tests of equal forecast accuracy and encompassing for nested models we examine the asymptotic and finite sample properties of tests for equal forecast accuracy and encompassing applied to 1 step ahead forecasts from nested linear models we first derive the asymptotic distributions of two standard tests and one new test of encompassing and provide tables of asymptotically valid critical values monte carlo methods are then used to evaluate the size and power of tests of equal forecast accuracy and encompassing the simulations indicate that post sample tests can be reasonably well sized of the post sample tests considered the encompassing test proposed in this paper is the most powerful we conclude with an empirical application regarding the predictive content of unemployment for inflation

reconstruction guarantee analysis of basis pursuit for binary measurement matrices in compressed sensing recently binary 0 1 measurement matrices especially those from coding theory were introduced to compressed sensing dimakis et al found that the linear programming lp decoding of ldpc codes is very similar to the lp reconstruction of compressed sensing and they further showed that the sparse binary parity check matrices of good ldpc codes can be used as provably good measurement matrices for compressed sensing under basis pursuit bp moreover khajehnejad et al made use of girth to certify the good performances of sparse binary measurement matrices in this paper we examine the performance of binary measurement matrices with uniform column weight and arbitrary girth under bp for a fixed measurement matrix we first introduce a performance indicator w bp min called minimum bp weight and show that any k sparse signals could be exactly recovered by bp if and only if k le w bp min 1 2 then lower bounds of w bp min are studied borrowing ideas from the tree bound for the ldpc codes we obtain several explicit lower bounds of w bp min which improve on the previous results in some cases these lower bounds also imply explicit ell 1 ell 1 ell 2 ell 1 and ell infty ell 1 sparse approximation guarantees and further confirm that large girth has positive impacts on the performance of binary measurement matrices under bp

statistical tests for multiple forecast comparison abstract we consider a multivariate version of the diebold mariano test for equal predictive ability of three or more forecasting models the wald type test s which has a null distribution that is asymptotically chi squared is shown to be generally invariant with respect to the ordering of the models being compared finite sample corrections for the test are also developed monte carlo simulations indicate that s has reasonable size properties in large samples but tends to be oversized in moderate samples the finite sample correction succeeds in correcting for size but only partially for the size adjusted tests power increases with sample size as expected it is speculated that further finite sample improvements can be achieved using hotelling s t 2 or bootstrap critical values

link delay estimation via expander graphs one of the purposes of network tomography is to infer the status of parameters e g delay for the links inside a network through end to end probing between external boundary nodes along predetermined routes in this work we apply concepts from compressed sensing and expander graphs to the delay estimation problem we first show that a relative majority of network topologies are not expanders for existing expansion criteria motivated by this challenge we then relax such criteria enabling us to acquire simulation evidence that link delays can be estimated for 30 more networks that is our relaxation expands the list of identifiable networks with bounded estimation error by 30 we conduct a simulation performance analysis of delay estimation and congestion detection on the basis of l1 minimization demonstrating that accurate estimation is feasible for an increasing proportion of networks

efficient and robust compressed sensing using optimized expander graphs expander graphs have been recently proposed to construct efficient compressed sensing algorithms in particular it has been shown that any n dimensional vector that is k sparse can be fully recovered using o klogn measurements and only o klogn simple recovery iterations in this paper we improve upon this result by considering expander graphs with expansion coefficient beyond 3 4 and show that with the same number of measurements only o k recovery iterations are required which is a significant improvement when n is large in fact full recovery can be accomplished by at most 2k very simple iterations the number of iterations can be reduced arbitrarily close to k and the recovery algorithm can be implemented very efficiently using a simple priority queue with total recovery time o nlog n k we also show that by tolerating a small penalty on the number of measurements and not on the number of recovery iterations one can use the efficient construction of a family of expander graphs to come up with explicit measurement matrices for this method we compare our result with other recently developed expander graph based methods and argue that it compares favorably both in terms of the number of required measurements and in terms of the time complexity and the simplicity of recovery finally we will show how our analysis extends to give a robust algorithm that finds the position and sign of the k significant elements of an almost k sparse signal and then using very simple optimization techniques finds a k sparse signal which is close to the best k term approximation of the original signal

predictor selection for positive autoregressive processes let observations y1 yn be generated from a first order autoregressive ar model with positive errors in both the stationary and unit root cases we derive moment bounds and limiting distributions of an extreme value estimator of the ar coefficient these results enable us to provide asymptotic expressions for the mean squared error mse of and the mean squared prediction error mspe of the corresponding predictor of yn 1 based on these expressions we compare the relative performance of and the least squares predictor estimator from the mspe mse point of view our comparison reveals that the better predictor estimator is determined not only by whether a unit root exists but also by the behavior of the underlying error distribution near the origin and hence is difficult to identify in practice to circumvent this difficulty we suggest choosing the predictor estimator with the smaller accumulated prediction error and show that the predictor estimator chosen in this way is as

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

sensitivity to basis mismatch in compressed sensing the theory of compressed sensing suggests that successful inversion of an image of the physical world broadly defined to include speech signals radar sonar returns vibration records sensor array snapshot vectors 2 d images and so on for its source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis provided that the image is sparse in an apriori known basis for imaging problems in spectrum analysis and passive and active radar sonar this basis is usually taken to be a dft basis however in reality no physical field is sparse in the dft basis or in any apriori known basis no matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and consequently there is mismatch between the assumed and the actual bases for sparsity in this paper we study the sensitivity of compressed sensing to mismatch between the assumed and the actual sparsity bases we start by analyzing the effect of basis mismatch on the best k term approximation error which is central to providing exact sparse recovery guarantees we establish achievable bounds for the l1 error of the best k term approximation and show that these bounds grow linearly with the image or grid dimension and the mismatch level between the assumed and actual bases for sparsity we then derive bounds with similar growth behavior for the basis pursuit l1 recovery error indicating that the sparse recovery may suffer large errors in the presence of basis mismatch although we present our results in the context of basis pursuit our analysis applies to any sparse recovery principle that relies on the accuracy of best k term approximations for its performance guarantees we particularly highlight the problematic nature of basis mismatch in fourier imaging where spillage from off grid dft components turns a sparse representation into an incompressible one we substantiate our mathematical analysis by numerical examples that demonstrate a considerable performance degradation for image inversion from compressed sensing measurements in the presence of basis mismatch for problem sizes common to radar and sonar

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

another heteroskedasticity and autocorrelation consistent covariance matrix estimator abstract a t consistent estimator of a heteroskedasticity and autocorrelation consistent covariance matrix estimator is proposed and evaluated the relevant applications are ones in which the regression disturbance follows a moving average process of known order in a system of l equations this am l estimator entails estimation of the moving average coefficients of an l dimensional vector simulations indicate that the ma l estimator s finite sample performance is better than that of the estimators of andrews and monahan 1992 and newey and west 1994 when cross products of instruments and disturbances are sharply negatively autocorrelated comparable or slightly worse otherwise

unbalanced expanders and randomness extractors from parvaresh vardy codes we give an improved explicit construction of highly unbalanced bipartite expander graphs with expansion arbitrarily close to the degree which is polylogarithmic in the number of vertices both the degree and the number of right hand vertices are polynomially close to optimal whereas the previous constructions of ta shma et al 2007 required at least one of these to be quasipolynomial in the optimal our expanders have a short and self contained description and analysis based on the ideas underlying the recent list decodable error correcting codes of parvaresh and vardy 2005 our expanders can be interpreted as near optimal randomness condensers that reduce the task of extracting randomness from sources of arbitrary min entropy rate to extracting randomness from sources of min entropy rate arbitrarily close to 1 which is a much easier task using this connection we obtain a new self contained construction of randomness extractors that is optimal up to constant factors while being much simpler than the previous construction of lu et al 2003 and improving upon it when the error parameter is small e g 1 poly n

parameter estimation and forecasting for multiplicative log normal cascades we study the well known multiplicative log normal cascade process in which the multiplication of gaussian and log normally distributed random variables yields time series with intermittent bursts of activity due to the nonstationarity of this process and the combinatorial nature of such a formalism its parameters have been estimated mostly by fitting the numerical approximation of the associated non gaussian probability density function to empirical data cf castaing et al physica d 46 177 1990 more recently alternative estimators based upon various moments have been proposed by beck physica d 193 195 2004 and kiyono et al phys rev e 76 041113 2007 in this paper we pursue this moment based approach further and develop a more rigorous generalized method of moments gmm estimation procedure to cope with the documented difficulties of previous methodologies we show that even under uncertainty about the actual number of cascade steps our methodology yields very reliable results for the estimated intermittency parameter employing the levinson durbin algorithm for best linear forecasts we also show that estimated parameters can be used for forecasting the evolution of the turbulent flow we compare forecasting results from the gmm and kiyono et al s procedure via monte carlo simulations we finally test the applicability of our approach by estimating the intermittency parameter and forecasting of volatility for a sample of financial data from stock and foreign exchange markets

the spectra of infinite hypertrees a model of regular infinite hypertrees is developed to mimic for hypergraphs what infinite trees do for graphs two notions of spectra or first eigenvalue are then examined for the infinite tree obtaining a precise value for the first notion and obtaining some estimates for the second the results indicate agreement of the first eigenvalue of the infinite hypertree with the second eigenvalue of a random hypergraph of the same degree to within logarithmic factors at least for the first notion of first eigenvalue author s abstract

properties of predictors in misspecified autoregressive time series models abstract this article investigates major effects of misspecification in stationary linear time series models when we fit a pth order autoregressive model the true model can be an autoregressive moving average model we derive the formulas of bias and mean squared error mse of the least squares estimator and the hth period ahead prediction mse in the time domain contrary to previous studies the process in estimation is not necessarily independent of the process in prediction and the distribution of process is not necessarily gaussian we examine the effects of this dependence and nonnormality on prediction in misspecified models

a predictability test for a small number of nested models we introduce quasi likelihood ratio tests for one sided multivariate hypotheses to evaluate the null that a parsimonious model performs equally well as a small number of models which nest the benchmark the limiting distributions of the test statistics are non standard for critical values we consider i bootstrapping and ii simulations assuming normality of the mean square prediction error difference the proposed tests have good size and power properties compared with existing equal and superior predictive ability tests for multiple model comparison we apply our tests to study the predictive ability of a phillips curve type for the us core inflation

a note on adiabatic invariance in hamiltonian systems depending singularly on the slow time abstract a one degree of freedom oscillatory hamiltonian system with a parameter depending singularly on the slow time is considered it is shown that the system possesses an adiabatic invariant and its asymptotics is estimated for a rather general type of singularity the leading term of the asymptotics turns out to be given by an integral of fresnel type and the order of asymptotics is related to the type of singularity stronger singularities cause larger change of adiabatic invariant the result is applied to estimate the reflection coefficient in the problem of scattering electromagnetic wave on an obstacle with refraction index depending singularly on the coordinate the relation to the stationary phase method is outlined

breakdown of stability of motion in superquadratic potentials based on the kam theory investigation of the equation of motion of a classical particle in a one dimensional superquadratic potential well under the influence of an external time periodic forcing raised a hope that all the solutions are bounded indeed due to the superquadraticity of the potential the frequency of oscillations of the solutions in the system tends to infinity as the amplitude increases therefore because of this relationship between the frequency and the amplitude intuitively one might expect that all resonances that could cause the accumulation of energy would be destroyed and thus all solutions would stay bounded for all time more formally according to moser s twist theorem this could mean the existence of invariant tubes in the extended phase space and therefore would result in the boundedness of the solutions

vanishingly sparse matrices and expander graphs with application to compressed sensing we revisit the probabilistic construction of sparse random matrices where each column has a fixed number of nonzeros whose row indices are drawn uniformly at random with replacement these matrices have a one to one correspondence with the adjacency matrices of fixed left degree expander graphs we present formulas for the expected cardinality of the set of neighbors for these graphs and present tail bounds on the probability that this cardinality will be less than the expected value deducible from these bounds are similar bounds for the expansion of the graph which is of interest in many applications these bounds are derived through a more detailed analysis of collisions in unions of sets key to this analysis is a novel dyadic splitting technique the analysis led to the derivation of better order constants that allow for quantitative theorems on existence of lossless expander graphs and hence the sparse random matrices we consider and also quantitative compressed sensing sampling theorems when using sparse nonmean zero measurement matrices

how to compute the volume in high dimension in some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones

approximately normal tests for equal predictive accuracy in nested models abstract forecast evaluation often compares a parsimonious null model to a larger model that nests the null model under the null that the parsimonious model generates the data the larger model introduces noise into its forecasts by estimating parameters whose population values are zero we observe that the mean squared prediction error mspe from the parsimonious model is therefore expected to be smaller than that of the larger model we describe how to adjust mspes to account for this noise we propose applying standard methods west k d 1996 asymptotic inference about predictive ability econometrica 64 1067 1084 to test whether the adjusted mean squared error difference is zero we refer to nonstandard limiting distributions derived in clark and mccracken 2001 tests of equal forecast accuracy and encompassing for nested models journal of econometrics 105 85 110 2005a evaluating direct multistep forecasts econometric reviews 24 369 404 to argue that use of standard normal critical values will yield actual sizes close to but a little less than nominal size simulation evidence supports our recommended procedure

a consistent test for nonlinear out of sample predictive accuracy abstract in this paper we draw on both the consistent specification testing and the predictive ability testing literatures and propose an integrated conditional moment type predictive accuracy test that is similar in spirit to that developed by bierens j econometr 20 1982 105 econometrica 58 1990 1443 and bierens and ploberger econometrica 65 1997 1129 the test is consistent against generic nonlinear alternatives and is designed for comparing nested models one important feature of our approach is that the same loss function is used for in sample estimation and out of sample prediction in this way we rule out the possibility that the null model can outperform the nesting generic alternative model it turns out that the limiting distribution of the icm type test statistic that we propose is a functional of a gaussian process with a covariance kernel that reflects both the time series structure of the data as well as the contribution of parameter estimation error as a consequence critical values that are data dependent and cannot be directly tabulated one approach in this case is to obtain critical value upper bounds using the approach of bierens and ploberger econometrica 65 1997 1129 here we establish the validity of a conditional p value method for constructing critical values the method is similar in spirit to that proposed by hansen econometrica 64 1996 413 and inoue econometric theory 17 2001 156 although we additionally account for parameter estimation error in a series of monte carlo experiments the finite sample properties of three variants of the predictive accuracy test are examined our findings suggest that all three variants of the test have good finite sample properties when quadratic loss is specified even for samples as small as 600 observations however non quadratic loss functions such as linex loss require larger sample sizes of 1000 observations or more in order to ensure reasonable finite sample performance

using out of sample mean squared prediction errors to test the martingale difference hypothesis abstract we consider using out of sample mean squared prediction errors mspes to evaluate the null that a given series follows a zero mean martingale difference against the alternative that it is linearly predictable under the null of no predictability the population mspe of the null no change model equals that of the linear alternative we show analytically and via simulations that despite this equality the alternative model s sample mspe is expected to be greater than the null s for rolling regression estimators of the alternative model s parameters we propose and evaluate an asymptotically normal test that properly accounts for the upward shift of the sample mspe of the alternative model our simulations indicate that our proposed procedure works well