Asymptotic Distribution (asymptotic + distribution)

Distribution by Scientific Domains

Selected Abstracts

Asymptotic Distribution of Score Statistics for Spatial Cluster Detection with Censored Data

BIOMETRICS, Issue 4 2008
Daniel Commenges
SummaryCook, Gold, and Li (2007, Biometrics 63, 540,549) extended the Kulldorff (1997, Communications in Statistics 26, 1481,1496) scan statistic for spatial cluster detection to survival-type observations. Their approach was based on the score statistic and they proposed a permutation distribution for the maximum of score tests. The score statistic makes it possible to apply the scan statistic idea to models including explanatory variables. However, we show that the permutation distribution requires strong assumptions of independence between potential cluster and both censoring and explanatory variables. In contrast, we present an approach using the asymptotic distribution of the maximum of score statistics in a manner not requiring these assumptions. [source]

Identification of Persistent Cycles in Non-Gaussian Long-Memory Time Series

Mohamed Boutahar
Abstract., Asymptotic distribution is derived for the least squares estimates (LSE) in the unstable AR(p) process driven by a non-Gaussian long-memory disturbance. The characteristic polynomial of the autoregressive process is assumed to have pairs of complex roots on the unit circle. In order to describe the limiting distribution of the LSE, two limit theorems involving long-memory processes are established in this article. The first theorem gives the limiting distribution of the weighted sum, is a non-Gaussian long-memory moving-average process and (cn,k,1 , k , n) is a given sequence of weights; the second theorem is a functional central limit theorem for the sine and cosine Fourier transforms [source]

A Point Estimator for the Time Course of Drug Release

Stephan Koehne-Voss
Abstract Procedures for deconvolution of pharmacokinetic data are routinely used in the pharmaceutical industry to determine drug release and absorption which is essential in designing optimized drug formulations. Although these procedures are described extensively in the pharmacokinetic literature, they have been studied less from a statistical point of view and variance estimation has not been addressed. We discuss the statistical properties of a numerical procedure for deconvolution. Based on a point-area deconvolution method we define an estimator for the function that describes the time course of drug release from a drug formulation. Asymptotic distributions are derived and several methods of variance and interval estimation are compared (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Estimating the unknown change point in the parameters of the lognormal distribution

V. K. Jandhyala
Abstract We develop change-point methodology for identifying dynamic trends in the parameters of a two-parameter lognormal distribution. The methodology primarily considers the asymptotic distribution of the maximum likelihood estimate of the unknown change point. Among others, the asymptotic distribution enables one to construct confidence interval estimates for the unknown change point. The methodology is applied to identify changes in the monthly water discharges of the Nacetinsky Creek in the German part of the Ergebirge Mountains. Copyright © 2006 John Wiley & Sons, Ltd. [source]

The exponentiated Gumbel distribution with climate application

Saralees Nadarajah
Abstract The Gumbel distribution is perhaps the most widely applied statistical distribution for climate modeling. In this article we introduce a distribution that generalizes the standard Gumbel distribution in the same way the exponentiated exponential distribution generalizes the standard exponential distribution. We refer to this new distribution as the exponentiated Gumbel distribution. We provide a comprehensive treatment of the mathematical properties of this new distribution and illustrate its use for modeling rainfall data from Orlando, Florida. Among the mathematical properties, we derive the analytical shapes of the corresponding probability density function and the hazard rate function, calculate expressions for the nth moment and the asymptotic distribution of the extreme order statistics, and investigate the variation of the skewness and kurtosis measures. We also discuss estimation by the method of maximum likelihood. Copyright © 2005 John Wiley & Sons, Ltd. [source]

Genetic association tests in the presence of epistasis or gene-environment interaction

Kai WangArticle first published online: 24 APR 200
Abstract A genetic variant is very likely to manifest its effect on disease through its main effect as well as through its interaction with other genetic variants or environmental factors. Power to detect genetic variants can be greatly improved by modeling their main effects and their interaction effects through a common set of parameters or "generalized association parameters" (Chatterjee et al. [2006] Am. J. Hum. Genet. 79:1002,1016) because of the reduced number of degrees of freedom. Following this idea, I propose two models that extend the work by Chatterjee and colleagues. Particularly, I consider not only the case of relatively weak interaction effect compared to the main effect but also the case of relatively weak main effect. This latter case is perhaps more relevant to genetic association studies. The proposed methods are invariant to the choice of the allele for scoring genotypes or the choice of the reference genotype score. For each model, the asymptotic distribution of the likelihood ratio statistic is derived. Simulation studies suggest that the proposed methods are more powerful than existing ones under certain circumstances. Genet. Epidemiol. 2008. © 2008 Wiley-Liss, Inc. [source]

Affected-sib-pair test for linkage based on constraints for identical-by-descent distributions corresponding to disease models with imprinting,

Michael Knapp
Abstract Holmans' possible triangle test for affected sib pairs has proven to be a powerful tool for linkage analysis. This test is a likelihood-ratio test for which maximization is restricted to the set of possible sharing probabilities. Here, we extend the possible triangle test to take into account genomic imprinting, which is also known as parent-of-origin effect. While the classical test without imprinting looks at whether affected sib pairs share 0, 1, or 2 alleles identical-by-descent, the likelihood-ratio test allowing for imprinting further distinguishes whether the sharing of exactly one allele is through the father or mother. Thus, if the disease gene is indeed subject to imprinting, the extended test presented here can take into account that affecteds will have inherited the mutant allele preferentially from one particular parent. We calculate the sharing probabilities at a marker locus linked to a disease susceptibility locus. Using our formulation, the constraints on these probabilities given by Dudoit and Speed ([1999] Statistics in Genetics; New York: Springer) can easily be verified. Next, we derive the asymptotic distribution of the restricted likelihood-ratio test statistic under the null hypothesis of no linkage, and give LOD-score criteria for various test sizes. We show, for various disease models, that the test allowing for imprinting has significantly higher power to detect linkage if imprinting is indeed present, at the cost of only a small reduction in power in case of no imprinting. Altogether, unlike many methods currently available, our novel model-free sib-pair test adequately models the epigenetic parent-of-origin effect, and will hopefully prove to be a useful tool for the genetic mapping of complex traits. © 2004 Wiley-Liss, Inc. [source]


Andres Aradillas-Lopez
This article extends the pairwise difference estimators for various semilinear limited dependent variable models proposed by Honoré and Powell (Identification and Inference in Econometric Models. Essays in Honor of Thomas Rothenberg Cambridge: Cambridge University Press, 2005) to permit the regressor appearing in the nonparametric component to itself depend upon a conditional expectation that is nonparametrically estimated. This permits the estimation approach to be applied to nonlinear models with sample selectivity and/or endogeneity, in which a "control variable" for selectivity or endogeneity is nonparametrically estimated. We develop the relevant asymptotic theory for the proposed estimators and we illustrate the theory to derive the asymptotic distribution of the estimator for the partially linear logit model. [source]

Asymmetric power distribution: Theory and applications to risk measurement

Ivana Komunjer
Theoretical literature in finance has shown that the risk of financial time series can be well quantified by their expected shortfall, also known as the tail value-at-risk. In this paper, I construct a parametric estimator for the expected shortfall based on a flexible family of densities, called the asymmetric power distribution (APD). The APD family extends the generalized power distribution to cases where the data exhibits asymmetry. The first contribution of the paper is to provide a detailed description of the properties of an APD random variable, such as its quantiles and expected shortfall. The second contribution of the paper is to derive the asymptotic distribution of the APD maximum likelihood estimator (MLE) and construct a consistent estimator for its asymptotic covariance matrix. The latter is based on the APD score whose analytic expression is also provided. A small Monte Carlo experiment examines the small sample properties of the MLE and the empirical coverage of its confidence intervals. An empirical application to four daily financial market series reveals that returns tend to be asymmetric, with innovations which cannot be modeled by either Laplace (double-exponential) or Gaussian distribution, even if we allow the latter to be asymmetric. In an out-of-sample exercise, I compare the performances of the expected shortfall forecasts based on the APD-GARCH, Skew- t -GARCH and GPD-EGARCH models. While the GPD-EGARCH 1% expected shortfall forecasts seem to outperform the competitors, all three models perform equally well at forecasting the 5% and 10% expected shortfall. Copyright © 2007 John Wiley & Sons, Ltd. [source]

Testing against smooth stochastic trends

Jukka Nyblom
A trend estimated from an unobserved components model tends to be smoother when it is modelled as an integrated random walk rather than a random walk with drift. This article derives a test of the null hypothesis that the trend is deterministic against the alternative that it is an integrated random walk. It is assumed that the other component in the model is normally distributed white noise. Critical values are tabulated, the asymptotic distribution is derived and the performance of the test is compared with the test against a trend specified as a random walk with drift. The test is extended to allow for serially correlated and evolving seasonal components. When there is a stationary process containing a single autoregressive unit root close to one, a bounds test can be applied. In the case of a first-order autoregressive disturbance, it is shown that a consistent test can still be obtained by carrying out estimation of the nuisance parameters under the null hypothesis. The overall conclusion is that the most effective test against an integrated random walk is a parametric one based on the random walk plus drift test statistic, constructed from innovations, with the nuisance parameters estimated in the unrestricted model. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Detecting changes in the mean of functional observations

István Berkes
Summary., Principal component analysis has become a fundamental tool of functional data analysis. It represents the functional data as Xi(t)=,(t)+,1,l<,,i, l+ vl(t), where , is the common mean, vl are the eigenfunctions of the covariance operator and the ,i, l are the scores. Inferential procedures assume that the mean function ,(t) is the same for all values of i. If, in fact, the observations do not come from one population, but rather their mean changes at some point(s), the results of principal component analysis are confounded by the change(s). It is therefore important to develop a methodology to test the assumption of a common functional mean. We develop such a test using quantities which can be readily computed in the R package fda. The null distribution of the test statistic is asymptotically pivotal with a well-known asymptotic distribution. The asymptotic test has excellent finite sample performance. Its application is illustrated on temperature data from England. [source]

Non-parametric confidence bands in deconvolution density estimation

Nicolai Bissantz
Summary., Uniform confidence bands for densities f via non-parametric kernel estimates were first constructed by Bickel and Rosenblatt. In this paper this is extended to confidence bands in the deconvolution problem g=f*, for an ordinary smooth error density ,. Under certain regularity conditions, we obtain asymptotic uniform confidence bands based on the asymptotic distribution of the maximal deviation (L, -distance) between a deconvolution kernel estimator and f. Further consistency of the simple non-parametric bootstrap is proved. For our theoretical developments the bias is simply corrected by choosing an undersmoothing bandwidth. For practical purposes we propose a new data-driven bandwidth selector that is based on heuristic arguments, which aims at minimizing the L, -distance between and f. Although not constructed explicitly to undersmooth the estimator, a simulation study reveals that the bandwidth selector suggested performs well in finite samples, in terms of both area and coverage probability of the resulting confidence bands. Finally the methodology is applied to measurements of the metallicity of local F and G dwarf stars. Our results confirm the ,G dwarf problem', i.e. the lack of metal poor G dwarfs relative to predictions from ,closed box models' of stellar formation. [source]

Likelihood inference for a class of latent Markov models under linear hypotheses on the transition probabilities

Francesco Bartolucci
Summary., For a class of latent Markov models for discrete variables having a longitudinal structure, we introduce an approach for formulating and testing linear hypotheses on the transition probabilities of the latent process. For the maximum likelihood estimation of a latent Markov model under hypotheses of this type, we outline an EM algorithm that is based on well-known recursions in the hidden Markov literature. We also show that, under certain assumptions, the asymptotic null distribution of the likelihood ratio statistic for testing a linear hypothesis on the transition probabilities of a latent Markov model, against a less stringent linear hypothesis on the transition probabilities of the same model, is of type. As a particular case, we derive the asymptotic distribution of the likelihood ratio statistic between a latent class model and its latent Markov version, which may be used to test the hypothesis of absence of transition between latent states. The approach is illustrated through a series of simulations and two applications, the first of which is based on educational testing data that have been collected within the National Assessment of Educational Progress 1996, and the second on data, concerning the use of marijuana, which have been collected within the National Youth Survey 1976,1980. [source]

The restricted likelihood ratio test at the boundary in autoregressive series

Willa W. Chen
Abstract., The restricted likelihood ratio test, RLRT, for the autoregressive coefficient in autoregressive models has recently been shown to be second-order pivotal when the autoregressive coefficient is in the interior of the parameter space and so is very well approximated by the distribution. In this article, the non-standard asymptotic distribution of the RLRT for the unit root boundary value is obtained and is found to be almost identical to that of the in the right tail. Together, these two results imply that the distribution approximates the RLRT distribution very well even for near unit root series and transitions smoothly to the unit root distribution. [source]

Bootstrapping a weighted linear estimator of the ARCH parameters

Arup Bose
Abstract., A standard assumption while deriving the asymptotic distribution of the quasi maximum likelihood estimator in ARCH models is that all ARCH parameters must be strictly positive. This assumption is also crucial in deriving the limit distribution of appropriate linear estimators (LE). We propose a weighted linear estimator (WLE) of the ARCH parameters in the classical ARCH model and show that its limit distribution is multivariate normal even when some of the ARCH coefficients are zero. The asymptotic dispersion matrix involves unknown quantities. We consider appropriate bootstrapped version of this WLE and prove that it is asymptotically valid in the sense that the bootstrapped distribution (given the data) is a consistent estimate (in probability) of the distribution of the WLE. Although we do not show theoretically that the bootstrap outperforms the normal approximation, our simulations demonstrate that it yields better approximations than the limiting normal. [source]

Semiparametric inference on a class of Wiener processes

Xiao Wang
Abstract., This article studies the estimation of a nonhomogeneous Wiener process model for degradation data. A pseudo-likelihood method is proposed to estimate the unknown parameters. An attractive algorithm is established to compute the estimator under this pseudo-likelihood formulation. We establish the asymptotic properties of the estimator, including consistency, convergence rate and asymptotic distribution. Random effects can be incorporated into the model to represent the heterogeneity of degradation paths by letting the mean function be random. The Wiener process model is extended naturally to a normal inverse Gaussian process model and similar pseudo-likelihood inference is developed. A score test is used to test the presence of the random effects. Simulation studies are conducted to validate the method and we apply our method to a real data set in the area of health structure monitoring. [source]

A Generalized Portmanteau Test For Independence Of Two Infinite-Order Vector Autoregressive Series

Chafik Bouhaddioui
Primary 62M10; secondary 62M15 Abstract., In many situations, we want to verify the existence of a relationship between multivariate time series. Here, we propose a semiparametric approach for testing the independence between two infinite-order vector autoregressive (VAR(,)) series, which is an extension of Hong's [Biometrika (1996c) vol. 83, 615,625] univariate results. We first filter each series by a finite-order autoregression and the test statistic is a standardized version of a weighted sum of quadratic forms in the residual cross-correlation matrices at all possible lags. The weights depend on a kernel function and on a truncation parameter. Using a result of Lewis and Reinsel [Journal of Multivariate Analysis (1985) Vol. 16, pp. 393,411], the asymptotic distribution of the test statistic is derived under the null hypothesis and its consistency is also established for a fixed alternative of serial cross-correlation of unknown form. Apart from standardization factors, the multivariate portmanteau statistic proposed by Bouhaddioui and Roy [Statistics and Probability Letters (2006) vol. 76, pp. 58,68] that takes into account a fixed number of lags can be viewed as a special case by using the truncated uniform kernel. However, many kernels lead to a greater power, as shown in an asymptotic power analysis and by a small simulation study in finite samples. A numerical example with real data is also presented. [source]

Estimating the Rank of the Spectral Density Matrix

Gonzalo Camba-Mendez
Abstract., The rank of the spectral density matrix conveys relevant information in a variety of statistical modelling scenarios. This note shows how to estimate the rank of the spectral density matrix at any given frequency. The method presented is valid for any hermitian positive definite matrix estimate that has a normal asymptotic distribution with a covariance matrix the rank of which is known. [source]

Nonparametric Estimation and Testing in Panels of Intercorrelated Time Series

Vidar Hjellvik
Abstract., We consider nonparametric estimation and testing of linearity in a panel of intercorrelated time series. We place the emphasis on the situation where there are many time series in the panel but few observations for each of the series. The intercorrelation is described by a latent process, and a conditioning argument involving this process plays an important role in deriving the asymptotic theory. To be accurate the asymptotic distribution of the test functional of linearity requires a very large number of observations, and bootstrapping gives much better finite sample results. A number of simulation experiments and an illustration on a real data set are included. [source]

The asymptotic variance of the estimated roots in a cointegrated vector autoregressive model

Søren Johansen
Abstract., We show that the asymptotic distribution of the estimated stationary roots in a vector autoregressive model is Gaussian. A simple expression for the asymptotic variance in terms of the roots and the eigenvectors of the companion matrix is derived. The results are extended to the cointegrated vector autoregressive model and we discuss the implementation of the results for complex roots. [source]

A note on estimation by least squares for harmonic component models

A. M. Walker
Abstract. Let observations (X1,,,Xn) be generated by a harmonic model such that Xt=A0 cos ,0t+B0 sin ,0t+,t, where A0,B0,,0 are constants and (,t) is a stationary process with zero mean and finite variance. The estimation of A0,B0,,0 by the method of least squares is considered. It is shown that, without any restriction on , in the minimization procedure, the estimate is an n -consistent estimate of ,0, and hence () has the usual asymptotic distribution. The extension to a harmonic model with k>1 components is discussed. The case k=2 is considered in detail, but it was only found possible to establish the result under the restriction that both angular frequencies lie in the interval [source]

Spectral Regression For Cointegrated Time Series With Long-Memory Innovations

D. Marinucci
Spectral regression is considered for cointegrated time series with long-memory innovations. The estimates we advocate are shown to be consistent when cointegrating relationships among stationary variables are investigated, while ordinary least squares are inconsistent due to correlation between the regressors and the cointegrating residuals; in the presence of unit roots, these estimates share the same asymptotic distribution as ordinary least squares. As a corollary of the main result, we provide a functional central limit theorem for quadratic forms in non-stationary fractionally integrated processes. [source]

Estimation of the Dominating Frequency for Stationary and Nonstationary Fractional Autoregressive Models

Jan Beran
This paper was motivated by the investigation of certain physiological series for premature infants. The question was whether the series exhibit periodic fluctuations with a certain dominating period. The observed series are nonstationary and/or have long-range dependence. The assumed model is a Gaussian process Xt whose mth difference Yt = (1 ,B)mXt is stationary with a spectral density f that may have a pole (or a zero) at the origin. the problem addressed in this paper is the estimation of the frequency ,max where f achieves the largest local maximum in the open interval (0, ,). The process Xt is assumed to belong to a class of parametric models, characterized by a parameter vector ,, defined in Beran (1995). An estimator of ,max is proposed and its asymptotic distribution is derived, with , being estimated by maximum likelihood. In particular, m and a fractional differencing parameter that models long memory are estimated from the data. Model choice is also incorporated. Thus, within the proposed framework, a data driven procedure is obtained that can be applied in situations where the primary interest is in estimating a dominating frequency. A simulation study illustrates the finite sample properties of the method. In particular, for short series, estimation of ,max is difficult, if the local maximum occurs close to the origin. The results are illustrated by two of the data examples that motivated this research. [source]

Residual Autocorrelation Distribution in the Validation Data Set

Alessandro Fasso
Testing model performance on a data set other than the data set used for estimation is common practice in econometrics, technological stochastic modelling and environmetrics. In this paper, using an ARMAX model, the asymptotic distribution of the residual autocorrelations in the validation data set is given and a ,2 test for overall residual incorrelation is considered. [source]

A Pareto model for classical systems

Saralees Nadarajah
Abstract A new Pareto distribution is introduced for pooling knowledge about classical systems. It takes the form of the product of two Pareto probability density functions (pdfs). Various structural properties of this distribution are derived, including its cumulative distribution function (cdf), moments, mean deviation about the mean, mean deviation about the median, entropy, asymptotic distribution of the extreme order statistics, maximum likelihood estimates and the Fisher information matrix. Copyright © 2007 John Wiley & Sons, Ltd. [source]

On the distribution of the estimated process yield index Spk

J. C. Lee
Abstract This paper considers an asymptotic distribution for an estimate of the process yield index proposed by Boyles (1994). The asymptotic distribution of is useful in statistical inferences for . An illustrative example is given for hypothesis testing and for interval estimation on the yield index . Copyright © 2002 John Wiley & Sons, Ltd. [source]

Random trees and general branching processes

Anna Rudas
Abstract We consider a tree that grows randomly in time. Each time a new vertex appears, it chooses exactly one of the existing vertices and attaches to it. The probability that the new vertex chooses vertex x is proportional to w(deg(x)), a weight function of the actual degree of x. The weight function w : , , ,+ is the parameter of the model. In 4 and 11 the authors derive the asymptotic degree distribution for a model that is equivalent to the special case, when the weight function is linear. The proof therein strongly relies on the linear choice of w. Using well-established results from the theory of general branching processes we give the asymptotical degree distribution for a wide range of weight functions. Moreover, we provide the asymptotic distribution of the tree itself as seen from a randomly selected vertex. The latter approach gives greater insight to the limiting structure of the tree. Our proof is robust and we believe that the method may be used to answer several other questions related to the model. It relies on the fact that considering the evolution of the random tree in continuous time, the process may be viewed as a general branching process, this way classical results can be applied. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007 [source]

Projection estimators of Pickands dependence functions

Amélie Fils-Villetard
Abstract The authors consider the construction of intrinsic estimators for the Pickands dependence function of an extreme-value copula. They show how an arbitrary initial estimator can be modified to satisfy the required shape constraints. Their solution consists in projecting this estimator in the space of Pickands functions, which forms a closed and convex subset of a Hilbert space. As the solution is not explicit, they replace this functional parameter space by a sieve of finite-dimensional subsets. They establish the asymptotic distribution of the projection estimator and its finite-dimensional approximations, from which they conclude that the projected estimator is at least as efficient as the initial one. Estimation par projection de la fonction de dépendance de Pickands Les auteurs s'intéressent à la construction d'estimateurs intrinsèques de la fonction de dépendance de Pickands d'une copule des valeurs extrêmes. Ils montrent comment un estimateur initial quelconque peut être modifié pour satisfaire les contraintes de forme voulues. Leur solution consiste à projeter cet estimateur dans l'espace des fonctions de Pickands, qui forme un sous-ensemble convexe fermé d'un espace de Hilbert. Comme la solution n'est pas explicite, ils remplacent cet espace paramétrique fonctionnel par une succession d'approximations de dimension finie. Ils établissent la distribution asymptotique de la projection de l'estimateur et de ses approximations de dimension finie, ce qui leur permet de conclure que l'estimateur projeté est au moins aussi efficace que l'estimateur initial. [source]

Specification and estimation of social interaction models with network structures

Lung-fei Lee
Summary, This paper considers the specification and estimation of social interaction models with network structures and the presence of endogenous, contextual and correlated effects. With macro group settings, group-specific fixed effects are also incorporated in the model. The network structure provides information on the identification of the various interaction effects. We propose a quasi-maximum likelihood approach for the estimation of the model. We derive the asymptotic distribution of the proposed estimator, and provide Monte Carlo evidence on its small sample performance. [source]

The Tobit model with a non-zero threshold

Richard T. Carson
Summary, The standard Tobit maximum likelihood estimator under zero censoring threshold produces inconsistent parameter estimates, when the constant censoring threshold , is non-zero and unknown. Unfortunately, the recording of a zero rather than the actual censoring threshold value is typical of economic data. Non-trivial minimum purchase prices for most goods, fixed cost for doing business or trading, social customs such as those involving charitable donations, and informal administrative recording practices represent common examples of non-zero constant censoring threshold where the constant threshold is not readily available to the econometrician. Monte Carlo results show that this bias can be extremely large in practice. A new estimator is proposed to estimate the unknown censoring threshold. It is shown that the estimator is superconsistent and follows an exponential distribution in large samples. Due to the superconsistency, the asymptotic distribution of the maximum likelihood estimator of other parameters is not affected by the estimation uncertainty of the censoring threshold. [source]