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

robust bounded influence tests in general parametric models abstract we introduce robust tests for testing hypotheses in a general parametric model these are robust versions of the wald scores and likelihood ratio tests and are based on general m estimators their asymptotic properties and influence functions are derived it is shown that the stability of the level is obtained by bounding the self standardized sensitivity of the corresponding m estimator furthermore optimally bounded influence tests are derived for the wald and scores type tests applications to real and simulated data sets are given to illustrate the tests performance

large deformations of rotating polygonal space structures the effects of rotation on space structures are very important nowadays because the structures are large and thus relatively flexible unacceptable stresses or deformations may occur due to centrifugal forces in a recent paper we considered the effects of rotation on an unbalanced circular ring the present note studies a related problem i e the free rotation of polygonal frames these shapes are basic in structural design

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

a moderate deviation principle for 2 d stochastic navier stokes equations in this paper we prove a central limit theorem and establish a moderate deviation principle for twodimensional stochastic navier stokes equations with multiplicative noise the weak convergence method plays an important role

saddlepoint approximations for regression models summary this paper uses the techniques of saddlepoint or tilted exponential approximation to develop an approximation to the small sample distribution of estimators defined by a system of estimating equations when observations are independently but not identically distributed this allows for the explicit treatment of models with explanatory variables the approximation is applied to l1 regression and the accuracy of the approximation is demonstrated in a case in which the usual normal approximation is very poor an analysis of the example indicates the usefulness of a new variant of exponential tilting in calculating the transformation measure a quantity which plays an important role in assuring the accuracy of the approximation

asymptotic optimality of data driven neyman s tests for uniformity data driven neyman s tests resulting from a combination of neyman s smooth tests for uniformity and schwarz s selection procedure are investigated asymptotic intermediate efficiency of those tests with respect to the neyman pearson test is shown to be 1 for a large set of converging alternatives the result shows that data driven neyman s tests contrary to classical goodness of fit tests are indeed omnibus tests adapting well to the data at hand

empirical likelihood ratio confidence intervals for a single functional summary the empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken in this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions it is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean for a class of m estimates that includes quantiles and for differentiable statistical functionals the results are nonparametric extensions of wilks s 1938 theorem for parametric likelihood ratios the intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on student s t statistic a hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced

on the density of minimum contrast estimators conditions for the existence of the density of a minimum contrast estimator in a parametric statistical family are given together with a formula for this density the formula is exact if multiple local minima cannot occur otherwise the formula is an exact expression for the point process of local minima of the contrast function although it is not in general feasible to compute the expression for the density the formula can be used as a basis for further expansion of the large deviation type when the estimate is sufficient either in the original model or after conditioning on an approximate or exact ancillary the formula simplifies drastically in particular it is shown how barndorff nielsen s formula for the density of the maximum likelihood estimator given an ancillary statistic is derived from the formula given here in this way the nature of barndorff nielsen s formula as an asymptotic approximation and its appearance as an exact formula for certain cases are demonstrated

approximation metrics for discrete and continuous systems established system relationships for discrete systems such as language inclusion simulation and bisimulation require system observations to be identical when interacting with the physical world modeled by continuous or hybrid systems exact relationships are restrictive and not robust in this paper we develop the first framework of system approximation that applies to both discrete and continuous systems by developing notions of approximate language inclusion approximate simulation and approximate bisimulation relations we define a hierarchy of approximation pseudo metrics between two systems that quantify the quality of the approximation and capture the established exact relationships as zero sections our approximation framework is compositional for a synchronous composition operator algorithms are developed for computing the proposed pseudo metrics both exactly and approximately the exact algorithms require the generalization of the fixed point algorithms for computing simulation and bisimulation relations or dually the solution of a static game whose cost is the so called branching distance between the systems approximations for the pseudo metrics can be obtained by considering lyapunov like functions called simulation and bisimulation functions we illustrate our approximation framework in reducing the complexity of safety verification problems for both deterministic and nondeterministic continuous systems

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

on the bootstrap saddlepoint approximations we compare saddlepoint approximations to the exact distributions of a studentized mean and to its bootstrap approximation we show that on bounded sets these empirical saddlepoint approximations achieve second order relative errors uniformly we also consider the relative errors for larger deviations it follows that the studentized t bootstrap p value and the coverage of the bootstrap confidence interval have second order relative errors

opacity of nondeterministic transition systems a bi simulation relation approach in this paper we propose several opacity preserving bi simulation relations for nondeterministic transition systems ntss in terms of initial state opacity current state opacity k step opacity and infinite step opacity we also show how one can leverage quotient constructions to compute such relations as a result although the opacity verification problem for infinite ntss is generally undecidable if one can find such an opacity preserving relation from an infinite nts to a finite one the lack of opacity of the infinite nts can be easily verified over the finite one which is decidable

numerical linear algebra aspects of globally convergent homotopy methods probability one homotopy algorithms are a class of methods for solving nonlinear systems of equations that are globally convergent with probability one these methods are theoretically powerful and if constructed and implemented properly are robust numerically stable accurate and practical the concomitant numerical linear algebra problems deal with rectangular matrices and good algorithms require a delicate balance not always achieved of accuracy robustness and efficiency in both space and time the author s experience with globally convergent homotopy algorithms is surveyed here and some of the linear algebra difficulties for dense and sparse problems are discussed

the likelihood ratio approximation to the conditional distribution of the maximum likelihood estimator in the discrete case summary the likelihood ratio approximation also called barndorff nielsen s approximation and often denoted by p provides a highly accurate approximation to the conditional density of a maximum likelihood estimator 0 given an ancillary statistic in this paper the properties of p are considered for the case in which the underlying random variables have a lattice distribution and 0 has a discrete but not necessarily lattice distribution if 0 has a lattice distribution then p provides a valid approximation to the density of 0 with respect to counting measure if the distribution of 0 is non lattice then p is still a valid approximation to the conditional density of 0 however the dominating measure is no longer counting measure

saddlepoint approximation for moment generating functions of truncated random variables we consider the problem of approximating the moment generating function mgf of a truncated random variable in terms of the mgf of the underlying i e untruncated random variable the purpose of approximating the mgf is to enable the application of saddlepoint approximations to certain distributions determined by truncated random variables two important statistical applications are the following the approximation of certain multivariate cumulative distribution functions and the approximation of passage time distributions in ion channel models which incorporate time interval omission we derive two types of representation for the mgf of a truncated random variable one of these representations is obtained by exponential tilting the second type of representation which has two versions is referred to as an exponential convolution representation each representation motivates a different approximation it turns out that each of the three approximations is extremely accurate in those cases to which it is suited moreover there is a simple rule of thumb for deciding which approximation to use in a given case and if this rule is followed then our numerical and theoretical results indicate that the resulting approximation will be extremely accurate

saddlepoint approximations for short and long memory time series a frequency domain approach abstract saddlepoint techniques provide numerically accurate small sample approximations to the distribution of estimators and test statistics except for a few simple models these approximations are not available in the framework of stationary time series we contribute to fill this gap under short or long range serial dependence for gaussian and non gaussian processes we show how to derive and implement saddlepoint approximations for two relevant classes of frequency domain statistics ratio statistics and whittle s estimator we compare our new approximations to the ones obtained by the standard asymptotic theory and by two widely applied bootstrap methods the numerical exercises for whittle s estimator show that our approximations yield accuracy s improvements while preserving analytical tractability a real data example concludes the paper

data driven version of neyman s smooth test of fit abstract neyman s smooth test for testing uniformity is a recognized goodness of fit procedure as stated by lariccia the test can be viewed as a compromise between omnibus test procedures with generally low power in all directions and procedures whose power is focused in the direction of a specific alternative the basic idea behind this test is to embed the null density into say a k dimensional exponential family and then to construct an asymptotically optimal test for the parametric testing problem the resulting procedure is neyman s test with k components the most difficult problem related with using this test is the choice of k recommendations in statistical literature are sometimes confusing some authors advocate a small number of components whereas others show that in some situations a larger number of components is profitable all existing suggestions concerning how to select k exploit in fact some preliminary knowledge about a possible alternative in this article a new data driven met

accurate and robust tests for indirect inference in this paper we propose accurate parameter and over identification tests for indirect inference under the null hypothesis the new tests are asymptotically 2 distributed with a relative error of order n 1 they exhibit better finite sample accuracy than classical tests for indirect inference which have the same asymptotic distribution but an absolute error of order n 1 2 robust versions of the tests are also provided we illustrate their accuracy in nonlinear regression poisson regression with overdispersion and diffusion models

moderate deviations of minimum contrast estimators under contamination since statistical models are simplifications of reality it is important in estimation theory to study the behavior of estimators also under distributions slightly different from the proposed model in testing theory when dealing with test statistics where nuisance parameters are estimated knowledge of the behavior of the estimators of the nuisance parameters is needed under alternatives to evaluate the power in this paper the moderate deviation behavior of minimum contrast estimators is investigated not only under the supposed model but also under distributions close to the model a particular example is the multivariate maximum likelihood estimator determined within the proposed model the set up is quite general including for instance also discrete distributions the rate of convergence under alternatives is determined both when comparing the minimum contrast estimator with a natural parameter in the parameter space and when comparing it with the proposed true value in the parameter space it turns out that under the model the asymptotic optimality of the maximum likelihood estimator in the local sense continues to hold in the moderate deviation area

saddlepoint approximations for the difference of order statistics and studentized sample quantiles summary for a sample from a given distribution the difference of two order statistics and the studentized quantile are statistics whose distribution is needed to obtain tests and confidence intervals for quantiles and quantile differences this paper gives saddlepoint approximations for densities and saddlepoint approximations of the lugannani rice form for tail probabilities of these statistics the relative errors of the approximations are n i uniformly in a neighbourhood of the parameters and this uniformity is global if the densities are log concave

saddlepoint approximations in resampling methods summary saddlepoint approximations are shown to be easy to use and accurate in a variety of simple bootstrap and randomization applications examples include mean estimation ratio estimation two sample comparisons and autoregressive estimation

indirect estimation of large conditionally heteroskedastic factor models with an application to the dow 30 stocks we derive indirect estimators of conditionally heteroskedastic factor models in which the volatilities of common and idiosyncratic factors depend on their past unobserved values by calibrating the score of a kalman filter approximation with inequality constraints on the auxiliary model parameters we also propose alternative indirect estimators for large scale models and explain how to apply our procedures to many other dynamic latent variable models we analyse the small sample behaviour of our indirect estimators and several likelihood based procedures through an extensive monte carlo experiment with empirically realistic designs finally we apply our procedures to weekly returns on the dow 30 stocks

information criteria for selecting possibly misspecified parametric models abstract we consider penalized likelihood criteria for selecting models of dependent processes the models may be strictly nested overlapping or nonnested linear or nonlinear and correctly specified or misspecified we provide sufficient conditions on the penalty to guarantee the selection with probability one or with probability approaching one of the model attaining the lower average kullback leibler information criterion klic or when both have the same klic the more parsimonious model as special cases our results describe the akaike schwarz and hannan quinn information criteria as examples we consider selection of armax garch and star models

target estimation for bias and mean square error reduction given a statistical functional t and a parametric family of distributions a bias reduced functional t is defined by setting the expected value of the statistic equal to the observed value under certain regularity conditions this new statistic called the target estimator will have smaller bias and mean square error than the original estimator the theoretical aspects are analyzed by using higher order von mises expansions several examples are given including m estimates of location and scale the procedure is applied to an autoregressive model the errors in variables model and the logistic regression model a comparison with the jackknife and the bootstrap estimators is also included

free rotation of a circular ring with an unbalanced mass large space structures are much more flexible than their terrestrial counterparts rotation of large space structures may be desirable for stability thermal or artificial gravity reasons previous literature includes the large deformations due to the free rotation of a slender rod and a ring about a diameter an important model of a space station is a ring rotating about its axis of symmetry if the ring is balanced it will be stable and remain circular the present note considers the case when the ring is unbalanced by a mass attached to a point on the ring

decidability of opacity verification problems in labeled petri net systems abstract a system is said to be opaque if an intruder that observes its evolution through a mask cannot infer that the system s evolution belongs to a given secret behavior opacity verification is the problem of determining whether the system is opaque with respect to a given secret or not in this paper we address the decidability of the opacity verification problem using reduction approaches we show that verification of initial state current state and language opacity is undecidable in labeled petri nets

data driven smooth tests when the hypothesis is composite in recent years several authors have recommended smooth tests for testing goodness of fit however the number of components in the smooth test statistic should be chosen well otherwise considerable loss of power may occur schwarz s selection rule provides one such good choice earlier results on simple null hypotheses are extended here to composite hypotheses which tend to be of more practical interest for general composite hypotheses consistency of the data driven smooth tests holds at essentially any alternative monte carlo experiments on testing exponentiality and normality show that the data driven version of neyman s test compares well to other even specialized tests over a wide range of alternatives

data driven smooth tests for composite hypotheses the classical problem of testing goodness of fit of a parametric family is reconsidered a new test for this problem is proposed and investigated the new test statistic is a combination of the smooth test statistic and schwarz s selection rule more precisely as the sample size increases an increasing family of exponential models describing departures from the null model is introduced and schwarz s selection rule is presented to select among them schwarz s rule provides the right dimension given by the data while the smooth test in the right dimension finishes the job theoretical properties of the selection rules are derived under null and alternative hypotheses they imply consistency of data driven smooth tests for composite hypotheses at essentially any alternative

the density of multivariate m estimates when a unique m estimate exists its density is obtained as a corollary to a more general theorem which asserts that under mild conditions the intensity function of the point process of solutions of the estimating equations exists and is given by the density of the estimating function standardized by multiplying it by the inverse of its derivative we apply the results to give a result for huber s proposal 2 applied to regression and scale estimates we also give a saddlepoint approximation for the density and use this to give approximations for tail areas for smooth functions of the m estimates

symbolic models for nonlinear control systems without stability assumptions finite state models of control systems were proposed by several researchers as a convenient mechanism to synthesize controllers enforcing complex specifications most techniques for the construction of such symbolic models have two main drawbacks either they can only be applied to restrictive classes of systems or they require the exact computation of reachable sets in this paper we propose a new abstraction technique that is applicable to any nonlinear sampled data control system as long as we are only interested in its behavior in a compact set moreover the exact computation of reachable sets is not required the effectiveness of the proposed results is illustrated by synthesizing a controller to steer a vehicle

bounds for probabilities of small relative errors for empirical saddlepoint and bootstrap tail approximations to obtain test probabilities based on empirical approximations to the distribution of a studentized function of a mean we need the approximations to be accurate with sufficiently high probability in particular when these test probabilities are small it is best to consider relative errors here we show that in the case of univariate standardized means and in the general case of tests based on smooth functions of means the empirical approximations have asymptotically small relative errors on sets with probability differing from 1 by an exponentially small quantity and that these error rates hold for moderately large deviations in particular for standardized deviations of order n 1 6 the probabilities approximated are exponentially small with exponents of order n 1 3 and the corresponding relative errors tend to zero on sets whose complements have probabilities of the order of the probabilities being approximated

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

approximate marginal densities of nonlinear functions this paper presents an asymptotic approximation for the marginal density of a nonlinear function g 0 that is applicable when the joint density of 0 is dominated by a single mode and the jacobian of g is of full rank near that mode the approximation is based on laplace s method and its asymptotic properties are similar to those of the saddlepoint approximation the approximation is applied to the computation of a marginal posterior density a marginal sampling density and a marginal density based on a multivariate saddlepoint approximation to a joint density

general saddlepoint approximations of marginal densities and tail probabilities saddlepoint approximations of marginal densities and tail probabilities of general nonlinear statistics are derived these are based on the expansion of the statistic up to the second order their accuracy is shown in a variety of examples including logit and probit models and rank estimators for regression

a new approach for the verification of infinite step and k step opacity using two way observers abstract in the context of security analysis for information flow properties where a potentially malicious observer intruder tracks the observed behavior of a given system infinite step opacity respectively k step opacity holds if the intruder can never determine for sure that the system was in a secret state for any instant within infinite steps respectively k steps prior to that particular instant we present new algorithms for the verification of the properties of infinite step opacity and k step opacity for partially observed discrete event systems modeled as finite state automata our new algorithms are based on a novel separation principle for state estimates that characterizes the information dependence in opacity verification problems and they have lower computational complexity than previously proposed ones in the literature specifically we propose a new information structure called the two way observer that is used for the verification of infinite step and k step opacity based on the two way observer a new upper bound for the delay in k step opacity is derived which also improves previously known results

a fully automated framework for control of linear systems from temporal logic specifications we consider the following problem given a linear system and a linear temporal logic ltl formula over a set of linear predicates in its state variables find a feedback control law with polyhedral bounds and a set of initial states so that all trajectories of the closed loop system satisfy the formula our solution to this problem consists of three main steps first we partition the state space in accordance with the predicates in the formula and construct a transition system over the partition quotient which captures our capability of designing controllers second using a procedure resembling model checking we determine runs of the transition system satisfying the formula third we generate the control strategy illustrative examples are included

saddlepoint approximations for marginal and conditional probabilities of transformed variables the lugannani and rice formula for tail areas in the univariate case has recently been extended to tail areas for marginal distributions and for conditional distributions in certain multivariate settings however the results on relative order of errors are given only formally or under the strong continuity assumptions necessary to obtain density approximations this paper attempts to give a unified treatment of these results for smooth transformations of multivariate means under weaker conditions appropriate to indirect edgeworth approximations for probabilities

testing the fit of a parametric function abstract general methods for testing the fit of a parametric function are proposed the idea underlying each method is to accept the prescribed parametric model if and only if it is chosen by a model selection criterion several different selection criteria are considered including one based on a modified version of the akaike information criterion and others based on various score statistics the tests have a connection with nonparametric smoothing because they use orthogonal series estimators to detect departures from a parametric model an important aspect of the tests is that they can be applied in a wide variety of settings including generalized linear models spectral analysis the goodness of fit problem and longitudinal data analysis implementation using standard statistical software is straightforward asymptotic distribution theory for several test statistics is described and the tests are shown to be consistent against essentially any alternative hypothesis simulations and a data exampl

high breakdown inference for mixed linear models mixed linear models are used to analyze data in many settings these models have a multivariate normal formulation in most cases the maximum likelihood estimator mle or the residual mle reml is usually chosen to estimate the parameters however the latter are based on the strong assumption of exact multivariate normality welsh and richardson have shown that these estimators are not robust to small deviations from multivariate normality this means that in practice a small proportion of data even only one can drive the value of the estimates on their own because the model is multivariate we propose a high breakdown robust estimator for very general mixed linear models that include for example covariates this robust estimator belongs to the class of s estimators from which we can derive asymptotic properties for inference we also use it as a diagnostic tool to detect outlying subjects we discuss the advantages of this estimator compared with other robust estimators proposed previously and i

practical small sample asymptotics for regression problems abstract saddlepoint approximations are derived for sums of independent but not necessarily identically distributed random variables along with generalizations to estimating equations and multivariate problems these results are particularly useful for accurately approximating the distribution of regression coefficients general formulas are given for the distribution of the coefficients arising out of a generalized linear model with both canonical and noncanonical link functions we illustrate the case of logistic regression with a real data example and show how the gibbs sampler may be used to obtain confidence sets for each regression parameter based on the saddlepoint approximation

simulation estimation of time series models abstract a formal econometric treatment of the estimation of the parameters of a fully specified stochastic equilibrium model is proposed the method estimation by simulation yields an estimator which is shown to have an asymptotic normal distribution a goodness of fit test based on a chi square statistic is also derived

general saddlepoint approximations with applications to l statistics abstract saddlepoint approximations are extended to general statistics the technique is applied to derive approximations to the density of linear combinations of order statistics including trimmed means a comparison with exact results shows the accuracy of these approximations even in very small sample sizes

saddlepoint approximations and tests based on multivariate m estimates we consider multidimensional m functional parameters defined by expectations of score functions associated with multivariate m estimators and tests for hypotheses concerning multidimensional smooth functions of these parameters we propose a test statistic suggested by the exponent in the saddlepoint approximation to the density of the function of the m estimates this statistic is analogous to the log likelihood ratio in the parametric case we show that this statistic is approximately distributed as a chi squared variate and obtain a lugannani rice style adjustment giving a relative error of order we propose an empirical exponential likelihood statistic and consider a test based on this statistic finally we present numerical results for three examples including one in robust regression

empirical saddlepoint approximations for multivariate m estimators summary in this paper we investigate the use of the empirical distribution function in place of the underlying distribution function f to construct an empirical saddlepoint approximation to the density fn of a general multivariate m estimator we obtain an explicit form for the error term in the approximation investigate the effect of renormalizing the estimator carry out some numerical comparisons and discuss the regression problem the sampling distribution of a statistic is a basic tool for evaluating the properties of the statistic and for constructing inference procedures based on the statistic the density of the sampling distribution may be used to compute moments to develop approximations or to compute the distribution function and hence various tail prob abilities which can be used to construct tests or confidence intervals when the exact density is intractable it may be possible to approximate it by a normal density an edgeworth expansion or a saddlepoint approximation daniels 1954 the saddle point approximation is derived from an asymptotic expansion but is often very accurate even in quite small samples to highlight this property hampel 1973 described these and related techniques as small sample asymptotic approximations as can be seen from equation 3 later the main property of this approximation and its major advantage over edgeworth expansions is that it is always non negative and the relative error is uniformly of order n 1 for recent reviews see reid 1988 barndorff nielsen and cox 1989 and field and ronchetti 1990 the construction of these approximations typically requires knowledge of the underlying distribution however just as we can estimate the variance of a normal approxima tion or estimate the terms in an edgeworth expansion we can estimate the saddle point approximation in this paper we examine the properties of an estimate of the saddlepoint approximation our investigation is motivated by three considerations firstly the estimated approximation may be useful for comparing various estimators on a given data set if the sampling distribution is non normal at least in principle we need to compare the whole distribution and comparisons based on the estimated asymptotic variance alone may be misleading secondly the estimated approximation may be useful for evaluating the quality of simpler approximations including the normal approxima tion finally it may be useful for developing inference procedures methods like

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

saddlepoint test in measurement error models we develop second order hypothesis testing procedures in functional measurement error models for small or moderate sample sizes where the classical first order asymptotic analysis often fails to provide accurate results in functional models no distributional assumptions are made on the unobservable covariates and this leads to semiparametric models our testing procedure is derived using saddlepoint techniques and is based on an empirical distribution estimation subject to the null hypothesis constraints in combination with a set of estimating equations which avoid a distribution approximation the validity of the method is proved in theorems for both simple and composite hypothesis tests and is demonstrated through simulation and a farm size data analysis

simple and accurate one sided inference based on a class of m estimators summary this paper presents a simple method of calculating one sided p values which yield accurate inferences for a simple null hypothesis or equivalently accurate one sided con fidence limits regarding a scalar parameter in the presence of nuisance parameters the method which extends the work of diciccio et al 2001 and lee young 2005 in the context of adjusted likelihood estimation is based on a test statistic analogous to the signed root of the loglikelihood but derived from the objective function of an m estimator from a certain class monte carlo simulation is used to avoid the need for onerous analytical calculations typical of competing procedures based on edgeworth or saddle point approximations the specific class of m estimators under consideration is character ised by the requirement that the associated test statistic has constant variance to second order this class contains among others the maximum likelihood estimator as well as variants of commonly used m estimators of location such as huber s 1964 proposal 2

on the relationship between empirical likelihood and empirical saddlepoint approximation for multivariate m estimators by comparing the expansions of the empirical log likelihood ratio and the empirical cumulant generating function calculated at the saddlepoint we investigate the relationship between empirical likelihood and empirical saddlepoint approximations this leads to a nonparametric approximation of the density of a multivariate m estimator based on the empirical likelihood and on the other hand it provides nonparametric confidence regions based on the empirical cumulant generating function some examples illustrate the use of the empirical likelihood in saddlepoint approximations and vice versa

a second order investigation of asymptotic ancillarity the paper deals with approximate ancillarity as discussed by efron and hinkley 1978 in the multivariate i i d case we derive the second order edgeworth expansion of the mle given a normalized version of the second derivative of the log likelihood at its maximum the expansion agrees with the one derived by amari 1982a for curved exponential families but holds for any family satisfying the regularity conditions given in the paper it is shown that the fisher information lost by reducing the data to the mle is recovered by the conditioning and it is sketched how the loss of information relates to the deficiency as defined by lecam finally we investigate some properties of three test statistics proving a conjecture by efron and hinkley 1978 concerning the conditional null distribution of the likelihood ratio test statistic and establishing a kind of superiority of the observed fisher information over the expected one as estimate of the inverse variance of the mle

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

generalized laguerre expansions of multivariate probability densities with moments we generalize the well known laguerre series approach to approximate multivariate probability density functions pdfs using multidimensional laguerre polynomials the generalized laguerre series which is defined around a gamma pdf is suited for simulating high complex natural phenomena that deviate from gaussianity combining the multivariate laguerre approximation and bayes theorem an approximation to the conditional pdfs is derived numerical results first showed the superiority of the gamma expansion over other numerical methods the ability of the gamma expansion to fit mixtures of gaussian ans super gaussian pdfs univariate and multivariate lognormal pdfs and complex geologic media is shown through different examples

verification of initial state opacity in security applications of discrete event systems in this paper we formulate and analyze methodologies for verifying the notion of initial state opacity in discrete event systems that are modeled as non deterministic finite automata with partial observation on their transitions a system is initial state opaque if the membership of its true initial state to a set of secret states remains opaque i e uncertain to an intruder who observes system activity through some projection map initial state opacity can be used to characterize security requirements in a variety of applications including tracking problems in sensor networks in order to model and analyze the intruder capabilities regarding initial state opacity we first address the initial state estimation problem in a non deterministic finite automaton via the construction of an initial state estimator we analyze the properties and complexity of the initial state estimator and show how the complexity of the verification method can be greatly reduced in the special case when the set of secret states is invariant we also establish that the verification of initial state opacity is a pspace complete problem

delta method in large deviations and moderate deviations for estimators the delta method is a popular and elementary tool for deriving limiting distributions of transformed statistics while applications of asymptotic distributions do not allow one to obtain desirable accuracy of approximation for tail probabilities the large and moderate deviation theory can achieve this goal motivated by the delta method in weak convergence a general delta method in large deviations is proposed the new method can be widely applied to driving the moderate deviations of estimators and is illustrated by examples including the wilcoxon statistic the kaplan meier estimator the empirical quantile processes and the empirical copula function we also improve the existing moderate deviations results for m estimators and l statistics by the new method some applications of moderate deviations to statistical hypothesis testing are provided

the modified signed likelihood statistic and saddlepoint approximations summary for a number of tests in exponential families we show that the use of a normal approximation to the modified signed likelihood ratio statistic r is equivalent to the use of a saddlepoint approximation this is also true in a large deviation region where the signed likelihood ratio statistic r is of order in

small sample confidence intervals abstract in this article we present a technique for constructing one or two sided confidence intervals which are second order correct in terms of coverage for either parametric or nonparametric problems the construction is valid in the presence of nuisance parameters the situation we consider is this there are p parameters and we want a confidence interval for some function of them possibly one of the parameters itself the p parameters are estimated by m estimates which means they are obtained as a solution of a system of equations maximum likelihood estimates are included as a special case the essential intermediate result given in equation 3 4 says that the estimated parameter of interest can be written as a mean up to order o p 1 n the representation 3 4 is attained when is a smooth function of the parameters and is the solution of a well behaved system of equations we avoid the use of pivots and strive to obtain accurate coverage confidence intervals are construc

large deviations of estimators the performance of a sequence of estimators t n of g theta can be measured by its inaccuracy rate lim inf n rightarrow infty n 1 log mathbb p theta t n g theta varepsilon for fixed varepsilon 0 optimality of consistent estimators operatorname wrt the inaccuracy rate is investigated it is shown that for exponential families in standard representation with a convex parameter space the maximum likelihood estimator is optimal if the parameter space is not convex which occurs for instance in curved exponential families in general no optimal estimator exists for the location problem the inaccuracy rate of m estimators is established if the underlying density is sufficiently smooth an optimal m estimator is obtained within the class of translation equivariant estimators tail behaviour of location estimators is studied a connection is made between gross error and inaccuracy rate optimality

small sample asymptotic distributions of m estimators of location summary asymptotic formulae for the distribution of m estimators i e maximum likelihood type estimators of location including the arithmetic mean are derived which numerical studies show to give relative errors for densities and tail areas of the order of magnitude of 1 down to sample sizes 3 and 4 even in the extreme tails the paper is the continuation of earlier work by the second author and is also closely related to daniels s work on the saddlepoint approximation the method consists in expanding the derivative of the logarithm of the unstandardized density of the estimator in powers of 1 n at each point using recentring by means of conjugate distributions this method yields a unified point of view for the comparison of other asymptotic methods namely saddlepoint method edgeworth expansion and large deviations approach which are also compared numerically

a conditional saddlepoint approximation for testing problems a saddlepoint approximation is provided for the distribution function of one m statistic conditional on another m statistic many interesting statistics based on dependent quantities e g spacings multinomial frequencies rank differences can be expressed in terms of independent identically distributed random variables conditioned on their sum so that this conditional saddlepoint approximation yields accurate approximations for the distribution of such statistics this saddlepoint approximation can also be used in conditional testing where nuisance parameters are eliminated by conditioning on sufficient statistics

a moderate deviation principle for 2 d stochastic navier stokes equations driven by multiplicative levy noises abstract in this paper we establish a moderate deviation principle for two dimensional stochastic navier stokes equations driven by multiplicative levy noises the weak convergence method introduced by budhiraja dupuis and ganguly in 3 plays a key role

high order accurate methods for retrospective sampling problems in this paper we discuss the relationship between prospective and retrospective sampling problems estimates of the parameter of interest may be obtained by solving suitable estimating equations under both sampling schemes most common examples of such estimates include the maximum likelihood estimates some classical results and more recent development of the first order asymptotic relationship between the estimators are reviewed high order expansions are given for the distributions of the retrospective estimators expansions for the marginal distributions of interest are obtained for both prospective and retrospective data furthermore it is shown that the two expansions are asymptotically equal at least up to order o n 1 this implies that readily available prospective saddlepoint methods may be applied to the analysis of retrospective data without loss of high order accuracy the results are also briefly illustrated numerically

infereni on full or partial parameters based on the standardized signed log likelihood ratio on etudie des tests et des limites de confiance bases sur la statistique du rapport de log vraisemblance signee en ajustant cette statistique de telle sorte qu a un ordre superieur d approximation la distribution normale standard soit valable

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

higher order asymptotics and its application to testing the equality of the examinee ability over two sets of items in educational and psychological measurement researchers and or practitioners are often interested in examining whether the ability of an examinee is the same over two sets of items such problems can arise in measurement of change detection of cheating on unproctored tests erasure analysis detection of item preknowledge etc traditional frequentist approaches that are used in such problems include the wald test the likelihood ratio test and the score test e g fischer appl psychol meas 27 3 26 2003 finkelman weiss kim kang appl psychol meas 34 238 254 2010 glas dagohoy psychometrika 72 159 180 2007 guo drasgow int j sel assess 18 351 364 2010 klauer rettig br j math stat psychol 43 193 206 1990 sinharay j educ behav stat 42 46 68 2017 this paper shows that approaches based on higher order asymptotics e g barndorff nielsen cox inference and asymptotics springer london 1994 ghosh higher order asymptotics institute of mathematical statistics hayward 1994 can also be used to test for the equality of the examinee ability over two sets of items the modified signed likelihood ratio test e g barndorff nielsen biometrika 73 307 322 1986 and the lugannani rice approximation lugannani rice adv appl prob 12 475 490 1980 both of which are based on higher order asymptotics are shown to provide some improvement over the traditional frequentist approaches in three simulations two real data examples are also provided