Empirical Characteristic Function Research Articles

A review of testing procedures based on the empirical characteristic function

The empirical characteristic function (ECF) has been in use in statistical inference for nearly fifty years now. We provide an overview of testing procedures based on the ECF within certain statistical models. Specifically our emphasis is on recent developments of ECF procedures for goodness-of-fit testing for parametric families of distributions with structured data, and for the two-sample and the k-sample problem.

Simple Nonparametric Estimators for the Bid-Ask Spread in the Roll Model

We propose new methods for estimating the bid-ask spread from observed transaction prices alone. Our methods are based on the empirical characteristic function instead of the sample autocovariance function like the method of Roll (1984). As in Roll (1984), we have a closed form expression for the spread, but this is only based on a limited amount of the model-implied identification restrictions. We also provide methods that take account of more identification information. We compare our methods theoretically and numerically with the Roll method as well as with its best known competitor, the Hasbrouck (2004) method, which uses a Bayesian Gibbs methodology under a Gaussian assumption. Our estimators are competitive with Roll’s and Hasbrouck’s when the latent true fundamental return distribution is Gaussian, and perform much better when this distribution is far from Gaussian. Our methods are applied to the E-mini futures contract on the S&P 500 during the Flash Crash of May 6, 2010. Extensions to models allowing for unbalanced order flow or Hidden Markov trade direction indicators or trade direction indicators having general asymmetric support or adverse selection are also presented, without requiring additional data.

Transformations to symmetry based on the probability weighted characteristic function

We suggest a nonparametric version of the probability weighted empirical characteristic function (PWECF) introduced by Meintanis et al. [10] and use this PWECF in order to estimate the parameters of arbitrary transformations to symmetry. The almost sure consistency of the resulting estimators is shown. Finite-sample results for i.i.d. data are presented and are subsequently extended to the regression setting. A real data illustration is also included.

Nonparametric probability weighted empirical characteristic function and applications

We study certain properties of probability weighted quantities and suggest a nonparametric estimator of the probability weighted characteristic function introduced by Meintanis et al. (2014). Some properties of this estimator are studied and corresponding inference procedures for symmetry and the two-sample problem are proposed. Monte Carlo results on the finite-sample behavior of the procedures are also included.

Fourier-type estimation of the power GARCH model with stable-Paretian innovations

We consider estimation for general power GARCH models under stable-Paretian innovations. Exploiting the simple structure of the conditional characteristic function of the observations driven by these models we propose minimum distance estimation based on the empirical characteristic function of corresponding residuals. Consistency of the estimators is proved, and the asymptotic distribution of the estimator is studied. Efficiency issues are explored and finite-sample results are presented as well as applications of the proposed procedures to real data from the financial markets. A multivariate extension is also considered.

Jump activity estimation for pure-jump semimartingales via self-normalized statistics

We derive a nonparametric estimator of the jump-activity index $\beta$ of a "locally-stable" pure-jump It\^{o} semimartingale from discrete observations of the process on a fixed time interval with mesh of the observation grid shrinking to zero. The estimator is based on the empirical characteristic function of the increments of the process scaled by local power variations formed from blocks of increments spanning shrinking time intervals preceding the increments to be scaled. The scaling serves two purposes: (1) it controls for the time variation in the jump compensator around zero, and (2) it ensures self-normalization, that is, that the limit of the characteristic function-based estimator converges to a nondegenerate limit which depends only on $\beta$. The proposed estimator leads to nontrivial efficiency gains over existing estimators based on power variations. In the L\'{e}vy case, the asymptotic variance decreases multiple times for higher values of $\beta$. The limiting asymptotic variance of the proposed estimator, unlike that of the existing power variation based estimators, is constant. This leads to further efficiency gains in the case when the characteristics of the semimartingale are stochastic. Finally, in the limiting case of $\beta=2$, which corresponds to jump-diffusion, our estimator of $\beta$ can achieve a faster rate than existing estimators.

Research on Parameter Estimation Methods for Alpha Stable Noise in a Laser Gyroscope's Random Error.

Alpha stable noise, determined by four parameters, has been found in the random error of a laser gyroscope. Accurate estimation of the four parameters is the key process for analyzing the properties of alpha stable noise. Three widely used estimation methods—quantile, empirical characteristic function (ECF) and logarithmic moment method—are analyzed in contrast with Monte Carlo simulation in this paper. The estimation accuracy and the application conditions of all methods, as well as the causes of poor estimation accuracy, are illustrated. Finally, the highest precision method, ECF, is applied to 27 groups of experimental data to estimate the parameters of alpha stable noise in a laser gyroscope’s random error. The cumulative probability density curve of the experimental data fitted by an alpha stable distribution is better than that by a Gaussian distribution, which verifies the existence of alpha stable noise in a laser gyroscope’s random error.

A test for equality of two distributions via jackknife empirical likelihood and characteristic functions

The two-sample problem: testing the equality of two distributions is investigated. A jackknife empirical likelihood (JEL) test is proposed through incorporating characteristic functions, which reduces to a two-sample U-statistic. When the dimension of data is fixed, the nonparametric Wilks’s theorem for the proposed JEL ratio statistics is established. When the dimension diverges with the sample size at a moderate rate, p=o(n1/3), it is proved that under some mild conditions the normalized JEL ratio statistic has a standard normal limit. Moreover, when the dimension exceeds the sample size, p>n, an alternative version of JEL test is proposed. It is verified that under the null hypothesis this alternative version of JEL test has an asymptotical chi-squared distribution with two degrees of freedom. Some numerical results via simulation study and an analysis of a microarray dataset are presented and both confirm theoretical results empirically.

Bootstrap Procedures for Online Monitoring of Changes in Autoregressive Models

We compare the behavior of several bootstrap procedures for monitoring changes in the error distribution of autoregressive time series. The proposed procedures are designed to control the overall significance level and include classical tests based on the empirical distribution function as well as Fourier-type methods that utilize the empirical characteristic function, both functions being computed on the basis of properly estimated residuals. The Monte Carlo study incorporates different estimators and a variety of sampling situations with and without outliers.

Goodness-of-fit tests for multivariate stable distributions based on the empirical characteristic function

We consider goodness-of-fit testing for multivariate stable distributions. The proposed test statistics exploit a characterizing property of the characteristic function of these distributions and are consistent under some conditions. The asymptotic distribution is derived under the null hypothesis as well as under local alternatives. Conditions for an asymptotic null distribution free of parameters and for affine invariance are provided. Computational issues are discussed in detail and simulations show that with proper choice of the user parameters involved, the new tests lead to powerful omnibus procedures for the problem at hand.

Estimation of Affine Jump-Diffusions Using Realized Variance and Bipower Variation in Empirical Characteristic Function Method

Extensions of Empirical Characteristic Function (ECF) method for estimating parameters of affine jump-diffusions with unobserved stochastic volatility (SV) are considered. We develop a new approach based on a bias-corrected ECF for the Realized Variance (in the case of diffusions) and Bipower Variation or second generation jump-robust estimators of integrated stochastic variance (in the case of jumps in the underlying). Effective numerical implementation of Unconditional and Conditional ECF methods through a special configuration of grid points in the frequency domain is proposed. The method is illustrated based on a multifactor jump-diffusion SV model with exponential Poisson jumps in the volatility and underlying correlated by a new ''Gamma-factor copula'' that allows for analytically tractable joint characteristic function. A closed form Lauricella-Kummer-type density is derived for the stationary SV distribution. This distribution extends in a certain way a Generalized Gamma Convolution family of Thorin, and it is proven to be infinitely divisible, but not always self-decomposable. Numerical results for S&P 500 Index, VIX Index and rigorous Monte-Carlo study for a number of SV models are presented.

Fast goodness-of-fit tests based on the characteristic function

A class of goodness-of-fit tests whose test statistic is an L2 norm of the difference of the empirical characteristic function of the sample and a parametric estimate of the characteristic function in the null hypothesis, is considered. The null distribution is usually estimated through a parametric bootstrap. Although very easy to implement, the parametric bootstrap can become very computationally expensive as the sample size, the number of parameters or the dimension of the data increase. It is proposed to approximate the null distribution through a weighted bootstrap. The method is studied both theoretically and numerically. It provides a consistent estimator of the null distribution. In the numerical examples carried out, the estimated type I errors are close to the nominal values. The asymptotic properties are similar to those of the parametric bootstrap but, from a computational point of view, it is more efficient.

Tests for the equality of conditional variance functions in nonparametric regression

In this paper we are interested in checking whether the conditional variances are equal in k ≥ 2 location-scale models. Our procedure is fully nonparametric and is based on the comparison of the error distributions under the null hypothesis of equality of variances and without making use of this null hypothesis. We propose four test statistic based on empirical distribution functions (Kolmogorov-Smirnovand and Cramer-von Mises type test statistics) and two test statistics based on empirical characteristic functions. The limiting distributions of these six test statistics are established under the null hypothesis and under a local alternative. We show how to approximate the critical values under null using either an estimated version of the asymptotic null distribution or a bootstrap procedure. Simulation studies are conducted to assess the finite sample performance of the proposed tests. We also apply our tests to data on monthly expenditure.

An approximation to the null distribution of a class of Cramér–von Mises statistics

A class of goodness-of-fit tests of the Cramér–von Mises type is considered. More specifically, the test statistic of each test is an L2-norm of the difference between the empirical characteristic function associated with a random sample and a parametric estimator of the characteristic function of the population in the null hypothesis. The null distribution of these statistics is unknown and we study a way of estimating it, which is based on approximating the asymptotic null distribution. The asymptotic null distribution is a linear combination of independent chi-squared variates, where the weights are the eigenvalues of certain operator. The calculation of these eigenvalues is, in most cases, a very difficult task. In order to bypass this computation we approximate the test statistic. The asymptotic null distribution of the approximation is again a linear combination of chi-squared variates, but now the weights can be easily approximated. A simulation study is carried out to examine the accuracy of the proposed approximation for finite sample sizes. Although we center our attention on the aforementioned class, the methodology exposed can be applied for approximating the null distribution of other Cramér–von Mises type test statistics.

The complex multinormal distribution, quadratic forms in complex random vectors and an omnibus goodness-of-fit test for the complex normal distribution

This paper first reviews some basic properties of the (noncircular) complex multinormal distribution and presents a few characterizations of it. The distribution of linear combinations of complex normally distributed random vectors is then obtained, as well as the behavior of quadratic forms in complex multinormal random vectors. We look into the problem of estimating the complex parameters of the complex normal distribution and give their asymptotic distribution. We then propose a virtually omnibus goodness-of-fit test for the complex normal distribution with unknown parameters, based on the empirical characteristic function. Monte Carlo simulation results show that our test behaves well against various alternative distributions. The test is then applied to an fMRI data set and we show how it can be used to “validate” the usual hypothesis of normality of the outside-brain signal. An R package that contains the functions to perform the test is available from the authors.

The Split-SV model

A modification of one of the most popular stochastic model in describing financial indexes dynamics is introduced. For describing a nonlinear behavior of volatility, a threshold noise indicator in the autoregressive time series of stochastic volatility is used. Toward this end, the model named the Split-SV model is introduced and its basic stochastic properties are investigated. Furthermore, the Empirical Characteristic Function (ECF) method is used for obtaining the parameter estimations of the model and a numerical simulation of the obtained estimates is given as well. Finally, the Split-SV model is applied for fitting the empirical data: the daily returns of the exchange rates of GBP and USD per euro.

Optimum linear regression in additive Cauchy–Gaussian noise

We study the estimation problem of linear regression in the presence of a new impulsive noise model, which is a sum of Cauchy and Gaussian random variables in time domain. The probability density function (PDF) of this mixture noise, referred to as the Voigt profile, is derived from the convolution of the Cauchy and Gaussian PDFs. To determine the linear regression parameters, the maximum likelihood estimator (MLE) is developed first. Since the Voigt profile suffers from a complicated analytical form, an M-estimator with the pseudo-Voigt function is also derived. In our algorithm development, both scenarios of known and unknown density parameters are considered. For the latter case, we estimate the density parameters by utilizing the empirical characteristic function prior to applying the MLE. Simulation results show that the performance of both proposed methods can attain the Cramér–Rao lower bound.

Reducing the computational cost of the ECF using a nuFFT: A fast and objective probability density estimation method

A nonuniform, fast Fourier transform can be used to reduce the computational cost of the empirical characteristic function (ECF) by a factor of 100. This fast ECF calculation method is applied to a new, objective, and robust method for estimating the probability distribution of univariate data, which effectively modulates and filters the ECF of a dataset in a way that yields an optimal estimate of the (Fourier transformed) underlying distribution. This improvement in computational efficiency is leveraged to estimate probability densities from a large ensemble of atmospheric velocity increments (gradients), with the purpose of characterizing the statistical and fractal properties of the velocity field. It is shown that the distribution of velocity increments depends on location in an atmospheric model and that the increments are clearly not normally distributed. The estimated increment distributions exhibit self-similar and distinctly multifractal behavior, as shown by structure functions that exhibit power-law scaling with a non-linear dependence of the power-law exponent on the structure function order.

Efficient estimation of integrated volatility in presence of infinite variation jumps

We propose new nonparametric estimators of the integrated volatility of an It\^{o} semimartingale observed at discrete times on a fixed time interval with mesh of the observation grid shrinking to zero. The proposed estimators achieve the optimal rate and variance of estimating integrated volatility even in the presence of infinite variation jumps when the latter are stochastic integrals with respect to locally "stable" L\'{e}vy processes, that is, processes whose L\'{e}vy measure around zero behaves like that of a stable process. On a first step, we estimate locally volatility from the empirical characteristic function of the increments of the process over blocks of shrinking length and then we sum these estimates to form initial estimators of the integrated volatility. The estimators contain bias when jumps of infinite variation are present, and on a second step we estimate and remove this bias by using integrated volatility estimators formed from the empirical characteristic function of the high-frequency increments for different values of its argument. The second step debiased estimators achieve efficiency and we derive a feasible central limit theorem for them.

An Empirical Characteristic Function Approach to Selecting a Transformation to Normality

In this paper, we study the problem of transforming to normality. We propose to estimate the transformation parameter by minimizing a weighted squared distance between the empirical characteristic function of transformed data and the characteristic function of the normal distribution. Our approach also allows for other symmetric target characteristic functions. Asymptotics are established for a random sample selected from an unknown distribution. The proofs show that the weight function <TEX>$t^{-2}$</TEX> needs to be modified to have thinner tails. We also propose the method to compute the influence function for M-equation taking the form of U-statistics. The influence function calculations and a small Monte Carlo simulation show that our estimates are less sensitive to a few outliers than the maximum likelihood estimates.