robust adaptive rate optimal testing for the white noise hypothesis abstract a new test is proposed for the weak white noise null hypothesis the test is based on a new automatic selection of the order for a box pierce 1970 test statistic or the test statistic of hong 1996 the heteroskedasticity and autocorrelation consistent hac critical values from lee 2007 are used allowing for estimation of the error term the data driven order selection is tailored to detect a new class of alternatives with autocorrelation coefficients which can be o n 1 2 provided there are sufficiently many of such coefficients a simulation experiment illustrates the good statistical properties of the test both under the weak white noise null and the alternative

testing for high dimensional white noise using maximum cross correlations we propose a new omnibus test for vector white noise using the maximum absolute autocorrelations and cross correlations of the component series based on an approximation by the l norm of a normal random vector the critical value of the test can be evaluated by bootstrapping from a multivariate normal distribution in contrast to the conventional white noise test the new method is proved to be valid for testing departure from white noise that is not independent and identically distributed we illustrate the accuracy and the power of the proposed test by simulation which also shows that the new test outperforms several commonly used methods including the lagrange multiplier test and the multivariate box pierce portmanteau tests especially when the dimension of the time series is high in relation to the sample size the numerical results also indicate that the performance of the new test can be further enhanced when it is applied to pre transformed data obtained via the time series principal component analysis proposed by j chang b guo and q yao arxiv 1410 2323 the proposed procedures have been implemented in an r package

robust m tests without consistent estimation of the asymptotic covariance matrix we extend the kvb approach of kiefer vogelsang and bunzel 2000 to constructing robust m tests without consistent estimation of the asymptotic covariance matrix we demonstrate that when model parameters have to be estimated the normalizing matrix computed using a full sample estimator is able to eliminate the nuisance parameters when there is no estimation effect but not otherwise to circumvent the problem of estimation effect we propose using recursive estimators to compute the normalizing matrix and show that the resulting m test is asymptotically pivotal this m test is thus robust not only to heteroscedasticity and serial correlations of unknown form but also to the presence of an estimation effect as examples we consider robust tests for serial correlations and robust information matrix tests the former tests extend that of lobato 2001 and are applicable to model residuals for testing higher order moments we find that the latter tests are also robust when a lower order moment is miss

fixed smoothing asymptotics for time series 1 in this paper we derive higher order edgeworth expansions for the finite sample distributions of the subsampling based t statistic and the wald statistic in the gaussian location model under the so called fixed smoothing paradigm in particular we show that the error of asymptotic approximation is at the order of the reciprocal of the sample size and obtain explicit forms for the leading error terms in the expansions the results are used to justify the second order correctness of a new bootstrap method the gaussian dependent bootstrap in the context of gaussian location model 1 introduction many economic and financial applications involve time series data with autocorrelation and heteroskedasticity properties often the unknown dependence structure is not the chief object of interest but the inference on the parameter of interest involves the estimation of unknown dependence in stationary time series models estimated by generalized method of moments gmm robust inference is typically accomplished by consistently estimating the asymptotic covariance matrix which is proportional to the long run variance lrv matrix of the estimating equations or moment conditions defining the estimator using a kernel smoothing method in the econometrics and statistics literature the bandwidth parameter truncation lag involved in the kernel smoothing method is assumed to grow slowly with sample size in order to achieve consistency the inference is conducted by plugging in a covariance matrix estimator that is consistent under heteroskedasticity and autocorrelation this approach dates back to newey and west 25 and andrews 1 recently kiefer and vogelsang 13 kv hereafter developed an alternative first order asymptotic theory for the hac heteroskedasticity and autocorrelation consistent based robust inference where the proportion of the bandwidth involved in the hac estimator to the sample size t denoted as b is held fixed in the asymptotics under the fixed b asymptotics the hac estimator converges to a nondegenerate yet nonstandard limiting distribution the tests based on the fi xed b asymptotic approximation were shown to enjoy better finite sample properties than the tests based on the small b asymptotic theory

the jackknife and the bootstrap for general stationary observations we extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence we do not attempt a reduction to i i d values the jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of l consecutive data once in the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero under appropriate conditions consistency is obtained if l l n rightarrow infty and l n n rightarrow 0 general statistics are approximated by an arithmetic mean in regular cases this approximation determines the asymptotic behavior bootstrap replicates are constructed by selecting blocks of length l randomly with replacement among the blocks of observations the procedures are illustrated by using the sunspot numbers and some simulated data

a bootstrapped spectral test for adequacy in weak arma models this paper proposes a cramer von mises cm test statistic to check the adequacy of weak arma models without posing a martingale difference assumption on the error terms the asymptotic null distribution of the cm test is obtained moreover this cm test is consistent and has nontrivial power against the local alternative of order n 1 2 due to the unknown dependence of error terms and the estimation effects a new block wise random weighting method is constructed to bootstrap the critical values of the test statistic the new method is easy to implement and its validity is justified the theory is illustrated by a small simulation study and an application to s p 500 stock index

testing that a dependent process is uncorrelated an analysis is presented of a new testing procedure for the null hypothesis that a stochastic process is uncorrelated when the process is possibly dependent unlike with existing procedures the user does not need to choose any arbitrary number to implement the proposed test the asymptotic null distribution of the proposed test statistic is not standard but it is tabulated by means of simulations the test is compared with two alternative test procedures that require selection of user chosen numbers on the basis of asymptotic local power and finite sample behavior although the asymptotic local power of the proposed test is lower than those corresponding to the alternative tests in a monte carlo study i show that in small samples the test typically better controls the type i error and that the loss of power is not substantial

limit theorems for mixing sequences without rate assumptions we extend levy s classical criterion for a sequence of independent identically distributed random variables to belong to the domain of partial attraction of a nondegenerate gaussian law to stationary mixing sequences we also extend some results of kesten and of kuelbs and zinn on the lil behavior of independent identically distributed random variables to stationary mixing sequences no assumptions on the rate of decay for the mixing coefficient are made

the limit distribution in the q clt for is unique and can not have a compact support in a paper by umarov et al 2008 milan j math 76 307 28 a generalization of the fourier transform called the q fourier transform was introduced and applied for the proof of a q generalized central limit theorem q clt subsequently hilhorst illustrated 2009 braz j phys 39 371 9 2010 j stat mech p10023 that the q fourier transform for is not invertible in the space of density functions indeed using an invariance principle he constructed a family of densities with the same q fourier transform and noted that as a consequence the q clt falls short of achieving its stated goal the distributions constructed there have compact support we prove now that the limit distribution in the q clt is unique and can not have a compact support this result excludes all the possible counterexamples which can be constructed using the invariance principle and fills the gap mentioned by hilhorst

unsupervised self normalized change point testing for time series abstractwe propose a new self normalized method for testing change points in the time series setting self normalization has been celebrated for its ability to avoid direct estimation of the nuisance asymptotic variance and its flexibility of being generalized to handle quantities other than the mean however it was developed and mainly studied for constructing confidence intervals for quantities associated with a stationary time series and its adaptation to change point testing can be nontrivial as direct implementation can lead to tests with nonmonotonic power compared with existing results on using self normalization in this direction the current article proposes a new self normalized change point test that does not require prespecifying the number of total change points and is thus unsupervised in addition we propose a new contrast based approach in generalizing self normalized statistics to handle quantities other than the mean which is specifically tailored for change point testing monte car

a nonstandard empirical likelihood for time series standard blockwise empirical likelihood bel for stationary weakly dependent time series requires specifying a fixed block length as a tuning parameter for setting confidence regions this aspect can be difficult and impacts coverage accuracy as an alternative this paper proposes a new version of bel based on a simple though non standard data blocking rule which uses a data block of every possible length consequently the method involves no block selection and is also anticipated to exhibit better coverage performance its non standard blocking scheme however induces non standard asymptotics and requires a significantly different development compared to standard bel we establish the large sample distribution of log ratio statistics from the new bel method for calibrating confidence regions for mean or smooth function parameters of time series this limit law is not the usual chi square one but is distribution free and can be reproduced through straightforward simulations numerical studies indicate that the proposed method generally exhibits better coverage accuracy than standard bel

inference for autocorrelations under weak assumptions abstract in this article we consider the large sample behavior of estimates of autocorrelations and autoregressive moving average arma coefficients as well as their distributions under weak conditions specifically the usual text book formulas for variances of these estimates are based on strong assumptions and should not be routinely applied without careful consideration such is the case when the time series follows an arma process with uncorrelated innovations that may not be assumed to be independent and identically distributed as a specific case it is well known that if the process is independent and identically distributed then the sample autocorrelation estimates scaled by the square root of the sample size are asymptotically standard normal this result is used extensively as a diagnostic check on the residuals of a fitted model or as an initial test on the observed time series to determine whether further model fitting is warranted in this article we show that this result can be quite

automatic optimal batch size selection for recursive estimators of time average covariance matrix abstractthe time average covariance matrix tacm k z k where k is the auto covariance function is an important quantity for the inference of the mean of an rd valued stationary process d 1 this article proposes two recursive estimators for with optimal asymptotic mean square error amse under different strengths of serial dependence the optimal estimator involves a batch size selection which requires knowledge of a smoothness parameter k z k k for some this article also develops recursive estimators for combining these two estimators we obtain a fully automatic procedure for optimal online estimation for consistency and convergence rates of the proposed estimators are derived applications to confidence region construction and markov chain monte carlo convergence diagnosis are discussed supplementary materials for this article are available online

precise rates in complete moment convergence for ρ mixing sequences abstract let x 1 x 2 be a strictly stationary sequence of mixing random variables with mean zeros and positive finite variances set s n x 1 x n suppose that lim n e s n 2 n 2 0 n 1 2 q 2 n where q 2 2 we prove that if e x 1 2 log x 1 for any 0 1 then lim 0 2 n 2 log n 1 n 2 e s n 2 i s n n log n e n 2 2 where n is the standard normal random variable

central limit theorems for additive functionals of markov chains central limit theorems and invariance principles are obtained for additive functionals of a stationary ergodic markov chain say s n g x 1 g x n where e g x 1 0 and e g x 1 2 the conditions imposed restrict the moments of g and the growth of the conditional means e s n x 1 no other restrictions on the dependence structure of the chain are required when specialized to shift processes the conditions are implied by simple integral tests involving g

subsampling for heteroskedastic time series abstract in this article a general theory for the construction of confidence intervals or regions in the context of heteroskedastic dependent data is presented the basic idea is to approximate the sampling distribution of a statistic based on the values of the statistic computed over smaller subsets of the data this method was first proposed by politis and romano 1994b for stationary observations we extend their results to heteroskedastic observations and prove a general asymptotic validity result under minimal conditions in contrast the usual bootstrap and moving blocks bootstrap are typically valid only for asymptotically linear statistics and their justification requires a case by case analysis our general asymptotic results are applied to a regression setting with dependent heteroskedastic errors

tsallis entropy maximization procedure revisited abstract the proper way of averaging is an important question with regards to tsallis thermostatistics three different procedures have been thus far employed in the pertinent literature the third one i e the tsallis mendes plastino tmp physica a 261 1998 534 normalization procedure exhibits clear advantages with respect to earlier ones in this work we advance a distinct from the tmp one way of handling the lagrange multipliers involved in the extremization process that leads to tsallis statistical operator it is seen that the new approach considerably simplifies the pertinent analysis without losing the beautiful properties of the tsallis mendes plastino formalism

bootstrapping the box pierce q test a robust test of uncorrelatedness abstract this paper describes a test of the null hypothesis that the first k autocorrelations of a covariance stationary time series are zero in the presence of statistical dependence the test is based on the box pierce q statistic with bootstrap based p values the bootstrap is implemented using a double blocks of blocks procedure with prewhitening the finite sample performance of the bootstrap q test is investigated by simulation in our experiments the performance is satisfactory for samples of n 500 at this sample size the differences between the empirical and nominal rejection probabilities are essentially eliminated

diagnostic checking in multivariate arma models with dependent errors using normalized residual autocorrelations in this paper we derive the asymptotic distribution of normalized residual empirical autocovariances and autocorrelations under weak assumptions on the noise we propose new portmanteau statistics

a central limit theorem for stationary rho mixing sequences with infinite variance on etend un resultat classique a des suites strictement stationnaires de variables aleatoires a valeurs reelles qui sont melangeantes avec un certain taux de melange logarithmique

trend function hypothesis testing in the presence of serial correlation in this paper test statistics are proposed that can be used to test hypotheses about the parameters of the deterministic trend function of a univariate time series the tests are valid in the presence of general forms of serial correlation in the errors and can be used without having to estimate the serial correlation parameters either parametrically or nonparametrically the tests are valid for i 0 and i 1 errors trend functions that are permitted include general linear polynomial trend functions that may have breaks at either known or unknown locations asymptotic distributions are derived and consistency of the tests is established the general results are applied to a model with a simple linear trend a local asymptotic analysis is used to compute asymptotic size and power of the tests for this example size is well controlled and is relatively unaffected by the variance of the initial condition asymptotic power curves are computed for the simple linear trend model and are compared to existing tests it is shown that the new tests have nontrivial asymptotic power a simulation study shows that the asymptotic approximations are adequate for sample sizes typically used in economics the tests are used to construct confidence intervals for average gnp growth rates for eight industrialized countries using post war data

green functions based on tsallis nonextensive statistical mechanics normalized q expectation value formulation in this paper the green function theory of quantum many particle systems recently presented is reworked within the framework of nonextensive statistical mechanics with a new normalized q expectation values this reformulation introduces a renormalization of temperature of the earlier theory and a self consistency condition the importance of these two features is nontrivial and to emphasize this we explicitly contrast the maximum entropy density matrices derived for an exactly solvable model based on the two types of the constraints the linear response theory is also presented along with its two particle green function version in order to emphasize the importance of the new formalism we collect here the results where both the formalisms have been used to examine the same set of problems this reveals clearly that the new formalism is the method of choice because the numerical results are much more physically meaningful than those found in the old version even though the general features or the answers retain the same characteristics in certain cases in the case where thermodynamic entities are to be examined as in the case of the q dependence of bose einstein condensation the self consistent requirement in the new formalism is numerically much more subtle and thus the earlier results are modified as shown in fig 1

simple robust testing of regression hypotheses in this paper we consider the problem of hypothesis testing in models with errors that have serial correlation or heteroskedasticity of unknown form this situation is often encountered in regression models applied to economic time series data it is a classic textbook result that while ordinary least squares ols estimates of regression parameters remain consistent and asymptotically normal when errors are heteroskedastic or autocorrelated provided usual regularity conditions hold and no lagged dependent variables are in the model standard tests are no longer valid if the true form of serial correlation heteroskedasticity were known then generalized least squares gls provides efficient estimates and standard inference can be conducted on the gls transformed model but in practice the form of serial correlation heteroskedasticity is often unknown and this has led to the development of techniques that permit valid asymptotic inference without having to specify a model of the serial correlation or heteroskedasticity the most common approach is to estimate the variance covariance matrix of the ols estimates nonparametrically using spectral methods heteroskedasticity and autocorrelation consistent hac estimators and construct standard tests using the asymptotic normality of the ols estimates hac estimators have been extensively analyzed in the econometrics literature and important contributions are given by andrews 1991 andrews and monahan 1992 gallant 1987 hansen 1992 newey and west 1987 robinson 1991 1998 and white 1984 among others the benefit of hac estimator tests is asymptotically valid inference that is robust to general forms of serial correlation heteroskedasticity in the errors we propose an alternative method of constructing robust test statistics we apply a nonsingular data dependent stochastic transformation to the ols estimates the asymptotic distribution of the transformed estimates does not depend on nuisance parameters then test statistics that are asymptotically invariant to nuisance parameters asymptotic pivotal statistics are constructed the resulting test statistics have nonstandard asymptotic distributions that only depend on the number of restrictions being tested and critical values are easy to simulate using standard techniques the main advantage of our approach compared to standard approaches is that estimates of the variance covariance matrix are not explicitly required to construct the tests this is potentially important for two reasons first with the exception of the estimator proposed by robinson 1998 2 consistent nonparametric estimates of variance covariance matrices in models with serial

diagnostic checking in arma models with uncorrelated errors we consider tests for lack of fit in arma models with nonindependent innovations in this framework the standard box pierce and ljung box portmanteau tests can perform poorly specifically the usual text book formulas for asymptotic distributions are based on strong assumptions and should not be applied without careful consideration in this article we derive the asymptotic covariance matrix of a vector of autocorrelations for residuals of arma models under weak assumptions on the noise the asymptotic distribution of the portmanteau statistics follows a consistent estimator of and a modification of the portmanteau tests are proposed this allows us to construct valid asymptotic significance limits for the residual autocorrelations and asymptotically valid goodness of fit tests when the underlying noise process is assumed to be noncorrelated rather than independent or a martingale difference a set of monte carlo experiments and an application to the standard poor 500 returns illustrate the p

two time green function technique in tsallis classical statistical mechanics abstract the two time green function technique is formulated in nonextensive classical statistical mechanics within the optimal lagrange multiplier framework we introduce also the nonextensive classical version of the spectral density method for a direct calculation of the green functions and its effectiveness is tested exploring the equilibrium properties of the classical ferromagnetic heisenberg spin chain

unified inference for sparse and dense longitudinal models in longitudinal data analysis statistical inference for sparse data and dense data could be substantially different for kernel smoothing estimate of the mean function the convergence rates and limiting variance functions are different under the two scenarios the latter phenomenon poses challenges for statistical inference as a subjective choice between the sparse and dense cases may lead to wrong conclusions we develop self normalization based methods that can adapt to the sparse and dense cases in a unified framework simulations show that the proposed methods outperform some existing methods

the cross quantilogram measuring quantile dependence and testing directional predictability between time series this paper proposes the cross quantilogram to measure the quantile dependence between two time series we apply it to test the hypothesis that one time series has no directional predictability to another time series we establish the asymptotic distribution of the cross quantilogram and the corresponding test statistic the limiting distributions depend on nuisance parameters to construct consistent confidence intervals we employ the stationary bootstrap procedure we show the consistency of this bootstrap also we consider the self normalized approach which is shown to be asymptotically pivotal under the null hypothesis of no predictability we provide simulation studies and two empirical applications first we use the cross quantilogram to detect predictability from stock variance to excess stock return compared to existing tools used in the literature of stock return predictability our method provides a more complete relationship between a predictor and stock return second we investigate the systemic risk of individual financial institutions such as jp morgan chase goldman sachs and aig this article has supplementary materials online

on singular semilinear elliptic equations abstract for the semilinear elliptic equation u p x u 0 x r n n 3 0 we show via the barrier method the existence of a positive entire solution behaving like x 2 n near

bootstrapping the portmanteau tests in weak auto regressive moving average models this paper uses a random weighting rw method to bootstrap the critical values for the ljung box monti portmanteau tests and weighted ljung box monti portmanteau tests in weak arma models unlike the existing methods no user chosen parameter is needed to implement the rw method as an application these four tests are used to check the model adequacy in power garch models simulation evidence indicates that the weighted portmanteau tests have the power advantage over other existing tests a real example on s p 500 index illustrates the merits of our testing procedure as one extension work the block wise rw method is also studied

on the invariance principle for stationary ϕ mixing triangular arrays with infinitely divisible limits given a stationary mixing triangular array of banach space valued random vectors whose row sums converge weakly to an infinitely divisible probability measure necessary and sufficient conditions for the validity of the corresponding invariance principle in distribution are given

inference for linear models with dependent errors summary the paper is concerned with inference for linear models with fixed regressors and weakly dependent stationary time series errors theoretically we obtain asymptotic normality for the m estimator of the regression parameter under mild conditions and establish a uniform bahadur representation for recursive m estimators methodologically we extend the recently proposed self normalized approach of shao from stationary time series to the regression set up where the sequence of response variables is typically non stationary in mean since the limiting distribution of the self normalized statistic depends on the design matrix and its corresponding critical values are case dependent we develop a simulation based approach to approximate the critical values consistently through a simulation study we demonstrate favourable finite sample performance of our method in comparison with a block bootstrap based approach empirical illustrations using two real data sets are also provided

a bootstrap assisted spectral test of white noise under unknown dependence abstract to test for the white noise null hypothesis we study the cramer von mises test statistic that is based on the sample spectral distribution function since the critical values of the test statistic are difficult to obtain we propose a blockwise wild bootstrap procedure to approximate its asymptotic null distribution using a hilbert space approach we establish the weak convergence of the difference between the sample spectral distribution function and the true spectral distribution function as well as the consistency of bootstrap approximation under mild assumptions finite sample results from a simulation study and an empirical data analysis are also reported

convergence rates of the strong law for stationary mixing sequences in this note we estimate the rate of convergence in marcinkiewicz zygmung strong law for partial sumss n of strong stationary mixing sequences of random variables the results improve the corresponding ones obtained by tze leung lai 1977 and christian hipp 1979

central limit theorems for mixing sequences of random variables under minimal conditions soit xj j 1 une suite strictement stationnaire de variables aleatoires de moyenne nulle de variance finie et satisfaisant une condition de melange forte soit s n la nieme somme partielle et supposons var s n variant regulierement et d ordre 1 on demontre que si s n var s n 1 2 ne converge pas vers 0 en l 1 alors xj j 1 est dans le domaine d attraction partielle d une loi gaussienne

distribution free tests for time series models specification abstract we consider a class of time series specification tests based on quadratic forms of weighted sums of residuals autocorrelations asymptotically distribution free tests in the presence of estimated parameters are obtained by suitably transforming the weights which can be optimally chosen to maximize the power function when testing in the direction of local alternatives we discuss in detail an asymptotically optimal distribution free alternative to the popular box pierce when testing in the direction of ar or ma alternatives the performance of the test with small samples is studied by means of a monte carlo experiment

confidence intervals for spectral mean and ratio statistics summary we propose a new method to construct confldence intervals for spectral mean and related ratio statistics of a stationary process that avoids direct estimation of their asymptotic variances by introducing a bandwidth a self normalization procedure is adopted and the distribution of the new statistic is asymptotically nuisance parameter free the bandwidth is chosen using information criteria and a moving average sieve approximation through a simulation study we demonstrate good flnite sample performance of our method when the sample size is moderate while a comparison with empirical likelihood based method for ratio statistics is made conflrming wider applicability of our method

maximal inequalities for partial sums of rho mixing sequences a rosenthal type inequality for maximum partial sums of p mixing sequences is obtained applications to the complete convergence and almost sure summability of partial sums are also discussed

multi scale tests for serial correlation this paper introduces a new family of portmanteau tests for serial correlation using the wavelet transform we decompose the variance of the underlying process into the variance of its low frequency and of its high frequency components and we design a variance ratio test of no serial correlation in the presence of dependence such decomposition can be carried out iteratively each wavelet filter leading to a rich family of tests whose joint limiting null distribution is a multivariate normal we illustrate the size and power properties of the proposed tests through monte carlo simulations

stationary strongly mixing sequences not satisfying the central limit theorem for every sequence varepsilon n n in n in 0 1 there exists a strictly stationary orthonormal sequence x n n in n of random variables with p a cap b p a p b leq varepsilon n for all a in sigma x 1 cdots x k b in sigma x k n x k n 1 cdots k in mathbb n n in mathbb n such that the distribution of n 1 2 sum n i 1 x i is not weakly convergent to the standard normal distribution