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

saddlepoint approximations for the generalized variance and wilks statistic saddlepoint approximations to densities of sums of random variables were introduced by daniels 1954 these methods were further developed by barndorff nielsen cox 1979 to approximate joint and conditional densities for canonical sufficient statistics in exponential family settings approximations for the cumulative distributions of sums were developed by lugannani rice 1980 and robinson 1982 while the approximation of conditional cumulative distribution functions was addressed by skovgaard 1987 the modifications necessary to apply the lugannani rice and skovgaard approximations to conditional distributions of canonical sufficient statistics in exponential family settings have been summarized by davison 1988 reid 1988 provides an overall review of the saddlepoint literature in the present paper the lugannani rice and skovgaard approximations are used to reproduce the distributions of the generalized variance and wilks likelihood ratio statistic in a multivatiate analysis of variance representing the degree of freedom in a gamma regression model as an appropriate linear function the generalized variance distribution arises as the marginal distribution of a canonical sufficient statistic to which the lugannani rice approximation may be applied the more recent approximations of fraser 1990 and barndorff nielsen 1990 are exactly the same for this setting comparison of its accuracy with the approximation of hoel 1937 as well as a normal approximation johnson kotz 1972 p 198 shows the lugannani rice approximation to be highly accurate and uniformly better than its competitors the distribution of wilks likelihood ratio statistic has two distinct characterizations in terms of products of independent beta variates anderson 1958 ch 8 in conjunction with these characterizations we use skovgaard s approximation to construct two distinct

laplace approximation for bessel functions of matrix argument we derive laplace approximations to three functions of matrix argument which arise in statistics and elsewhere matrix bessel av matrix bessel bv and the type ii confluent hypergeometric function of matrix argument we examine the theoretical and numerical properties of the approximations on the theoretical side it is shown that the laplace approximations to av bv and given here together with the laplace approximations to the matrix argument functions 1f1 and 2f1 presented in butler and wood laplace approximations to hyper geometric functions with matrix argument ann statist 2002 satisfy all the important confluence relations and symmetry relations enjoyed by the original functions

saddlepoint approximations for the bartlett nanda pillai trace statistic in multivariate analysis summary the null distribution of the bartlett nanda pillai trace statistic is shown to be equivalent to the conditional distribution of canonical sufficient statistics in a multiparameter exponential family setting as such sequential and double saddlepoint approximations are used to approximate the distribution the existence of tabulated percentiles for the distribution allows for comparison of the accuracy of the various saddlepoint approximations the ability of the approximations to accommodate large numbers of nuisance parameters is also revealed with numerical studies the bartlett nanda pillai trace statistic is one of the four principal test statistics used in the multivariate analysis of variance p values of tests based on it are computed from its null distribution we show in 2 that this null distribution is equivalent to a conditional distribution involving components of the canonical sufficient statistic in the context of a certain multiparameter regular exponential family such equivalence allows for the possibility of saddlepoint approximation for both the null density and distribution function so that highly accurate approximations for p values can be easily obtained section 3 reviews the various saddlepoint methods that we shall base approximations upon sequential saddlepoint methods introduced by fraser reid wong 1991 are used to compute a sequential density approximation as well as distributional approximations based on the lugannani rice 1980 formula and the r approximation of bamdorff nielsen 1986 1991 double saddlepoint analogues to the above methods are also computed a double saddlepoint density approximation bamdorff nielsen cox 1979 takes the explicit form in 3 8 below as a modification to a rescaled beta density this explicit form is due to the explicit solution 4 11 that arises from the double saddlepoint equation double saddlepoint distributional approximations based on skovgaard 1987 and the r approximation bamdorff nielsen 1986 1991 are also computed the numerical comparisons of 6 reveal that the double saddlepoint distributional approximations of skovgaard 1987 and bamdorff nielsen 1986 1991 are grossly

a numerically robust state space approach to stable predictive control strategies recent work with predictive control strategies has shown that the stable gpc sgpc approach has significant computational and numerical advantages however sgpc is cast in the transfer function framework which limits its application here we develop a means of extending the results to state space models and show that improved numerical conditioning can be obtained for many stable predictive control strategies

exponential linear models a two pass procedure for saddlepoint approximation for an exponential linear model the saddlepoint method gives accurate approximations for the density of the minimal sufficient statistic or maximum likelihood estimate and for the corresponding distribution functions in this paper we describe a simple numerical procedure that constructs such approximations for a real parameter in an exponential linear model using only a two pass calculation on the observed likelihood function for the original data simple examples of the numerical procedure are discussed but we take the general accuracy of the saddlepoint procedure as given an immediate application of this is to exponential family models where inference for a component of the canonical parameter is to be based on the conditional density of the corresponding component of the sufficient statistic given the remaining components this conditional density is also of exponential family form but its functional form and cumulant generating function may not be accessible the procedure is applied to the corresponding likelihood approximated as the full likelihood divided by an approximate marginal likelihood obtained from barndorff nielsen s formula a double saddlepoint approximation provides another means of bypassing this difficulty the computational procedure is also examined as a numerical procedure for obtaining the saddlepoint approximation to the fourier inversion of a characteristic function as such it is a two pass calculation on a table of the cumulant generating function

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

statistical eigen inference from large wishart matrices we consider settings where the observations are drawn from a zero mean multivariate real or complex normal distribution with the population covariance matrix having eigenvalues of arbitrary multiplicity we assume that the eigenvectors of the population covariance matrix are unknown and focus on inferential procedures that are based on the sample eigenvalues alone i e eigen inference results found in the literature establish the asymptotic normality of the fluctuation in the trace of powers of the sample covariance matrix we develop concrete algorithms for analytically computing the limiting quantities and the covariance of the fluctuations we exploit the asymptotic normality of the trace of powers of the sample covariance matrix to develop eigenvalue based procedures for testing and estimation specifically we formulate a simple test of hypotheses for the population eigenvalues and a technique for estimating the population eigenvalues in settings where the cumulative distribution function of the nonrandom population eigenvalues has a staircase structure monte carlo simulations are used to demonstrate the superiority of the proposed methodologies over classical techniques and the robustness of the proposed techniques in high dimensional relatively small sample size settings the improved performance results from the fact that the proposed inference procedures are global in a sense that we describe and exploit global information thereby overcoming the inherent biases that cripple classical inference procedures which are local and rely on local information

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

asymptotic power of sphericity tests for high dimensional data this paper studies the asymptotic power of tests of sphericity against perturbations in a single unknown direction as both the dimensionality of the data and the number of observations go to infinity we establish the convergence under the null hypothesis and the alternative of the log ratio of the joint densities of the sample covariance eigenvalues to a gaussian process indexed by the norm of the perturbation when the perturbation norm is larger than the phase transition threshold studied in baik et al 2005 the limiting process is degenerate and discrimination between the null and the alternative is asymptotically certain when the norm is below the threshold the process is non degenerate so that the joint eigenvalue densities under the null and alternative hypotheses are mutually contiguous using the asymptotic theory of statistical experiments we obtain asymptotic power envelopes and derive the asymptotic power for various sphericity tests in the contiguity region in particular we show that the asymptotic power of the tracy widom type tests is trivial whereas that of the eigenvalue based likelihood ratio test is strictly larger than the size and close to the power envelope

some pseudoprimes and related numbers having special forms we give an example of a pseudoprime which is itself of the form 2n 2 answering a question posed by a rotkiewicz show that lehmer s example of an even pseudoprime having three prime factors is not unique and answer a question of benkoski concerning the solutions of 2n 2 1 mod n

accurate and efficient evaluation of schur and jack functions we present new algorithms for computing the values of the schur s x 1 x 2 x n and jack j x 1 x 2 x n functions in floating point arithmetic these algorithms deliver guaranteed high relative accuracy for positive data x i a 0 and run in time that is only linear in n

on a problem of a rotkiewicz pour des entiers positifs a k 2 fixes arbitraires il existe une infinite d entiers composes n tels que a n k 1 mod n

on the congruence 2n k 1 mod n it is shown that there are infinitely many positive integers k such that the congruence 2n k 1 mod n has infinitely many solutions n

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

on the second order statistics of the multihop rayleigh fading channel second order statistics provides a dynamic representation of a fading channel and plays an important role in the evaluation and design of the wireless communication systems in this paper we present a novel analytical framework for the evaluation of important second order statistical parameters as the level crossing rate lcr and the average fade duration afd of the amplify and forward multihop rayleigh fading channel more specifically motivated by the fact that this channel is a cascaded one and can be modeled as the product of n fading amplitudes we derive novel analytical expressions for the average lcr and the afd of the product of n rayleigh fading envelopes or of the recently so called nrayleigh channel furthermore we derive simple and efficient closed form approximations to the aforementioned parameters using the multivariate laplace approximation theorem it is shown that our general results reduce to the corresponding ones of the specific dual hop case previously published numerical and computer simulation examples verify the accuracy of the presented mathematical analysis and show the tightness of the proposed approximations

m ary symbol error outage over nakagami m fading channels in shadowing environments this letter addresses the problem of finding a tractable expression for the symbol error outage seo in flat nakagami m fading and shadowing channels we deal with m ary phase shift keying m psk and quadrature amplitude modulation m qam which extends our previous results on bpsk signaling we propose a new tight approximation of the symbol error probability sep holding for m psk and m qam signals which is accurate over all signal to noise ratios snrs of interest we derive a new generic expression for the inverse sep which facilitates the derivation of a tight approximation of the seo in a lognormal shadowing environment

laplace approximations for hypergeometric functions with matrix argument in this paper we present laplace approximations for two functions of matrix argument the type i confluent hypergeometric function and the gauss hypergeometric function both of these functions play an important role in distribution theory in multivariate analysis but from a practical point of view they have proved challenging and they have acquired a reputation for being difficult to approximate appealing features of the approximations we present are i they are fully explicit and simple to evaluate in practice and ii typically they have excellent numerical accuracy the excellent numerical accuracy is demonstrated in the calculation of noncentral moments of wilks a and the likelihood ratio statistic for testing block independence and in the calculation of the cdf of the noncentral distribution of wilks a via a sequential saddlepoint approximation relative error properties of these approximations are also studied and it is noted that the approximations have uniformly bounded relative errors in important cases

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

a priori stability results for pfc abstractdespite its popularity in industry and obvious efficacy predictive functional control has few rigorous a priori stability results in the literature in many cases common sense and intuition with some trial and error are the main design tools this paper seeks to tackle that gap by providing some analysis of the control law and showing what forms of stability assurances can be given and how these depend on the user choices of coincidence horizon and desired closed loop pole the conditions are separated into necessary but not sufficient conditions for stability and conversely sufficient but not necessary conditions numerical examples demonstrate the efficacy of these conditions and the ease of use

mixtures of g priors for bayesian variable selection zellner s g prior remains a popular conventional prior for use in bayesian variable selection despite several undesirable consistency issues in this article we study mixtures of g priors as an alternative to default g priors that resolve many of the problems with the original formulation while maintaining the computational tractability that has made the g prior so popular we present theoretical properties of the mixture g priors and provide real and simulated examples to compare the mixture formulation with fixed g priors empirical bayes approaches and other default procedures please see arnold zellner s letter and the author s response

on the congruence 2 n 2 1 mod n there exist infinitely many positive integers n such that 2 2 1 mod n in the monograph 5 i proposed the following problem problem 18 p 138 let a k 1 be fixed positive integers do there exist infinitely many composite n such that a 1 put a 2 k 2 in the above problem since by fermat s theorem 2p l 1 mod p for odd primes p if 2 2 1 mod n and n 2 n must be composite r matuszewski and p rudnicki with the aid of the computer k 202 in warsaw checked that below 4208 such integers do not exist the following theorem holds congruence 2m 1 mod2m 1 it follows that 2m v v m 1 mod2m 1 22 1 3 s 1 mod2m 1 and 2 2 1 modn for n 2m 1 thus 2 2 1 mod n for n 2 1 where 0 4700063497 suppose now that 2 2 1 mod and n 8 let be a primitive factor of the number 2 2 1 a prime factor of 2 1 is said to be primitive if it does not divide any of the numbers 2m 1 for m 1 2 n 1 by a theorem of k zsigmondy 7 such a prime factor exists for any n 6 and is of the form nt 1 now we shall show that nx np is also a solution of the congruence 2 1 2 1 mod we have p 2 n 2 k 1 where k is a positive integer and p 2 3 n and 1 thus