A fast likelihood approximation for vector general linear processes with long series: application to fractional differencing

A formal comparison is made between direct standardization and three cross- classified data structures: tables of means which are linear additive; tables of means which are log-linear additive; and tables of frequencies which are log-linear addi tive and can be converted to tables of proportions which are logit-linear additive. Standardization is an appropriate method of summarizing the data if the differ ences between standardized means and so on are not affected by the choice of standard distribution. This condition occurs when there is no interaction between the predictor and control variables in their impact on the dependent variable. It is shown that the condition may also be expressed in the form of the general linear model with the corresponding interaction terms absent. Then, when standardiza tion is appropriate, differences between standardized quantities are estimates of differences between parameters in linear models. In some circumstances, e.g. when the cell sizes are small, if the specified interactions are believed absent then the cell entries may be fitted using the general linear model; standardization of the fitted entries would then be preferable to standardization of the observed entries.

In this paper, we derive Piterbarg's max-discretization theorem for two different grids considering centred stationary vector Gaussian processes. So far in the literature results in this direction have been derived for the joint distribution of the maximum of Gaussian processes over and over a grid . In this paper, we extend the recent findings by considering additionally the maximum over another grid . We derive the joint limiting distribution of maximum of stationary Gaussian vector processes for different choices of such grids by letting . As a by-product we find that the joint limiting distribution of the maximum over different grids, which we refer to as the Piterbarg distribution, is in the case of weakly dependent Gaussian processes a max-stable distribution.

: The structure of the asymptotic log-likelihood ratio decision procedure for the discrimination of alternative stationary zero mean multivariate Gaussian processes is developed. The log-likelihood ratio statistic is shown to be asymptotically normally distributed. New time and frequency domain formulas for the conditional mean (The Kullback-Liebler information measure) and variance of the log-likelihood ratio statistic under the alternative hypotheses are given and the probability of misclassification is shown to be bounded exponentially with n, the number of observations.

Let S(N,Z) be the sensitivity of the rate-distortion function (relative to the mean-square-error criterion) of a continuous-time stationary Gaussian process N=N(t) to non-Gaussian contamination Z=Z(t). We prove that for any entropy-regular process Z the equality S(N,Z)=S(N,Z*) holds, where Z*=Z*(t) is a stationary Gaussian process with the same autocorrelation function as Z. An explicit expression for S(N,Z) in the terms of the spectral densities of N and Z is also derived. We also prove that S(N,Z)=0 for any entropy-singular process Z.

Abstract. We propose an easy to implement method for making small sample parametric inference about the root of an estimating equation expressible as a quadratic form in normal random variables. It is based on saddlepoint approximations to the distribution of the estimating equation whose unique root is a parameter's maximum likelihood estimator (MLE), while substituting conditional MLEs for the remaining (nuisance) parameters. Monotoncity of the estimating equation in its parameter argument enables us to relate these approximations to those for the estimator of interest. The proposed method is equivalent to a parametric bootstrap percentile approach where Monte Carlo simulation is replaced by saddlepoint approximation. It finds applications in many areas of statistics including, nonlinear regression, time series analysis, inference on ratios of regression parameters in linear models and calibration. We demonstrate the method in the context of some classical examples from nonlinear regression models and ratios of regression parameter problems. Simulation results for these show that the proposed method, apart from being generally easier to implement, yields confidence intervals with lengths and coverage probabilities that compare favourably with those obtained from several competing methods proposed in the literature over the past half‐century.

We derive new necessary and sufficient conditions for exponentially convergent discrimination between two stationary vector Gaussian processes, and relate them to previously studied conditions for parameter identifiability and consistent discrimination.

13 - Matrices and related results

A large proportion of familial frontotemporal dementia is caused by TAR DNA-binding protein 43 (transactive response DNA-binding protein 43 kDa) proteinopathies. Accordingly, carriers of autosomal dominant mutations in the genes associated with TAR DNA-binding protein 43 aggregation, such as Chromosome 9 open reading frame 72 (C9orf72) or progranulin (GRN), are at risk of later developing frontotemporal dementia. Brain imaging abnormalities that develop before dementia onset in mutation carriers may serve as proxies for the presymptomatic stages of familial frontotemporal dementia due to a genetic cause. Our study objective was to investigate brain MRI-based white-matter changes in predementia participants carrying mutations in C9orf72 or GRN genes. We analysed mutation carriers and their family member controls (noncarriers) from the University of British Columbia familial frontotemporal dementia study. First, a total of 42 participants (8 GRN carriers; 11 C9orf72 carriers; 23 noncarriers) had longitudinal T1-weighted MRI over ∼2 years. White-matter signal hypointensities were segmented and volumes were calculated for each participant. General linear models were applied to compare the baseline burden and the annualized rate of accumulation of signal abnormalities among mutation carriers and noncarriers. Second, a total of 60 participants (9 GRN carriers; 17 C9orf72 carriers; 34 noncarriers) had cross-sectional diffusion tensor MRI available. For each participant, we calculated the average fractional anisotropy and mean, radial and axial diffusivity parameter values within the normal-appearing white-matter tissues. General linear models were applied to compare whether mutation carriers and noncarriers had different trends in diffusion tensor imaging parameter values as they neared the expected age of onset. Baseline volumes of white-matter signal abnormalities were not significantly different among mutation carriers and noncarriers. Longitudinally, GRN carriers had significantly higher annualized rates of accumulation (estimated mean: 15.87%/year) compared with C9orf72 carriers (3.69%/year) or noncarriers (2.64%/year). A significant relationship between diffusion tensor imaging parameter values and increasing expected age of onset was found in the periventricular normal-appearing white-matter region. Specifically, GRN carriers had a tendency of a faster increase of mean and radial diffusivity values and C9orf72 carriers had a tendency of a faster decline of fractional anisotropy values as they reached closer to the expected age of dementia onset. These findings suggest that white-matter changes may represent early markers of familial frontotemporal dementia due to genetic causes. However, GRN and C9orf72 mutation carriers may have different mechanisms leading to tissue abnormalities.

In the passage from fields to rings of coefficients quadratic forms with invertible matrices lose their decisive role. It turns out that if all quadratic forms over a ring are diagonalizable, then in effect this is always a local principal ideal ring with . The problem of the construction of a `normal' diagonal form of a quadratic form over a ring faces obstacles in the case of indices greater than 1. In the case of index 2 this problem has a solution given in Theorem 2.1 for (an extension of the law of inertia for real quadratic forms) and in Theorem 2.2 for containing an invertible non-square. Under the same conditions on a ring with nilpotent maximal ideal the number of classes of projectively congruent quadratic forms of the projective space associated with a free -module of rank is explicitly calculated (Proposition 3.2). Up to projectivities, the list of forms is presented for the projective plane over and also (Theorem 3.3) over the local ring with non-principal maximal ideal, where is a field with an invertible non-square in and . In the latter case the number of classes of non-diagonalizable quadratic forms of rank 0 depends on one's choice of the field and is not even always finite; all the other forms make up 21 classes.

Modeling Techniques for Fitting Healthcare Resource Use Costs Derived From Patient Reported Counts of Hospital, ER, And PCP Visits

In this paper, the authors address the problem of the minimax estimator of linear combinations of stochastic regression coefficients and parameters in the general normal linear model with random effects. Under a quadratic loss function, the minimax property of linear estimators is investigated. In the class of all estimators, the minimax estimator of estimable functions, which is unique with probability 1, is obtained under a multivariate normal distribution.

The traditional method for estimating the linear function of fixed parameters in mixed linear model is a two‐stage procedure. In the first stage of this procedure the variance components estimators are calculated and next in the second stage these estimators are taken as true values of variance components to estimating the linear function of fixed parameters according to generalized least squares method.In this paper the general mixed linear model is considered in which a matrix related to fixed parameters and or/a dispersion matrix of observation vector may be deficient in rank. It is shown that the estimators of a set of functions of fixed parameters obtained in second stage are unbiased if only the observation vector is symmetrically distributed about its expected value and the estimators of variance components from first stage are translation‐invariant and are even functions of the observation vector.

Journal Article Restricted unbiased iterative generalized least-squares estimation Get access HARVEY GOLDSTEIN HARVEY GOLDSTEIN Department of Mathematics, Statistics and Computing, Institute of Education, University of LondonLondon WC1H 0AL, U.K. Search for other works by this author on: Oxford Academic Google Scholar Biometrika, Volume 76, Issue 3, September 1989, Pages 622–623, https://doi.org/10.1093/biomet/76.3.622 Published: 01 September 1989 Article history Received: 01 December 1988 Revision received: 01 February 1989 Published: 01 September 1989

A Monte Carlo algorithm is developed for generating samples of arbitrary real-valued nonstationary Gaussian processes. The algorithm is based on representations of these processes by finite sums of harmonics with dependent Gaussian coefficients, in contract to similar representations available for stationary Gaussian processes that have independent Gaussian coefficients. The proposed algorithm is based on an idea in [Grigoriu, J. Eng. Mech., ASCE 119: 328–343, 1993]. It is shown that the covariance matrix of the random coefficients in the representation proposed for nonstationary Gaussian processes can be obtained simply from ordinates of the generalized spectral densities of these processes. Two numerical examples are presented to illustrate the application of the proposed algorithm and assess its performance. The examples are a stationary Gaussian process and a nonstationary Gaussian process obtained from a stationary process by distorting its time scale.

Introduction.The purpose of this paper is to study the Fourier transforms of the sample functions x(i) of random processes for which a moment E | x(i) |p is finite and bounded in t for some p > 0; for example, stationary processes with E | x(0) |p finite.x(t) will not in general approach 0 at oo and its Fourier transform will not usually be a function, but can be defined as a tempered distribution in Schwartz's theory [8].The main result is that if p > 1 the Fourier transform of x(i) is a "Stieltjes integral," i.e. it is obtained from a locally integrable function by differentiating once in the sense of distributions.Since local properties of the Fourier transform have little to do with local properties of x(i), it is not surprising that the same thing is true if x(f) is replaced by a "random distribution," i.e. a process possibly so irregular that values x(i0) at points i0 are not defined, but with averages " J"x(t)/(t)di" for smooth functions / defined ; we require pth moments of such averages bounded under translation of /.It is also possible to replace the real line as domain of values for t by a Euclidean space of arbitrary dimension k ; here a Stieltjes integral is of the form dkF¡Bty ••• dtk with F locally integrable, and this is the form of the Fourier transforms, again for p > 1.The results are stated in detail in §2.The positive assertions for p > 1 are proved in § §4-6, and the counter-examples for p :g 1 are given in §7.§ §1 and 3 deal with random distributions in general.§8 shows that the results are easily adapted to the case of processes with nonnegative or discrete parameter or stationary increments.For stationary Gaussian processes, the results appeared in my Princeton thesis, written with the helpful advice of G. A. Hunt and Edward Nelson.There the methods were based on Ito's "spectral measure," requiring both second moments and strict stationarity of the process or its increments ([4], [5]); here using other methods, essentially the Hausdorff-Young inequality, almost as much is proved from much more general hypotheses.1. Random distributions.A random distribution, as defined by Gel'fand [1], is a mapping f-*L(J) of a space S> = £$(Rk), i.e. the space of C°° functions with