desired eigensolution this is particularly relevant for example to the total energy calculations of solids

approximations of marginal tail probabilities for a class of smooth functions with applications to bayesian and conditional inference summary this paper presents an asymptotic approximation of marginal tail probabilities for a real valued function of a random vector where the function has continuous gradient that does not vanish at the mode of the joint density of the random vector this approximation has error 0 n 312 and improves upon a related standard normal approximation which has error 0 n 1 derivation involves the application of a tail probability formula given by diciccio field fraser 1990 to an approximation of a marginal density derived by tierney kass kadane 1989 the approximation can be applied for bayesian and conditional inference as well as for approximating sampling distributions and the accuracy of the approximation is illustrated through several numerical examples related to such applications in the context of conditional inference we develop refinements of the standard normal approximation to the distribution of two different signed root likelihood ratio statistics for a component of the natural parameter in exponential families

multivariate hermite interpolation by algebraic polynomials a survey abstract this is a survey of that theory of multivariate lagrange and hermite interpolation by algebraic polynomials which has been developed in the past 20 years its purpose is not to be encyclopedic but to present the basic concepts and techniques which have been developed in that period of time and to illustrate them with examples it takes classical hermite interpolation as a starting point but then successively broadens the assumptions so that finally interpolation of arbitrary functionals and the theory of singularities from algebraic geometry is discussed

improved likelihood inference for discrete data reference stat article 2006 001doi 10 1111 j 1467 9868 2006 00548 xview record in web of science record created on 2006 04 21 modified on 2017 05 12

highly accurate likelihood analysis for the seemingly unrelated regression problem the linear and nonlinear seemingly unrelated regression problem with general error distribution is analyzed using recent likelihood theory that arguably provides the definitive distribution for assessing a scalar parameter this involves implicit but well defined conditioning and marginalization for determining intrinsic measures of departure highly accurate p values are obtained for the key difference between two regression coefficients of central interest the p value gives the statistical position of the data with respect to the key parameter the theory and the results indicate that this methodology provides substantial improvement on first order likelihood procedures both in distributional accuracy and in precise measurement of the key parameter

approximate interval probabilities 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 a technique which in addition yields a useful expression for the error term the second approximation is based on an adjusted version of the signed log likelihood ratio statistic a third approximation is a modification of the first that avoids the specification of an ancillary

a unified approach to evaluation algorithms for multivariate polynomials we present a unified framework for most of the known and a few new evaluation algorithms for multivariate polynomials expressed in a wide variety of bases including the bezier multinomial or taylor lagrange and newton bases this unification is achieved by considering evaluation algorithms for multivariate polynomials expressed in terms of l bases a class of bases that include the bezier multinomial and a rich subclass of lagrange and newton bases all of the known evaluation algorithms can be generated either by considering recursive evaluation algorithms for l bases or by examining change of basis algorithms for l bases for polynomials of degree n in s variables the class of recursive evaluation algorithms includes a parallel up recurrence algorithm with computational complexity o n s 1 a nested multiplication algorithm with computational complexity o n s lgn and a ladder recurrence algorithm with computational complexity o n s these algorithms also generate a new generalization of the aitken neville algorithm for evaluation of multivariate polynomials expressed in terms of lagrange l bases the second class of algorithms based on certain change of basis algorithms between l bases include a dual nested multiplication algorithm with computational complexity o n s and a divided difference algorithm a forward difference algorithm and a lagrange evaluation algorithm with computational complexities o n s o 1 and o n per point respectively for the evaluation of multivariate polynomials at several points

a recursive algorithm for hermite interpolation over a triangular grid a recursive algorithm for hermite interpolation of bivariate data over triangular grids is presented this interpolation algorithm has a dynamic programming flavor and it computes a single polynomial that interpolates the full set of data the data we interpolate are partial derivatives and mixed partials up to some fixed order at the nodes of the grid the interpolant is a polynomial with minimal degree bound when the order is identical for all nodes the proposed interpolation algorithm is affinely invariant has at least linear precision is symmetric with respect to the grid directions and can reuse existing computations if points are added to the grid

regression analysis nonlinear or nonnormal simple and accurate p values from likelihood analysis we develop simple approximations for the p values to use with regression models having linear or nonlinear parameter structure and normal or nonnormal error distribution computer iteration then gives confidence intervals both frequentist and bayesian versions are given the approximations are derived from recent developments in likelihood analysis and have third order accuracy also for very small and medium sized samples the accuracy can typically be high the likelihood basis of the procedure seems to provide the grounds for this general accuracy examples are discussed and simulations record the distributional accuracy

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

on the common mean of several inverse gaussian distributions based on a higher order likelihood method an interval estimation method for the common mean of several heterogeneous inverse gaussian ig populations is discussed the proposed method is based on a higher order likelihood based procedure the merits of the proposed method are numerically compared with the signed log likelihood ratio statistic two generalized pivot quantities and the simple t test method with respect to their expected lengths coverage probabilities and type i errors numerical studies show that the coverage probabilities of the proposed method are very accurate and type i errors are close to the nominal level 05 even for very small samples the methods are also illustrated with two examples

accurate directional inference for vector parameters in linear exponential families we consider inference on a vector valued parameter of interest in a linear exponential family in the presence of a finite dimensional nuisance parameter based on higher order asymptotic theory for likelihood we propose a directional test whose p value is computed using one dimensional integration the work simplifies and develops earlier research on directional tests for continuous models and on higher order inference for discrete models and the examples include contingency tables and logistic regression examples and simulations illustrate the high accuracy of the method which we compare with the usual likelihood ratio test and with an adjusted version due to skovgaard in high dimensional settings such as covariance selection the approach works essentially perfectly whereas its competitors can fail catastrophically

default priors for bayesian and frequentist inference summary we investigate the choice of default priors for use with likelihood for bayesian and frequentist inference such a prior is a density or relative density that weights an observed likelihood function leading to the elimination of parameters that are not of interest and then a density type assessment for a parameter of interest for independent responses from a continuous model we develop a prior for the full parameter that is closely linked to the original bayes approach and provides an extension of the right invariant measure to general contexts we then develop a modified prior that is targeted on a component parameter of interest and by targeting avoids the marginalization paradoxes of dawid and co workers this modifies jeffreys s prior and provides extensions to the development of welch and peers these two approaches are combined to explore priors for a vector parameter of interest in the presence of a vector nuisance parameter examples are given to illustrate the computation of the priors