multivariate probability density deconvolution for stationary random processes the kernel type estimation of the joint probability density functions of stationary random processes from noisy observations is considered precise asymptotic expressions and bounds on the mean square estimation error are established along with rates of mean square convergence for processes satisfying a variety of mixing conditions the dependence of the convergence rates on the joint density of the noise process is studied

stochastic and mixed flower graphs stochasticity is introduced to a well studied class of recursively grown graphs u v flower nets which have power law degree distributions as well as small world properties when u 1 the stochastic variant interpolates between different deterministic flower graphs thus adding flexibility to the model the random multiplicative growth process involved however leads to a spread ensemble of networks with finite variance for the number of links nodes and loops nevertheless the degree exponent and loopiness exponent attain unique values in the thermodynamic limit of infinitely large graphs we also study a class of mixed flower networks closely related to the stochastic flowers but which are grown recursively in a deterministic way the deterministic growth of mixed flower nets eliminates ensemble spreads and their recursive growth allows for exact analysis of their uniquely defined mixed properties

network geometry with flavor from complexity to quantum geometry network geometry is attracting increasing attention because it has a wide range of applications ranging from data mining to routing protocols in the internet at the same time advances in the understanding of the geometrical properties of networks are essential for further progress in quantum gravity in network geometry simplicial complexes describing the interaction between two or more nodes play a special role in fact these structures can be used to discretize a geometrical d dimensional space and for this reason they have already been widely used in quantum gravity here we introduce the network geometry with flavor s 1 0 1 ngf describing simplicial complexes defined in arbitrary dimension d and evolving by a nonequilibrium dynamics the ngf can generate discrete geometries of different natures ranging from chains and higher dimensional manifolds to scale free networks with small world properties scale free degree distribution and nontrivial community structure the ngf admits as limiting cases both the bianconi barabasi models for complex networks the stochastic apollonian network and the recently introduced model for complex quantum network manifolds the thermodynamic properties of ngf reveal that ngf obeys a generalized area law opening a new scenario for formulating its coarse grained limit the structure of ngf is strongly dependent on the dimensionality d in d 1 ngfs grow complex networks for which the preferential attachment mechanism is necessary in order to obtain a scale free degree distribution instead for ngf with dimension d 1 it is not necessary to have an explicit preferential attachment rule to generate scale free topologies we also show that ngf admits a quantum mechanical description in terms of associated quantum network states quantum network states evolve by a markovian dynamics and a quantum network state at time t encodes all possible ngf evolutions up to time t interestingly the ngf remains fully classical but its statistical properties reveal the relation to its quantum mechanical description in fact the dimensional faces of the ngf have generalized degrees that follow either the fermi dirac boltzmann or bose einstein statistics depending on the flavor s and the dimensions d and

spectra and eigenvectors of scale free networks we study the spectra and eigenvectors of the adjacency matrices of scale free networks when bidirectional interaction is allowed so that the adjacency matrix is real and symmetric the spectral density shows an exponential decay around the center followed by power law long tails at both spectrum edges the largest eigenvalue ensuremath lambda 1 depends on system size n as ensuremath lambda 1 ensuremath sim n 1 4 for large n and the corresponding eigenfunction is strongly localized at the hub the vertex with largest degree the component of the normalized eigenfunction at the hub is of order unity we also find that the mass gap scales as n ensuremath 0 68

optimal rates of convergence for deconvolving a density abstract suppose that the sum of two independent random variables x and z is observed where z denotes measurement error and has a known distribution and where the unknown density f of x is to be estimated one application is the estimation of a prior density for a sequence of location parameters a second application arises in the errors in variables problem for nonlinear and generalized linear models when one attempts to model the distribution of the true but unobservable covariates this article shows that if z is normally distributed and f has k bounded derivatives then the fastest attainable convergence rate of any nonparametric estimator of f is only log n k 2 therefore deconvolution with normal errors may not be a practical proposition other error distributions are also treated stefanski carroll 1987a estimators achieve the optimal rates the results given have versions for multiplicative errors where they imply that even optimal rates are exceptionally slow

goodness of fit tests for kernel regression with an application to option implied volatilities this paper proposes a test of a restrictedspeci cation of regression basedon comparing residual sum of squares from kernel regression our main case is where both the restrictedspeci cation andthe general mod el are nonparametric with our test equivalently viewed as a test of dimension reduction we discuss practical features of implementing the test andvariations applicable to testing parametric mod els as the null hypothesis or semiparametric models that depend on a nite parameter vector as well as unknown functions we apply our testing procedure to option prices we reject a parametric version of the black scholes formula but fail to reject a semiparametric version against a general nonparametric regression 2001 elsevier science s a all rights reserved jel classi cation c12 c14

estimation of stochastic volatility models via monte carlo maximum likelihood this paper discusses the monte carlo maximum likelihood method of estimating stochastic volatility sv models the basic sv model can be expressed as a linear state space model with log chi square disturbances the likelihood function can be approximated arbitrarily accurately by decomposing it into a gaussian part constructed by the kalman filter and a remainder function whose expectation is evaluated by simulation no modifications of this estimation procedure are required when the basic sv model is extended in a number of directions likely to arise in applied empirical research this compares favorably with alternative approaches the finite sample performance of the new estimator is shown to be comparable to the monte carlo markov chain mcmc method

the realized empirical distribution function of stochastic variance with application to goodness of fit testing abstract we propose a nonparametric estimator of the empirical distribution function edf of the latent spot variance of the log price of a financial asset we show that over a fixed time span our realized edf or redf inferred from noisy high frequency data is consistent as the mesh of the observation grid goes to zero in a double asymptotic framework with time also increasing to infinity the redf converges to the cumulative distribution function of volatility if it exists we exploit these results to construct some new goodness of fit tests for stochastic volatility models in a monte carlo study the redf is found to be accurate over the entire support of volatility this leads to goodness of fit tests that are both correctly sized and relatively powerful against common alternatives in an empirical application we recover the redf from stock market high frequency data we inspect the goodness of fit of several two parameter marginal distributions that are inherent in standard stochastic volatility models the inverse gaussian offers the best overall description of random equity variation but the fit is less than perfect this suggests an extra parameter as available in e g the generalized inverse gaussian is required to model stochastic variance

scaling properties of scale free evolving networks continuous approach the scaling behavior of scale free evolving networks arising in areas such as communications scientific citations collaborations etc is studied we derive universal scaling relations describing properties of such networks and indicate the limits of their validity we show that the main properties of scale free evolving networks may be described in the framework of a simple continuous approach the simplest models of networks growing according to a mechanism of preferential attachment of links to nodes are used we consider different forms of this preference and demonstrate that the range of preferential attachments producing scale free networks is wide we also obtain scaling relations for networks with nonlinear accelerating growth and describe the temporal evolution of the arising distributions size effects char22 the cutoffs of these distributions char22 introduce restrictions for the observation of power law dependences mainly we discuss the so called degree distribution i e the distribution of the number of connections of nodes a scaling form of the distribution of links between pairs of individual nodes for a growing network of citations is also studied we describe the effects of differences between nodes the aging of nodes changes the exponents of the distributions the appearance of a single node with high fitness changes the degree distribution of a network dramatically if its fitness exceeds some threshold value this node captures a finite part of all links of the network we show that permanent random damage to a growing scale free network char22 a permanent deletion of some links char22 radically changes the values of the scaling exponents results of other kinds of permanent damage are described

nonparametric specification tests for stochastic volatility models based on volatility density this paper develops a specification test for stochastic volatility models by comparing the nonparametric kernel deconvolution density estimator of an integrated volatility density with its parametric counterpart l2 distance is used to measure the discrepancy the asymptotic null distributions of the test statistics are established and the asymptotic power functions are computed through monte carlo simulations the size and power properties of the test statistics are studied the tests are applied to an empirical example

noise dressing of financial correlation matrices we show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate time series the central result of the present study which focuses on the case of financial price fluctuations is the remarkable agreement between the theoretical prediction based on the assumption that the correlation matrix is random and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the s 500 or other major markets in particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management

root numbers of abelian varieties we generalize a theorem of d rohrlich concerning root numbers of elliptic curves over number fields our result applies to arbitrary abelian varieties namely under certain conditions which naturally extend the conditions used by d rohrlich we show that the root number w a associated to an abelian variety a over a number field f and a complex finite dimensional irreducible representation of gal f f with real valued character is equal to 1 we also show that our result is consistent with a refined version of the conjecture of birch and swinnerton dyer

a conditional kolmogorov test this paper introduces a conditional kolmogorov test of model specification for parametric models with covariates regressors the test is an extension of the kolmogorov test of goodness of fit for distribution functions the test is shown to have power against 1 n local alternatives and all fixed alternatives to the null hypothesis a parametric bootstrap procedure is used to obtain critical values for the test

relaxation properties of small world networks recently watts and strogatz introduced the so called small world networks in order to describe systems that combine simultaneously properties of regular and random lattices in this work we study diffusion processes defined on such structures by considering explicitly the probability for a random walker to be present at the origin the results are intermediate between the corresponding ones for fractals and cayley trees

exactly solvable random graph ensemble with extensively many short cycles we introduce and analyse ensembles of 2 regular random graphs with a tuneable distribution of short cycles the phenomenology of these graphs depends critically on the scaling of the ensembles control parameters relative to the number of nodes a phase diagram is presented showing a second order phase transition from a connected to a disconnected phase we study both the canonical formulation where the size is large but fixed and the grand canonical formulation where the size is sampled from a discrete distribution and show their equivalence in the thermodynamical limit we also compute analytically the spectral density which consists of a discrete set of isolated eigenvalues representing short cycles and a continuous part representing cycles of diverging size

random incidence matrices moments of the spectral density we study numerically and analytically the spectrum of incidence matrices of random labeled graphs on n vertices any pair of vertices is connected by an edge with probability p we give two algorithms to compute the moments of the eigenvalue distribution as explicit polynomials in n and p for large n and fixed p the spectrum contains a large eigenvalue at np and a semicircle of small eigenvalues for large n and fixed average connectivity pn dilute or sparse random matrices limit we show that the spectrum always contains a discrete component an anomaly in the spectrum near eigenvalue 0 for connectivity close to e is observed we develop recursion relations to compute the moments as explicit polynomials in pn their growth is slow enough so that they determine the spectrum the extension of our methods to the laplacian matrix is given in appendix

nonparametric tests of the markov hypothesis in continuous time models we propose several statistics to test the markov hypothesis for mixing stationary processes sampled at discrete time intervals our tests are based on the chapman kolmogorov equation we establish the asymptotic null distributions of the proposed test statistics showing that wilks s phenomenon holds we compute the power of the test and provide simulations to investigate the finite sample performance of the test statistics when the null model is a diffusion process with alternatives consisting of models with a stochastic mean reversion level stochastic volatility and jumps

universal and nonuniversal properties of cross correlations in financial time series we use methods of random matrix theory to analyze the cross correlation matrix c of stock price changes of the largest 1000 u s companies for the 2 year period 1994 1995 we find that the statistics of most of the eigenvalues in the spectrum of c agree with the predictions of random matrix theory but there are deviations for a few of the largest eigenvalues we find that c has the universal properties of the gaussian orthogonal ensemble of random matrices furthermore we analyze the eigenvectors of c through their inverse participation ratio and find eigenvectors with large ratios at both edges of the eigenvalue spectrum a situation reminiscent of localization theory results

spectra of real world graphs beyond the semicircle law results on the spectra of adjacency matrices corresponding to models of real world graphs we find that when the number of links grows as the number of nodes the spectral density of uncorrelated random matrices does not converge to the semicircle law furthermore the spectra of real world graphs have specific features depending on the details of the corresponding models in particular scale free graphs develop a trianglelike spectral density with a power law tail while small world graphs have a complex spectral density consisting of several sharp peaks these and further results indicate that the spectra of correlated graphs represent a practical tool for graph classification and can provide useful insight into the relevant structural properties of real networks

erratum small world networks evidence for a crossover picture phys rev lett 82 3180 1999 we have performed new calculations using the breadth first search algorithm 1 2 we are now able to study systems with sizes up to n 5500 as shown in fig 1 we now find t o 1 in agreement with the simple argument given in our letter but different from the originally reported numerical result st 0 67 6 0 10d the reason for the incorrect numerical result reported initially is the small system sizes we studied which did not allow us to reach the asymptotic regime we thank a barrat 3 and m e j newman and d j watts 4 for alerting us to the possibility of an error on our estimate of t we also thank m argollo de menezes for directing us to the breadth first search algorithm

finite plateau in spectral gap of polychromatic constrained random networks we consider critical behavior in the ensemble of polychromatic erd s renyi networks and regular random graphs where network vertices are painted in different colors the links can be randomly removed and added to the network subject to the condition of the vertex degree conservation in these constrained graphs we run the metropolis procedure which favors the connected unicolor triads of nodes changing the chemical potential of such triads for some wide region of we find the formation of a finite plateau in the number of intercolor links which exactly matches the finite plateau in the network algebraic connectivity the value of the first nonvanishing eigenvalue of the laplacian matrix 2 we claim that at the plateau the spontaneously broken z 2 symmetry is restored by the mechanism of modes collectivization in clusters of different colors the phenomena of a finite plateau formation holds also for polychromatic networks with m 2 colors the behavior of polychromatic networks is analyzed via the spectral properties of their adjacency and laplacian matrices

spectra of modular and small world matrices we compute spectra of symmetric random matrices describing graphs with general modular structure and arbitrary inter and intra module degree distributions subject only to the constraint of finite mean connectivities we also evaluate spectra of a certain class of small world matrices generated from random graphs by introducing shortcuts via additional random connectivity components both adjacency matrices and the associated graph laplacians are investigated for the laplacians we find lifshitz type singular behaviour of the spectral density in a localized region of small values in the case of modular networks we can identify contributions of local densities of state from individual modules for small world networks we find that the introduction of short cuts can lead to the creation of satellite bands outside the central band of extended states exhibiting only localized states in the band gaps results for the ensemble in the thermodynamic limit are in excellent agreement with those obtained via a cavity approach for large finite single instances and with direct diagonalization results

comparing nonparametric versus parametric regression fits in general there will be visible differences between a parametric and a nonparametric curve estimate it is therefore quite natural to compare these in order to decide whether the parametric model could be justified an asymptotic quantification is the distribution of the integrated squared difference between these curves we show that the standard way of bootstrapping this statistic fails we use and analyse a different form of bootstrapping for this task we call this method the wild bootstrap and apply it to fitting engel curves in expenditure data analysis

on the birch swinnerton dyer quotients modulo squares let a be an abelian variety over a number field k an identity between the l functions l a k i s for extensions k i of k induces a conjectural relation between the birch swinnerton dyer quotients we prove these relations modulo finiteness of lii and give an analogous statement for selmer groups based on this we develop a method for determining the parity of various combinations of ranks of a over extensions of k as one of the applications we establish the parity conjecture for elliptic curves assuming finiteness of iii e k e 2 6 and some restrictions on the reduction at primes above 2 and 3 the parity of the mordell weil rank of e k agrees with the parity of the analytic rank as determined by the root number we also prove the p parity conjecture for all elliptic curves over q and all primes p the parities of the p selmer rank and the analytic rank agree

nonparametric transition based tests for jump diffusions we develop a specification test for the transition density of a discretely sampled continuous time jump diffusion process based on a comparison of a nonparametric estimate of the transition density or distribution function with their corresponding parametric counterparts assumed by the null hypothesis as a special case our method applies to pure diffusions we provide a direct comparison of the two densities for an arbitrary specification of the null parametric model using three different discrepancy measures between the null and alternative transition density and distribution functions we establish the asymptotic null distributions of proposed test statistics and compute their power functions we investigate the finite sample properties through simulations and compare them with those of other tests this article has supplementary material online

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

ranks of twists of elliptic curves and hilbert s tenth problem in this paper we investigate the 2 selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields we give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2 selmer rank and we give lower bounds for the number of twists with bounded conductor that have a given 2 selmer rank as a consequence under appropriate hypotheses we can find many twists with trivial mordell weil group and assuming the shafarevich tate conjecture many others with infinite cyclic mordell weil group using work of poonen and shlapentokh it follows from our results that if the shafarevich tate conjecture holds then hilbert s tenth problem has a negative answer over the ring of integers of every number field

quasi maximum likelihood estimation of stochastic volatility models abstract changes in variance or volatility over time can be modelled using stochastic volatility sv models this approach is based on treating the volatility as an unobserved vatiable the logarithm of which is modelled as a linear stochastic process usually an autoregression this article analyses the asymptotic and finite sample properties of a quasi maximum likelihood qml estimator based on the kalman filter the relative efficiency of the qml estimator when compared with estimators based on the generalized method of moments is shown to be quite high for parameter values often found in empirical applications the qml estimator can still be employed when the sv model is generalized to allow for distributions with heavier tails than the normal sv models are finally fitted to daily observations on the yen dollar exchange rate

on a general class of models for interaction this paper develops a class of probability models for configurations of interacting points in a domain the distributions depend on a function which may be viewed as giving the potential energy of the configurations examples include models for interaction in a spatial region and on lattices and graphs new models in the general class arise naturally an example being a spatial model for points of different categories some general methods including series expansions and a simulation method known from statistical mechanics can be useful for many of these models several applications of this kind are considered and some connections between the models and with the statistical mechanics are explored

arch models as diffusion approximations abstract this paper investigates the convergence of stochastic difference equations e g arch to stochastic differential equations as the length of the discrete time intervals between observations goes to zero these results are applied to the garch 1 1 model of bollerslev 1986 and to the ar 1 exponential arch model of nelson 1989 in their continuous time limits the conditional variance processes in these models have stationary distributions that are inverted gamma and lognormal respectively in addition a class of diffusion approximations based on the exponential arch model is developed

goodness of fit testing and quadratic functional estimation from indirect observations we consider the convolution model where i i d random variables x i having unknown density f are observed with additive i i d noise independent of the x s we assume that the density f belongs to either a sobolev class or a class of supersmooth functions the noise distribution is known and its characteristic function decays either polynomially or exponentially asymptotically we consider the problem of goodness of fit testing in the convolution model we prove upper bounds for the risk of a test statistic derived from a kernel estimator of the quadratic functional f f 2 based on indirect observations when the unknown density is smoother enough than the noise density we prove that this estimator is n 1 2 consistent asymptotically normal and efficient for the variance we compute otherwise we give nonparametric upper bounds for the risk of the same estimator we give an approach unifying the proof of nonparametric minimax lower bounds for both problems we establish them for sobolev densities and for supersmooth densities less smooth than exponential noise in the two setups we obtain exact testing constants associated with the asymptotic minimax rates

bootstrap specification tests for diffusion processes this paper discusses specification tests for diffusion processes in the one dimensional case our proposed test is closest to the nonparametric test of ait sahalia rev financ stud 9 1996 385 however we compare cdfs instead of densities in the multidimensional and or multifactor case our proposed test is based on comparison of the empirical cdf of actual data and the empirical cdf of simulated data asymptotically valid critical values are obtained using an empirical process version of the block bootstrap which accounts for parameter estimation error an example based on a simple version of the cox et al econometrica 53 1985 385 model is outlined and related monte carlo experiments are carried out

mean field solution of the small world network model the small world network model is a simple model of the structure of social networks which possesses characteristics of both regular lattices and random graphs the model consists of a one dimensional lattice with a low density of shortcuts added between randomly selected pairs of points these shortcuts greatly reduce the typical path length between any two points on the lattice we present a mean field solution for the average path length and for the distribution of path lengths in the model this solution is exact in the limit of large system size and either a large or small number of shortcuts

a test for model specification of diffusion processes we propose a test for model specification of a parametric diffusion process based on a kernel estimation of the transitional density of the process the empirical likelihood is used to formulate a statistic for each kernel smoothing bandwidth which is effectively a studentized l2 distance between the kernel transitional density estimator and the parametric transitional density implied by the parametric process to reduce the sensitivity of the test on smoothing bandwidth choice the final test statistic is constructed by combining the empirical likelihood statistics over a set of smoothing bandwidths to better capture the finite sample distribution of the test statistic and data dependence the critical value of the test is obtained by a parametric bootstrap procedure properties of the test are evaluated asymptotically and numerically by simulation and by a real data example

predictive density construction and accuracy testing with multiple possibly misspecified diffusion models this paper develops tests for comparing the accuracy of predictive densities derived from possibly misspecified diffusion models in particular we first outline a simple simulation based framework for constructing predictive densities for one factor and stochastic volatility models then we construct accuracy assessment tests that are in the spirit of diebold and mariano 1995 and white 2000 in order to establish the asymptotic properties of our tests we also develop a recursive variant of the nonparametric simulated maximum

estimation of stochastic volatility models with diagnostics abstract efficient method of moments emm is used to fit the standard stochastic volatility model of various extensions to several daily financial time series emm matches to the score of the model determined by data analysis called the score generator discrepancies reveal characteristics of data that stochastic volatility models cannot approximate the two score generators employed here are semiparametric arch and nonlinear nonparametric with the first the standard model is rejected although some extensions are accepted with the second all versions are rejected the extensions required for an adequate fit are so elaborate that nonparametric specifications are probably more convenient

on the optimal rates of convergence for nonparametric deconvolution problems deconvolution problems arise in a variety of situations in statistics an interesting problem is to estimate the density f of a random variable x based on n i i d observations from y x varepsilon where varepsilon is a measurement error with a known distribution in this paper the effect of errors in variables of nonparametric deconvolution is examined insights are gained by showing that the difficulty of deconvolution depends on the smoothness of error distributions the smoother the harder in fact there are two types of optimal rates of convergence according to whether the error distribution is ordinary smooth or supersmooth it is shown that optimal rates of convergence can be achieved by deconvolution kernel density estimators

stationarity based specification tests for diffusions when the process is nonstationary abstract we analyze in this paper the asymptotic behavior of the specification test of ait sahalia 1996 for the stationary density of a diffusion process but when the diffusion is not stationary we consider integrated and explosive processes as well as nearly integrated ones in the spirit of the local to unity analysis in classical unit root theory we find that the behavior of the test predicted by the asymptotic distribution under an integrated process provides a better approximation to the small sample distribution of the test than that predicted by the asymptotic distribution under strict stationarity

eigenvalue tunneling and decay of quenched random network we consider the canonical ensemble of n vertex erd s renyi er random topological graphs with quenched vertex degree and with fugacity for each closed triple of bonds we claim complete defragmentation of large n graphs into the collection of p 1 almost full subgraphs cliques above critical fugacity c where p is the er bond formation probability evolution of the spectral density of the adjacency matrix with increasing leads to the formation of a multizonal support for c eigenvalue tunneling from the central zone to the side one means formation of a new clique in the defragmentation process the adjacency matrix of the network ground state has a block diagonal form where the number of vertices in blocks fluctuates around the mean value np the spectral density of the whole network in this regime has triangular shape we interpret the phenomena from the viewpoint of the conventional random matrix model and speculate about possible physical applications

nonlinearity and temporal dependence nonlinearities in the drift and diffusion coefficients influence temporal dependence in diffusion models we study this link using three measures of temporal dependence mixing mixing and mixing stationary diffusions that are mixing have mixing coefficients that decay exponentially to zero when they fail to be mixing they are still mixing and mixing but coefficient decay is slower than exponential for such processes we find transformations of the markov states that have finite variances but infinite spectral densities at frequency zero the resulting spectral densities behave like those of stochastic processes with long memory finally we show how state dependent poisson sampling alters the temporal dependence

bandwidth selection in nonparametric kernel testing we propose a sound approach to bandwidth selection in nonparametric kernel testing the main idea is to find an edgeworth expansion of the asymptotic distribution of the test concerned due to the involvement of a kernel bandwidth in the leading term of the edgeworth expansion we are able to establish closed form expressions to explicitly represent the leading terms of both the size and power functions and then determine how the bandwidth should be chosen according to certain requirements for both the size and power functions for example when a significance level is given we can choose the bandwidth such that the power function is maximized while the size function is controlled by the significance level both asymptotic theory and methodology are established in addition we develop an easy implementation procedure for the practical realization of the established methodology and illustrate this on two simulated examples and a real data example

semi nonparametric estimation and misspecification testing of diffusion models abstract novel transition based misspecification tests of semiparametric and fully parametric univariate diffusion models based on the estimators developed in kristensen d 2010 pseudo maximum likelihood estimation in two classes of semiparametric diffusion models journal of econometrics 156 239 259 are proposed it is demonstrated that transition based tests in general lack power in detecting certain departures from the null since they integrate out local features of the drift and volatility as a solution to this tests that directly compare drift and volatility estimators under the relevant null and alternative are also developed which exhibit better power against local alternatives

localization and centrality in networks eigenvector centrality is a common measure of the importance of nodes in a network here we show that under common conditions the eigenvector centrality displays a localization transition that causes most of the weight of the centrality to concentrate on a small number of nodes in the network in this regime the measure is no longer useful for distinguishing among the remaining nodes and its efficacy as a network metric is impaired as a remedy we propose an alternative centrality measure based on the nonbacktracking matrix which gives results closely similar to the standard eigenvector centrality in dense networks where the latter is well behaved but avoids localization and gives useful results in regimes where the standard centrality fails

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved