nonparametric instrumental variables estimation for efficiency frontier the paper investigates endogeneity issues in nonparametric frontier models it considers a nonseparable model for a cost function c y u where c and y are the cost and the output u is uniform in 0 1 and is increasing with respect to u the cost frontier corresponds to u 0 and u can be interpreted as a normalized level of inefficiency the endogeneity issue arises when y is dependent of u for identification and estimation we use a nonparametric instrumental variables estimator of the model for fixed value u and obtain an estimate of the quantile cost frontier y this involves the solution of a non linear integral equation if the true frontier y 0 is wanted it is then estimated by estimating the bias correction y 0 y under additional regularity conditions the procedure is illustrated through a simulated sample and with an empirical application to the efficiency of post offices

effective digital implementation of fuzzy control systems based on approximate discrete time models this paper addresses an effective digital implementation of fuzzy control systems via an intelligent digital redesign idr approach the purpose of idr is to effectively convert an existing continuous time fuzzy controller to an equivalent sampled data fuzzy controller in the sense of the state matching the authors show that under reasonable assumptions the idr based on the exact discrete time models can be reduced to the idr based on the approximate discrete time models the state matching error between the closed loop trajectories is carefully analyzed using the integral quadratic functional approach the estimation of the state matching error is presented using the linear matrix inequality lmi techniques the problem of designing the sampled data fuzzy controller to minimize the estimation as well as to guarantee the stability is formulated and solved as the convex optimization problem with lmi constraints it is also shown that the resulting sampled data fuzzy controller recovers the performance of the continuous time fuzzy controller as the sampling period approaches zero a numerical example is used to demonstrate the effectiveness of the proposed design technique

estimation of nonparametric models with simultaneity we introduce methods for estimating nonparametric nonadditive models with simultaneity the methods are developed by directly connecting the elements of the structural system to be estimated with features of the density of the observable variables such as ratios of derivatives or averages of products of derivatives of this density the estimators are therefore easily computed functionals of a nonparametric estimator of the density of the observable variables we consider in detail a model where to each structural equation there corresponds an exclusive regressor and a model with one equation of interest and one instrument that is included in a second equation for both models we provide new characterizations of observational equivalence on a set in terms of the density of the observable variables and derivatives of the structural functions based on those characterizations we develop two estimation methods in the first method the estimators of the structural derivatives are calculated by a simple matrix inversion and matrix multiplication analogous to a standard least squares estimator but with the elements of the matrices being averages of products of derivatives of nonparametric density estimators in the second method the estimators of the structural derivatives are calculated in two steps in a first step values of the instrument are found at which the density of the observable variables satisfies some properties in the second step the estimators are calculated directly from the values of derivatives of the density of the observable variables evaluated at the found values of the instrument we show that both pointwise estimators are consistent and asymptotically normal

record statistics for biased random walks with an application to financial data we consider the occurrence of record breaking events in random walks with asymmetric jump distributions the statistics of records in symmetric random walks was previously analyzed by majumdar and ziff phys rev lett 101 050601 2008 and is well understood unlike the case of symmetric jump distributions in the asymmetric case the statistics of records depends on the choice of the jump distribution we compute the record rate p n c defined as the probability for the nth value to be larger than all previous values for a gaussian jump distribution with standard deviation that is shifted by a constant drift c for small drift in the sense of c n 1 2 the correction to p n c grows proportional to arctan n and saturates at the value c 2 for large n the record rate approaches a constant which is approximately given by 1 2 c exp c 2 2 2 for c 1 these asymptotic results carry over to other continuous jump distributions with finite variance as an application we compare our analytical results to the record statistics of 366 daily stock prices from the standard poor s 500 index the biased random walk accounts quantitatively for the increase in the number of upper records due to the overall trend in the stock prices and after detrending the number of upper records is in good agreement with the symmetric random walk however the number of lower records in the detrended data is significantly reduced by a mechanism that remains to be identified

estimation of heterogeneous individual treatment effects with endogenous treatments this article estimates individual treatment effects ite and its probability distribution in a triangular model with binary valued endogenous treatments our estimation procedure takes two steps

some qualitative properties of sampled data control systems the authors consider sampled data control systems which consist of an interconnection of a continuous time nonlinear plant and a digital controller which has quantizers but is otherwise linear along with the required interface elements a d and d a converters in the present paper the authors study the qualitative effects of the quantizers and the plant nonlinearities of these systems

large deviation eigenvalue density for the soft edge laguerre and jacobi β ensembles we analyze the eigenvalue density for the laguerre and jacobi ensembles in the cases that the corresponding exponents are extensive in particular we obtain the asymptotic expansion up to terms o 1 in the large deviation regime outside the limiting interval of support as found in recent studies of the large deviation density for the gaussian ensemble and laguerre ensemble with fixed exponent there is a scaling from this asymptotic expansion to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge

index distribution of cauchy random matrices using a coulomb gas technique we compute analytically the probability that a large n n cauchy random matrix has n positive eigenvalues where n is called the index of the ensemble we show that this probability scales for large n as where is the dyson index of the ensemble the rate function c is computed in terms of single integrals that are easily evaluated numerically and amenable to an asymptotic analysis we find that the rate function around its minimum at 1 2 has a quadratic behavior modulated by a logarithmic singularity as a consequence the variance of the index scales for large n as var n cln n where c 2 2 is twice as large as the corresponding prefactor in the gaussian and wishart cases the analytical results are checked by numerical simulations and against an exact finite n formula which for 2 can be derived using orthogonal polynomials

instrumental variable estimation of nonseparable models there are many environments where knowledge of a structural relationship is required to answer questions of interest also nonseparability of a structural disturbance is a key feature of many models here we consider nonparametric identification and estimation of a model that is monotonic in a nonseparable scalar disturbance which disturbance is independent of instruments this model leads to conditional quantile restrictions we give local identification conditions for the structural equations from those quantile restrictions we find that a modified completeness condition is sufficient for local identification we also consider estimation via a nonparametric minimum distance estimator the estimator minimizes the sum of squares of predicted values from a nonparametric regression of the quantile residual on the instruments we show consistency of this estimator

testing a parametric quantile regression model with an endogenous explanatory variable against a nonparametric alternative abstract this paper is concerned with inference about a function g that is identified by a conditional quantile restriction involving instrumental variables the paper presents a test of the hypothesis that g belongs to a finite dimensional parametric family against a nonparametric alternative the test is not subject to the ill posed inverse problem of nonparametric instrumental variable estimation under mild conditions the test is consistent against any alternative model in large samples its power is arbitrarily close to 1 uniformly over a class of alternatives whose distance from the null hypothesis is proportional to n 1 2 where n is the sample size monte carlo simulations illustrate the finite sample performance of the test

on the testability of identification in some nonparametric models with endogeneity this paper examines three distinct hypothesis testing problems that arise in the context of identification of some nonparametric models with endogeneity the first hypothesis testing problem we study concerns testing necessary conditions for identification in some nonparametric models with endogeneity involving mean independence restrictions these conditions are typically referred to as completeness conditions the second and third hypothesis testing problems we examine concern testing for identification directly in some nonparametric models with endogeneity involving quantile independence restrictions for each of these hypothesis testing problems we provide conditions under which any test will have power no greater than size against any alternative in this sense we conclude that no nontrivial tests for these hypothesis testing problems exist publication abstract

phase transitions in the condition number distribution of gaussian random matrices we study the statistics of the condition number ensuremath kappa ensuremath lambda mathrm max ensuremath lambda mathrm min the ratio between largest and smallest squared singular values of n ifmmode times else texttimes fi m gaussian random matrices using a coulomb fluid technique we derive analytically and for large n the cumulative mathcal p ensuremath kappa lx and tail cumulative mathcal p ensuremath kappa gx distributions of ensuremath kappa we find that these distributions decay as mathcal p ensuremath kappa lx ensuremath approx exp ensuremath ensuremath beta n 2 ensuremath phi ensuremath x and mathcal p ensuremath kappa gx ensuremath approx exp ensuremath ensuremath beta n ensuremath phi x where ensuremath beta is the dyson index of the ensemble the left and right rate functions ensuremath phi ifmmode pm else textpm fi x are independent of ensuremath beta and calculated exactly for any choice of the rectangularity parameter ensuremath alpha m n ensuremath 1g0 interestingly they show a weak nonanalytic behavior at their minimum ensuremath langle ensuremath kappa ensuremath rangle corresponding to the average condition number a direct consequence of a phase transition in the associated coulomb fluid problem matching the behavior of the rate functions around ensuremath langle ensuremath kappa ensuremath rangle we determine exactly the scale of typical fluctuations ensuremath sim o n ensuremath 2 3 and the tails of the limiting distribution of ensuremath kappa the analytical results are in excellent agreement with numerical simulations

third order phase transition in random tilings we consider the domino tilings of an aztec diamond with a cut off corner of macroscopic square shape and given size and address the bulk properties of tilings as the size is varied we observe that the free energy exhibits a third order phase transition when the cut off square increasing in size reaches the arctic ellipse the phase separation curve of the original unmodified aztec diamond we obtain this result by studying the thermodynamic limit of a certain nonlocal correlation function of the underlying six vertex model with domain wall boundary conditions the so called emptiness formation probability efp we consider efp in two different representations as a function for toda chains and as a random matrix model integral the latter has a discrete measure and a linear potential with hard walls the observed phase transition shares properties with both gross witten wadia and douglas kazakov phase transitions

universal behaviour of extreme value statistics for selected observables of dynamical systems the main results of the extreme value theory developed for the investigation of the observables of dynamical systems rely up to now on the block maxima approach in this framework extremes are identified with the block maxima of the time series of the chosen observable in the limit of infinitely long blocks it has been proved that assuming suitable mixing conditions for the underlying dynamical systems the extremes of a specific class of observables are distributed according to the so called generalised extreme value gev distribution direct calculations show that in the case of quasi periodic dynamics the block maxima are not distributed according to the gev distribution in this paper we show that considering the exceedances over a given threshold instead of the block maxima approach it is possible to obtain a generalised pareto distribution also for extremes computed in systems which do not satisfy mixing conditions requiring that the invariant measure locally scales with a well defined exponent the local dimension we show that the limiting distribution for the exceedances of the observables previously studied with the block maxima approach is a generalised pareto distribution where the parameters depend only on the local dimensions and the values of the threshold but not on the number of observations considered we also provide connections with the results obtained with the block maxima approach in order to provide further support to our findings we present the results of numerical experiments carried out considering the well known chirikov standard map

applied nonparametric instrumental variables estimation instrumental variables are widely used in applied econometrics to achieve identification and carry out estimation and inference in models that contain endogenous explanatory variables in most applications the function of interest e g an engel curve or demand function is assumed to be known up to finitely many parameters e g a linear model and instrumental variables are used identify and estimate these parameters however linear and other finite dimensional parametric models make strong assumptions about the population being modeled that are rarely if ever justified by economic theory or other a priori reasoning and can lead to seriously erroneous conclusions if they are incorrect this paper explores what can be learned when the function of interest is identified through an instrumental variable but is not assumed to be known up to finitely many parameters the paper explains the differences between parametric and nonparametric estimators that are important for applied research describes an easily implemented nonparametric instrumental variables estimator and presents empirical examples in which nonparametric methods lead to substantive conclusions that are quite different from those obtained using standard parametric estimators

a closed form estimator for quantile treatment effects with endogeneity abstract this paper studies the estimation of quantile treatment effects based on the instrumental variable quantile regression ivqr model chernozhukov and hansen 2005 i develop a class of flexible plug in estimators based on closed form solutions derived from the ivqr moment conditions the proposed estimators remain tractable and root n consistent while allowing for rich patterns of effect heterogeneity functional central limit theorems and bootstrap validity results for the estimators of the quantile treatment effects and other functionals are provided monte carlo simulations demonstrate favorable finite sample properties of the proposed approach i apply my method to reanalyze the causal effect of 401 k plans

estimation of nonparametric conditional moment models with possibly nonsmooth moments this paper studies nonparametric estimation of conditional moment models in which the residual functions could be nonsmooth with respect to the unknown functions of endogenous variables it is a problem of nonparametric nonlinear instrumental variables iv estimation and a difficult nonlinear ill posed inverse problem with an unknown operator we first propose a penalized sieve minimum distance smd estimator of the unknown functions that are identified via the conditional moment models we then establish its consistency and convergence rate in strong metric allowing for possibly non compact function parameter spaces possibly non compact finite or infinite dimensional sieves with flexible lower semicompact or convex penalty or finite dimensional linear sieves without penalty under relatively low level sufficient conditions and for both mildly and severely ill posed problems we show that the convergence rates for the nonlinear ill posed inverse problems coincide with the known minimax optimal rates for the nonparametric mean iv regression we illustrate the theory by two important applications root n asymptotic normality of the plug in penalized smd estimator of a weighted average derivative of a nonparametric nonlinear iv regression and the convergence rate of a nonparametric additive quantile iv regression we also present a simulation study and an empirical estimation of a system of nonparametric quantile iv engel curves

functional linear regression with functional response abstract in this paper we develop new estimation results for functional regressions where both the regressor z t and the response y t are functions of hilbert spaces indexed by the time or a spatial location the model can be thought as a generalization of the multivariate regression where the regression coefficient is now an unknown operator we propose to estimate the operator by tikhonov regularization which amounts to apply a penalty on the l 2 norm of we derive the rate of convergence of the mean square error the asymptotic distribution of the estimator and develop tests on as trajectories are often not fully observed we consider the scenario where the data become more and more frequent infill asymptotics we also address the case where z is endogenous and instrumental variables are used to estimate an application to the electricity consumption completes the paper

nonparametric instrumental variables estimation of a quantile regression model we consider nonparametric estimation of a regression function that is identified by requiring a specified quantile of the regression error conditional on an instrumental variable to be zero the resulting estimating equation is a nonlinear integral equation of the first kind which generates an ill posed inverse problem the integral operator and distribution of the instrumental variable are unknown and must be estimated nonparametrically we show that the estimator is mean square consistent derive its rate of convergence in probability and give conditions under which this rate is optimal in a minimax sense the results of monte carlo experiments show that the estimator behaves well in finite samples

universal first passage properties of discrete time random walks and levy flights on a line statistics of the global maximum and records abstract in these lecture notes i will discuss the universal first passage properties of a simple correlated discrete time sequence x 0 0 x 1 x 2 x n up to n steps where x i represents the position at step i of a random walker hopping on a continuous line by drawing independently at each time step a random jump length from an arbitrary symmetric and continuous distribution it includes e g the levy flights i will focus on the statistics of two extreme observables associated with the sequence i its global maximum and the time step at which the maximum occurs and ii the number of records in the sequence and their ages i will demonstrate how the universal statistics of these observables emerge as a consequence of pollaczek spitzer formula and the associated sparre andersen theorem

new analysis on sparse solutions to random standard quadratic optimization problems and extensions the standard quadratic optimization problem stqp refers to the problem of minimizing a quadratic form over the standard simplex such a problem arises from numerous applications and is known to be np hard in a recent paper chen x peng j zhang s 2013 sparse solutions to random standard quadratic optimization problems math programming 141 1 2 273 293 chen et al showed that with a high probability close to 1 stqps with random data have sparse optimal solutions when the associated data matrix is randomly generated from a certain distribution such as uniform and exponential distributions in this paper we present a new analysis for random stqps combining probability inequalities derived from the first order and second order optimality conditions the new analysis allows us to significantly improve the probability bounds more important it allows us to handle normal distributions which is left open in chen et al 2013 the existence of sparse approximate solutions to convex stqps and extens

large deviations of extreme eigenvalues of random matrices we calculate analytically the probability of large deviations from its mean of the largest smallest eigenvalue of random matrices belonging to the gaussian orthogonal unitary and symplectic ensembles in particular we show that the probability that all the eigenvalues of an n ifmmode times else texttimes fi n random matrix are positive negative decreases for large n as ensuremath sim mathrm exp ensuremath ensuremath beta ensuremath theta 0 n 2 where the parameter ensuremath beta characterizes the ensemble and the exponent ensuremath theta 0 mathrm ln 3 4 0 274 text 653 dots is universal we also calculate exactly the average density of states in matrices whose eigenvalues are restricted to be larger than a fixed number ensuremath zeta thus generalizing the celebrated wigner semicircle law the density of states generically exhibits an inverse square root singularity at ensuremath zeta

inference in semiparametric conditional moment models with partial identification abstract this paper develops inference methods for conditional moment models in which the unknown parameter is possibly partially identified and may contain infinite dimensional components for a conjectured restriction on the parameter we consider testing the hypothesis that the restriction is satisfied by at least one element of the identified set we propose using the sieve minimum of a kolmogorov smirnov type statistic as the test statistic derive its asymptotic distribution and provide consistent bootstrap critical values in this way a broad family of restrictions can be consistently tested making the proposed procedure applicable to testing the model specification and constructing confidence set for any given component or some feature of the parameter our methods are robust to partial identification and allow for the moment functions to be nonsmooth as an illustration we apply the proposed inference methods to study the quantile instrumental variable engel curves for gasoline in brazil a monte carlo study demonstrates finite sample performance

generalized extreme value statistics and sum of correlated variables we show that generalized extreme value statistics the statistics of the kth largest value among a large set of random variables can be mapped onto a problem of random sums this allows us to identify classes of non identical and generally correlated random variables with a sum distributed according to one of the three k dependent asymptotic distributions of extreme value statistics namely the gumbel frechet and weibull distributions these classes as well as the limit distributions are naturally extended to real values of k thus providing a clear interpretation to the onset of gumbel distributions with non integer index k in the statistics of global observables this is one of the very few known generalizations of the central limit theorem to non independent random variables finally in the context of a simple physical model we relate the index k to the ratio of the correlation length to the system size which remains finite in strongly correlated systems

asymptotic stability of digitally redesigned control systems this paper analyses an asymptotic stability of a digitally redesigned control system when the states of the analogue and the digital control systems are approximately matched at every sampling point the digital redesign is a simple method of converting a given analogue controller to an equivalent digital controller in the sense of state matching the concerned state matching technique is to minimise the norm distance between the discretised closed loop system matrix of linear analogue control system and that of linear digital control system it is shown that i there exists an upper bound of the norm distance to guarantee the asymptotic stability of the digitally redesigned control system and ii the trajectories of the linear analogue and the linear digital control systems coincide at every sampling point if the norm distance is zero also a robustness result is provided in the case that nonlinear perturbations occur in the analogue and the digital control systems moreover design conditions for the

statistical distribution of quantum entanglement for a random bipartite state we compute analytically the statistics of the renyi and von neumann entropies standard measures of entanglement for a random pure state in a large bipartite quantum system the full probability distribution is computed by first mapping the problem to a random matrix model and then using a coulomb gas method we identify three different regimes in the entropy distribution which correspond to two phase transitions in the associated coulomb gas the two critical points correspond to sudden changes in the shape of the coulomb charge density the appearance of an integrable singularity at the origin for the first critical point and the detachment of the rightmost charge largest eigenvalue from the sea of the other charges at the second critical point analytical results are verified by monte carlo numerical simulations a short account of part of these results appeared recently in nadal et al phys rev lett 104 110501 2010

efficient estimation of semiparametric conditional moment models with possibly nonsmooth residuals this paper considers semiparametric efficient estimation ofconditional moment models with possibly nonsmooth residuals in unknown parametric components and unknown functions h of endogenous variables we show that 1 the penalized sieve minimum distance psmd estima tor can simultaneously achieve root n asymptotic normality ofand nonparametric optimal convergence rate of allowing for noncompact function parameter spaces 2 a simple weighted bootstrap procedure consistently estimates the limiting distribution of the psmd 3 the semi parametric efficiency bound formula of ai and chen 2003 remains valid for conditional models with nonsmooth residuals and the optimally weighted psmd estimator achieves the bound 4 the centered profiled optimally weighted psmd criterion is asymptotically chi square distributed we illustrate our theories using a partially linear quantile instrumental variables iv regression a monte carlo study and an empirical estimation of the shape invariant quantile iv engel curves

local identification of nonparametric and semiparametric models in parametric nonlinear structural models a classical sufficient condition for local identification like fisher 1966 and rothenberg 1971 is that the vector of moment conditions is differentiable at the true parameter with full rank derivative matrix we derive an analogous result for the nonparametric nonlinear structural models establishing conditions under which an infinite dimensional analog of the full rank condition is sufficient for local identification importantly we show that additional conditions are often needed in nonlinear nonparametric models to avoid nonlinearities overwhelming linear effects we give restrictions on a neighborhood of the true value that are sufficient for local identification we apply these results to obtain new primitive identification conditions in several important models including nonseparable quantile instrumental variable iv models and semiparametric consumption based asset pricing models

quasi bayesian analysis of nonparametric instrumental variables models this paper aims at developing a quasi bayesian analysis of the nonparametric instrumental variables model with a focus on the asymptotic properties of quasi posterior distributions in this paper instead of assuming a distributional assumption on the data generating process we consider a quasi likelihood induced from the conditional moment restriction and put priors on the function valued parameter we call the resulting posterior quasi posterior which corresponds to gibbs posterior in the literature here we focus on priors constructed on slowly growing finite dimensional sieves we derive rates of contraction and a nonparametric bernstein von mises type result for the quasi posterior distribution and rates of convergence for the quasi bayes estimator defined by the posterior expectation we show that with priors suitably chosen the quasi posterior distribution the quasi bayes estimator attains the minimax optimal rate of contraction convergence resp these results greatly sharpen the previous related work

time since maximum of brownian motion and asymmetric levy processes motivated by recent studies of record statistics in relation to strongly correlated time series we consider explicitly the drawdown time of a levy process which is defined as the time since it last achieved its running maximum when observed over a fixed time period we show that the density function of this drawdown time in the case of a completely asymmetric jump process may be factored as a function of t multiplied by a function of t t this extends a known result for the case of pure brownian motion we state the factors explicitly for the cases of exponential down jumps with drift and for the downward inverse gaussian levy process with drift

extreme value statistics of eigenvalues of gaussian random matrices we compute exact asymptotic results for the probability of the occurrence of large deviations of the largest smallest eigenvalue of random matrices belonging to the gaussian orthogonal unitary and symplectic ensembles in particular we show that the probability that all the eigenvalues of an n ifmmode times else texttimes fi n random matrix are positive negative decreases for large n as ensuremath sim text exp text ensuremath ensuremath beta ensuremath theta 0 n 2 where the dyson index ensuremath beta characterizes the ensemble and the exponent ensuremath theta 0 text ln text 3 4 0 274653 dots is universal we compute the probability that the eigenvalues lie in the interval ensuremath zeta 1 ensuremath zeta 2 which allows us to calculate the joint probability distribution of the minimum and the maximum eigenvalue as a by product we also obtain exactly the average density of states in gaussian ensembles whose eigenvalues are restricted to lie in the interval ensuremath zeta 1 ensuremath zeta 2 thus generalizing the celebrated wigner semi circle law to these restricted ensembles it is found that the density of states generically exhibits an inverse square root singularity at the location of the barriers these results are confirmed by numerical simulations some of the results presented in detail here were announced in a previous paper d s dean and s n majumdar phys rev lett 97 160201 2006

direct instrumental nonparametric estimation of inverse regression functions abstract this paper treats the estimation of the inverse g 1 of a monotonic function g satisfying e y g x w 0 where x w is continuously distributed using instrumental restrictions many parameters of interest in econometrics can be expressed as inverses of functions satisfying such a conditional moment as far as i know consistent estimators are available only if g x e y x which precludes endogenous models this and other technical concerns motivates a methodology for estimating g 1 in one step from the data when w contains components excluded from x the presented estimator achieves this objective by identifying g 1 as the solution of a nonlinear integral equation whose sample analog can be estimated nonparametrically the existence of a well behaved joint density function for x w produces an ill posed problem namely the solution is discontinuous in the data to solve this the estimator is regularized with the tikhonov technique if g 1 is smooth in a certain sense then the mse rate of convergence in a global metric is equal to n r 2 r 1 where is a continuous function with 0 0 and r 1 denotes the number of partial derivatives for the joint density of x w with respect to its second argument the analytical expression of depends on a link function characterizing the smoothness of g 1 this result holds for a broad class of link functions belonging to a suitable space

tikhonov regularization for nonparametric instrumental variable estimators abstract we study a tikhonov regularized tir estimator of a functional parameter identified by conditional moment restrictions in a linear model with both exogenous and endogenous regressors the nonparametric instrumental variable estimator is based on a minimum distance principle with penalization by the norms of the parameter and its derivatives after showing its consistency in the sobolev norm and uniform consistency under an embedding condition we derive the expression of the asymptotic mean integrated square error and the rate of convergence the optimal value of the regularization parameter is characterized in two examples we illustrate our theoretical findings and the small sample properties with simulation results finally we provide an empirical application to estimation of an engel curve and discuss a data driven selection procedure for the regularization parameter

iterative estimation of solutions to noisy nonlinear operator equations in nonparametric instrumental regression this paper discusses the solution of nonlinear integral equations with noisy integral kernels as they appear in nonparametric instrumental regression we propose a regularized newton type iteration and establish convergence and convergence rate results a particular emphasis is on instrumental regression models where the usual conditional mean assumption is replaced by a stronger independence assumption we demonstrate for the case of a binary instrument that our approach allows the correct estimation of regression functions which are not identifiable with the standard model this is illustrated in computed examples with simulated data

number statistics for β ensembles of random matrices applications to trapped fermions at zero temperature let mathcal p ensuremath beta v n mathcal i be the probability that a n ifmmode times else texttimes fi n phantom rule 4pt 0ex ensuremath beta ensemble of random matrices with confining potential v x has n mathcal i eigenvalues inside an interval mathcal i a b on the real line we introduce a general formalism based on the coulomb gas technique and the resolvent method to compute analytically mathcal p ensuremath beta v n mathcal i for large n we show that this probability scales for large n as mathcal p ensuremath beta v n mathcal i ensuremath approx exp left ensuremath ensuremath beta n 2 ensuremath psi v n mathcal i n right where ensuremath beta is the dyson index of the ensemble the rate function ensuremath psi v k mathcal i independent of ensuremath beta is computed in terms of single integrals that can be easily evaluated numerically the general formalism is then applied to the classical ensuremath beta gaussian mathcal i ensuremath l l ensuremath beta wishart mathcal i 1 l and ensuremath beta cauchy mathcal i ensuremath l l ensembles expanding the rate function around its minimum we find that generically the number variance var n mathcal i exhibits a nonmonotonic behavior as a function of the size of the interval with a maximum that can be precisely characterized these analytical results corroborated by numerical simulations provide the full counting statistics of many systems where random matrix models apply in particular we present results for the full counting statistics of zero temperature one dimensional spinless fermions in a harmonic trap

estimation of nonparametric conditional moment models with possibly nonsmooth generalized residuals this paper studies nonparametric estimation of conditional moment restrictions in which the generalized residual functions can be nonsmooth in the unknown functions of endogenous variables this is a nonparametric nonlinear instrumental variables iv problem we propose a class of penalized sieve minimum distance psmd estimators which are minimizers of a penalized empirical minimum distance criterion over a collection of sieve spaces that are dense in the infinite dimensional function parameter space some of the psmd procedures use slowly growing finite dimensional sieves with flexible penalties or without any penalty others use large dimensional sieves with lower semicompact and or convex penalties we establish their consistency and the convergence rates in banach space norms such as a sup norm or a root mean squared norm allowing for possibly noncompact infinite dimensional parameter spaces for both mildly and severely ill posed nonlinear inverse problems our convergence rates in hilbert space norms such as a root mean squared norm achieve the known minimax optimal rate for the nonparametric mean iv regression we illustrate the theory with a nonparametric additive quantile iv regression we present a simulation study and an empirical application of estimating nonparametric quantile iv engel curves

injectivity of a class of integral operators with compactly supported kernels abstract injectivity of integral operators is related to completeness conditions of their corresponding kernel functions completeness provides a useful way of obtaining nonparametric identification in various models including nonparametric regression models with instrumental variables nonclassical measurement error models and auction models etc however the condition is quite abstract for empirical work and lacks a proper economic interpretation we rely on known results regarding the volterra equation to provide sufficient conditions for completeness conditions for densities with compact support our conditions include various smoothness assumptions and monotonously moving support assumptions on the kernel function of the operator we apply our results to establish nonparametric identification in nonparametric iv regression models nonclassical measurement error models and auction models with an accessible interpretation and without specific functional form restrictions

a new instrumental method for dealing with endogenous selection this paper develops a new method for dealing with endogenous selection the usual instrumental strategy based on the independence between the outcome and the instrument is likely to fail when selection is directly driven by the dependent variable instead we suggest to rely on the independence between the instrument and the selection variable conditional on the outcome this approach may be particularly suitable for nonignorable nonresponse binary models with missing covariates or roy models with an unobserved sector the nonparametric identification of the joint distribution of the variables is obtained under a completeness assumption which has been used recently in several nonparametric instrumental problems even if the conditional independence between the instrument and the selection variable fails to hold the approach provides sharp bounds on parameters of interest under weaker monotonicity conditions apart from identification nonparametric and parametric estimations are also considered finally the method is applied to estimate the effect of grade retention in french primary schools

nonparametric instrumental variable estimation of structural quantile effects we study the asymptotic distribution of tikhonov regularized estimation of quantile structural effects implied by a nonseparable model the nonparametric instrumental variable estimator is based on a minimum distance principle we show that the min imum distance problem without regularization is locally ill posed and we consider penalization by the norms of the parameter and its derivatives we derive pointwise asymptotic normality and develop a consistent estimator of the asymptotic variance we study the small sample properties via simulation results and provide an empirical illustration of estimation of nonlinear pricing curves for telecommunications services in the united states

density of near extreme events we provide a quantitative analysis of the phenomenon of crowding of near extreme events by computing exactly the density of states dos near the maximum of a set of independent and identically distributed random variables we show that the mean dos converges to three different limiting forms depending on whether the tail of the distribution of the random variables decays slower than pure exponential faster than pure exponential or as a pure exponential function we argue that some of these results would remain valid even for certain correlated cases and verify it for power law correlated stationary gaussian sequences satisfactory agreement is found between the near maximum crowding in the summer temperature reconstruction data of western siberia and the theoretical prediction

examples of l2 complete and boundedly complete distributions abstract completeness and bounded completeness conditions are used increasingly in econometrics to obtain nonparametric identification in a variety of models from nonparametric instrumental variable regression to non classical measurement error models however distributions that are known to be complete or boundedly complete are somewhat scarce in this paper we consider an l 2 completeness condition that lies between completeness and bounded completeness we construct broad nonparametric classes of distributions that are l 2 complete and boundedly complete the distributions can have any marginal distributions and a wide range of strengths of dependence examples of l 2 incomplete distributions also are provided

quantile regression methods for recursive structural equation models abstract two classes of quantile regression estimation methods for the recursive structural equation models of chesher 2003 identification in nonseparable models econometrica 71 1405 1441 are investigated a class of weighted average derivative estimators based directly on the identification strategy of chesher is contrasted with a new control variate estimation method the latter imposes stronger restrictions achieving an asymptotic efficiency bound with respect to the former class an application of the methods to the study of the effect of class size on the performance of dutch primary school students shows that i reductions in class size are beneficial for good students in language and for weaker students in mathematics ii larger classes appear beneficial for weaker language students and iii the impact of class size on both mean and median performance is negligible

uniform confidence bands for functions estimated nonparametrically with instrumental variables abstract this paper is concerned with developing uniform confidence bands for functions estimated nonparametrically with instrumental variables we show that a sieve nonparametric instrumental variables estimator is pointwise asymptotically normally distributed the asymptotic normality result holds in both mildly and severely ill posed cases we present methods to obtain a uniform confidence band and show that the bootstrap can be used to obtain the required critical values monte carlo experiments illustrate the finite sample performance of the uniform confidence band

linear processes in high dimensions phase space and critical properties in this work we investigate the generic properties of a stochastic linear model in the regime of high dimensionality we consider in particular the vector autoregressive var model and the multivariate hawkes process we analyze both deterministic and random versions of these models showing the existence of a stable phase and an unstable phase we find that along the transition region separating the two regimes the correlations of the process decay slowly and we characterize the conditions under which these slow correlations are expected to become power laws we check our findings with numerical simulations showing remarkable agreement with our predictions we finally argue that real systems with a strong degree of self interaction are naturally characterized by this type of slow relaxation of the correlations

convergence rates of the spectral distributions of large random quaternion self dual hermitian matrices in this paper convergence rates of the spectral distributions of quaternion self dual hermitian matrices are investigated we show that under conditions of finite 6th moments the expected spectral distribution of a large quaternion self dual hermitian matrix converges to the semicircular law in a rate of o n 1 2 and the spectral distribution itself converges to the semicircular law in rates o p n 2 5 and o a s n 2 5 eta those results include gse as a special case

sieve wald and qlr inferences on semi nonparametric conditional moment models this paper considers inference on functionals of semi nonparametric conditional moment restrictions with possibly nonsmooth generalized residuals which include all of the nonlinear nonparametric instrumental variables iv as special cases these models are often ill posed and hence it is difficult to verify whether a possibly nonlinear functional is root n estimable or not we provide computationally simple unified inference procedures that are asymptotically valid regardless of whether a functional is root n estimable or not we establish the following new useful results 1 the asymptotic normality of a plug in penalized sieve minimum distance psmd estimator of a possibly nonlinear functional 2 the consistency of simple sieve variance estimators for the plug in psmd estimator and hence the asymptotic chi square distribution of the sieve wald statistic 3 the asymptotic chi square distribution of an optimally weighted sieve quasi likelihood ratio qlr test under the null hypothesis 4 the asymptotic tight distribution of a non optimally weighted sieve qlr statistic under the null 5 the consistency of generalized residual bootstrap sieve wald and qlr tests 6 local power properties of sieve wald and qlr tests and of their bootstrap versions 7 asymptotic properties of sieve wald and sqlr for functionals of increasing dimension simulation studies and an empirical illustration of a nonparametric quantile iv regression are presented

ages of records in random walks we consider random walks with continuous and symmetric step distributions we prove universal asymptotics for the average proportion of the age of the kth longest lasting record for k 1 2 ldots and for the probability that the record of the kth longest age is broken at step n due to the relation to the chinese restaurant process the ranked sequence of proportions of ages converges to the poisson dirichlet distribution

technical communique theoretical justification of approximate norm minimization method for intelligent digital redesign this paper investigates the theoretical justification of intelligent digital redesign idr methods based on the approximate discrete time models its main purpose is to design the digital fuzzy controller achieving the minimum norm distance between the trajectories of the approximate closed loop systems under the analog and the digital controls it is verified that if there exist the digital gains such that the approximate norm distance is sufficiently small the redesigned controller makes the trajectories of the exact closed loop systems close or at least asymptotically stabilizes the plant

digitalizing a fuzzy observer based output feedback control intelligent digital redesign approach this paper concerns an intelligent digital redesign idr technique for a takagi sugeno fuzzy observer based output feedback control fobofc system the term idr involves converting an existing analog control into an equivalent digital counterpart in the sense of state matching the idr problem is herein viewed as a minimization problem of the norm distances between nonlinearly interpolated linear operators to be matched its constructive condition with global rather than local state matching is formulated in terms of bilinear matrix inequalities the main features of the proposed method are that the state estimation error in the plant dynamics is considered in the idr condition that plays a crucial role in the performance improvement the stability property is preserved by the proposed idr algorithm the separation principle is shown when the premise variables are measurable finally the idr condition for a more general fobofc an estimated premise variable case is conducted a numerical example is demonstrated to visualize the feasibility of the developed methodology

posterior consistency of nonparametric conditional moment restricted models this paper addresses the estimation of the nonparametric conditional moment restricted model that involves an infinite dimensional parameter g0 we estimate it in a quasi bayesian way based on the limited information likelihood and investigate the impact of three types of priors on the posterior consistency i truncated prior priors supported on a bounded set ii thin tail prior a prior that has very thin tail outside a growing bounded set and iii normal prior with nonshrinking variance in addition g0 is allowed to be only partially identified in the frequentist sense and the parameter space does not need to be compact the posterior is regularized using a slowly growing sieve dimension and it is shown that the posterior converges to any small neighborhood of the identified region we then apply our results to the nonparametric instrumental regression model finally the posterior consistency using a random sieve dimension parameter is studied

first passage time statistics for two channel diffusion we present rigorous results for the mean first passage time and first passage time statistics for two channel markov additive diffusion in a 3 dimensional spherical domain inspired by biophysical examples we assume that the particle can only recognise the target in one of the modes which is shown to effect a non trivial first passage behaviour we also address the scenario of intermittent immobilisation in both cases we prove that despite the perfectly non recurrent motion of two channel markov additive diffusion in 3 dimensions the first passage statistics at long times do not display poisson like behaviour if none of the phases has a vanishing diffusion coefficient this stands in stark contrast to the standard one channel markov diffusion counterpart we also discuss the relevance of our results in the context of cellular signalling

how many eigenvalues of a gaussian random matrix are positive we study the probability distribution of the index n sub i e the number of positive eigenvalues of an nxn gaussian random matrix we show analytically that for large n and large n sub with the fraction 0 c n sub n 1 of positive eigenvalues fixed the index distribution p n sub cn n approx exp beta n sup 2 phi c where beta is the dyson index characterizing the gaussian ensemble the associated large deviation rate function phi c is computed explicitly for all 0 c 1 it is independent of beta and displays a quadratic form modulated by a logarithmic singularity around c 1 2 as a consequence the distribution of the index has a gaussian form near the peak but with a variance delta n of index fluctuations growing as delta n approx lnn beta pi sup 2 for large n for beta 2 this result is independently confirmed against an exact finite n formula yielding delta n lnn 2 pi sup 2 c o n sup 1 for large n where the constant c for even n has the nontrivial value c gamma 1 3ln2 2 pi sup 2 approx equal 0 185 248 and gamma 0 5772 is the euler constant we also determine for large n the probability that the interval zeta sub 1 zeta sub 2 is free of eigenvalues some of these results have been announced in a recent letter phys rev lett 103 220603 2009

identification in nonseparable models weak nonparametric restrictions are developed sufficient to identify the values of derivatives of structural functions in which latent random variables are nonseparable these derivatives can exhibit stochastic variation in a microeconometric context this allows the impact of a policy intervention as measured by the value of a structural derivative to vary across people who are identical as measured by covariates when the restrictions are satisfied quantiles of the distribution of a policy impact across people can be identified the identification restrictions are local in the sense that they are specific to the values of the covariates and the specific quantiles of latent variables at which identification is sought the conditions do not include the commonly required independence of latent variables and covariates they include local versions of the classical rank and order conditions and local quantile insensitivity conditions values of structural derivatives are identified by functionals of quantile regression functions and can be estimated using the same functionals applied to estimated quantile regression functions

two stage least absolute deviations estimators1 in this paper the method of least absolute deviations is applied to the estimation of the parameters of a structural equation in the simultaneous equations model a class of estimators called two stage least absolute deviations estimators is defined their asymptotic properties are derived and the problem of finding the optimal member of the class is considered in this paper we apply the method of least absolute deviations to the estimation of the parameters of a structural equation in the simultaneous equations model we define a class of estimators called two stage least absolute deviations estimators 2slad and derive their asymptotic properties they are so named as their relationship to the two stage least squares estimator 2sls is analogous to the relationship of the least absolute deviations estimator lad to the least squares estimator ls in the standard regression model the lad estimation has been extensively studied in the context of the standard regression model and its usefulness is universally recognized in this paper we show that the advantage of 2slad over 2sls in the simultaneous equations model can be as great as that of lad over ls in the standard regression model if 2slad is properly defined this last clause is very important since the results of this paper indicate that the lad analogue of 2sls that has been considered before in the literature is not an appropriate method

record statistics of financial time series and geometric random walks the study of record statistics of correlated series in physics such as random walks is gaining momentum and several analytical results have been obtained in the past few years in this work we study the record statistics of correlated empirical data for which random walk models have relevance we obtain results for the records statistics of select stock market data and the geometric random walk primarily through simulations we show that the distribution of the age of records is a power law with the exponent lying in the range 1 5 1 8 further the longest record ages follow the frechet distribution of extreme value theory the records statistics of geometric random walk series is in good agreement with that obtained from empirical stock data

truncated linear statistics associated with the eigenvalues of random matrices ii partial sums over proper time delays for chaotic quantum dots invariant ensembles of random matrices are characterized by the distribution of their eigenvalues lambda 1 ldots lambda n we study the distribution of truncated linear statistics of the form tilde l sum i 1 p f lambda i with p n this problem has been considered by us in a previous paper when the p eigenvalues are further constrained to be the largest ones or the smallest in this second paper we consider the same problem without this restriction which leads to a rather different analysis we introduce a new ensemble which is related but not equivalent to the thinned ensembles introduced by bohigas and pato this question is motivated by the study of partial sums of proper time delays in chaotic quantum dots which are characteristic times of the scattering process using the coulomb gas technique we derive the large deviation function for tilde l large deviations of linear statistics l sum i 1 n f lambda i are usually dominated by the energy of the coulomb gas which scales as sim n 2 implying that the relative fluctuations are of order 1 n for the truncated linear statistics considered here there is a whole region including the typical fluctuations region where the energy of the coulomb gas is frozen and the large deviation function is purely controlled by an entropic effect because the entropy scales as sim n the relative fluctuations are of order 1 sqrt n our analysis relies on the mapping on a problem of p fictitious non interacting fermions in n energy levels which can exhibit both positive and negative effective absolute temperatures we determine the large deviation function characterizing the distribution of the truncated linear statistics and show that for the case considered here f lambda 1 lambda the corresponding phase diagram is separated in three different phases

penalized sieve gel for weighted average derivatives of nonparametric quantile iv regressions abstract this paper considers estimation and inference for a weighted average derivative wad of a nonparametric quantile instrumental variables regression npqiv npqiv is a non separable and nonlinear ill posed inverse problem which might be why there is no published work on the asymptotic properties of any estimator of its wad we first characterize the semiparametric efficiency bound for a wad of a npqiv which unfortunately depends on an unknown conditional derivative operator and hence an unknown degree of ill posedness making it difficult to know if the information bound is singular or not in either case we propose a penalized sieve generalized empirical likelihood gel estimation and inference procedure which is based on the unconditional wad moment restriction and an increasing number of unconditional moments that are implied by the conditional npqiv restriction where the unknown quantile function is approximated by a penalized sieve under some regularity conditions we show that the self normalized penalized sieve gel estimator of the wad of a npqiv is asymptotically standard normal we also show that the quasi likelihood ratio statistic based on the penalized sieve gel criterion is asymptotically chi square distributed regardless of whether or not the information bound is singular

random convex hulls and extreme value statistics in this paper we study the statistical properties of convex hulls of n random points in a plane chosen according to a given distribution the points may be chosen independently or they may be correlated after a non exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem we present a unifying approach based on the notion of support function of a closed curve and the associated cauchy s formulae that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics as an example of correlated points we study here in detail the case when the points represent the vertices of n independent random walks in the continuum time limit this reduces to n independent planar brownian trajectories for which we compute exactly for all n the mean perimeter and the mean area of their global convex hull our results have relevant applications in ecology in estimating the home range of a herd of animals some of these results were announced recently in a short communication phys rev lett 103 140602 2009

an iv model of quantile treatment effects the ability of quantile regression models to characterize the heterogeneous impact of variables on different points of an outcome distribution makes them appealing in many economic applications however in observational studies the variables of interest e g education prices are often endogenous making conventional quantile regression inconsistent and hence inappropriate for recovering the causal effects of these variables on the quantiles of economic outcomes in order to address this problem we develop a model of quantile treatment effects qte in the presence of endogeneity and obtain conditions for identification of the qte without functional form assumptions the principal feature of the model is the imposition of conditions that restrict the evolution of ranks across treatment states this feature allows us to overcome the endogeneity problem and recover the true qte through the use of instrumental variables the proposed model can also be equivalently viewed as a structural simultaneous equation model with nonadditive errors where qte can be interpreted as the structural quantile effects sqe

lower tail of the kpz equation we provide the first tight bounds on the lower tail probability of the one point distribution of the kardar parisi zhang kpz equation with narrow wedge initial data our bounds hold for all sufficiently large times t and demonstrates a crossover between superexponential decay with exponent 52 and leading prefactor 415 t1 3 for tail depth greater than t2 3 and exponent 3 with leading prefactor at least 112 for tail depth less than t2 3

specification testing in nonparametric instrumental variable estimation abstract in nonparametric instrumental variable estimation the function being estimated is the solution to an integral equation a solution may not exist if for example the instrument is not valid this paper discusses the problem of testing the null hypothesis that a solution exists against the alternative that there is no solution we give necessary and sufficient conditions for existence of a solution and show that uniformly consistent testing of an unrestricted null hypothesis is not possible uniformly consistent testing is possible however if the null hypothesis is restricted by assuming that any solution to the integral equation is smooth many functions of interest in applied econometrics including demand functions and engel curves are expected to be smooth the paper presents a statistic for testing the null hypothesis that a smooth solution exists the test is consistent uniformly over a large class of probability distributions of the observable random variables for which the integral equation has no smooth solution the finite sample performance of the test is illustrated through monte carlo experiments

instrumental quantile regression inference for structural and treatment effect models we introduce a class of instrumental quantile regression methods for heterogeneous treatment effect models and simultaneous equations models with nonadditive errors and offer computable methods for estimation and inference these methods can be used to evaluate the impact of endogenous variables or treatments on the entire distribution of outcomes we describe an estimator of the instrumental variable quantile regression process and the set of inference procedures derived from it we focus our discussion of inference on tests of distributional equality constancy of effects conditional dominance and exogeneity we apply the procedures to characterize the returns to schooling in the u s

overidentification in regular models in the unconditional moment restriction model of hansen specification tests and more efficient estimators are both available whenever the number of moment restrictions exceeds the number of parameters of interest we show that a similar relationship between potential refutability of a model and existence of more efficient estimators is present in much broader settings specifically a condition we name local overidentification is shown to be equivalent to both the existence of specification tests with nontrivial local power and the existence of more efficient estimators of some smooth parameters in general semi nonparametric models under our notion of local overidentification various locally nontrivial specification tests such as hausman tests incremental sargan tests or optimally weighted quasi likelihood ratio tests naturally extend to general semi nonparametric settings we further obtain simple characterizations of local overidentification for general models of nonparametric conditional moment restrictions with possibly different conditioning sets the results are applied to determining when semi nonparametric models with endogeneity are locally testable and when nonparametric plug in and semiparametric two step gmm estimators are semiparametrically efficient examples of empirically relevant semi nonparametric structural models are presented

adaptive nonparametric instrumental variables estimation empirical choice of the regularization parameter abstract in nonparametric instrumental variables estimation the mapping that identifies the function of interest g is discontinuous and must be regularized to permit consistent estimation the optimal regularization parameter depends on population characteristics that are unknown in applications this paper presents a theoretically justified empirical method for choosing the regularization parameter in series estimation the method adapts to the unknown smoothness of g and other unknown functions the resulting estimator of g converges at least as fast as the optimal rate multiplied by log n 1 2 the asymptotic integrated mean square error aimse of the estimator is within a specified factor of the optimal aimse

exactly solvable record model for rainfall daily precipitation time series are composed of null entries corresponding to dry days and nonzero entries that describe the rainfall amounts on wet days assuming that wet days follow a bernoulli process with success probability p we show that the presence of dry days induces negative correlations between record breaking precipitation events the resulting nonmonotonic behavior of the fano factor of the record counting process is recovered in empirical data we derive the full probability distribution p r n of the number of records r n up to time n and show that for large n it converges to a poisson distribution with parameter ln pn we also study in detail the joint limit p 0 n which yields a random record model in continuous time t pn

nonequilibrium random matrix theory transition probabilities in this paper we present an analytic method for calculating the transition probability between two random gaussian matrices with given eigenvalue spectra in the context of dyson brownian motion we show that in the coulomb gas language in large n limit memory of the initial state is preserved in the form of a universal linear potential acting on the eigenvalues we compute the likelihood of any given transition as a function of time showing that as memory of the initial state is lost transition probabilities converge to those of the static ensemble

gradient based smoothing parameter selection for nonparametric regression estimation abstract estimating gradients is of crucial importance across a broad range of applied economic domains here we consider data driven bandwidth selection based on the gradient of an unknown regression function this is a difficult problem given that direct observation of the value of the gradient is typically not observed the procedure developed here delivers bandwidths which behave asymptotically as though they were selected knowing the true gradient simulated examples showcase the finite sample attraction of this new mechanism and confirm the theoretical predictions

on the frequencies of patterns of rises and falls abstract we investigate the probability of observing a given pattern of n rises and falls in a random stationary data series the data are modeled as a sequence of n 1 independent and identically distributed random numbers this probabilistic approach has a combinatorial equivalent where the data are modeled by a random permutation on n 1 objects the probability of observing a long pattern of rises and falls decays exponentially with its length n in general the associated decay rate is interpreted as the embedding entropy of the pattern this rate is evaluated exactly for all periodic patterns in the most general case it is expressed in terms of a determinant of generalized hyperbolic or trigonometric functions alternating patterns have the smallest rate min ln 2 0 451582 while other examples lead to arbitrarily large rates the probabilities of observing uniformly chosen random patterns are demonstrated to obey multifractal statistics the typical value 0 0 806361 of the rate plays the role of a lyapunov exponent a wide range of examples of patterns either deterministic or random is also investigated