Empirical Characteristic Function Research Articles

Conditional symmetry test based on empirical characteristic function

In this paper, we propose a method for testing the conditional symmetry of multivariate random variables, specifically testing whether or not the conditional distribution of one random variable is symmetric around zero given another. Both the conditional symmetry test for observable data and unobserved error terms in parametric models are considered. The proposed test statistics are built on the weighted L 2 -distance between empirical characteristic functions and require no distributional assumption about the data distribution. The asymptotic properties of the test statistics under the null and alternative hypotheses are investigated. Since the limiting null distributions are intractable, a nonparametric Monte Carlo resampling method is used to determine critical values. Simulation results show the good performance of the proposed method in various scenarios. Two real datasets are analysed to illustrate the practical usefulness of the proposed tests.

Robust tests for equality of regression curves based on characteristic functions

This paper focuses on the problem of testing the null hypothesis that the regression functions of several populations are equal under a general nonparametric homoscedastic regression model. It is well known that linear kernel regression estimators are sensitive to atypical responses. These distorted estimates will influence the test statistic constructed from them so the conclusions obtained when testing equality of several regression functions may also be affected. In recent years, the use of testing procedures based on empirical characteristic functions has shown good practical properties. For that reason, to provide more reliable inferences, we construct a test statistic that combines characteristic functions and residuals obtained from a robust smoother under the null hypothesis. The asymptotic distribution of the test statistic is studied under the null hypothesis and under root − n contiguous alternatives. A Monte Carlo study is performed to compare the finite sample behaviour of the proposed test with the classical one obtained using local averages. The reported numerical experiments show the advantage of the proposed methodology over the one based on Nadaraya–Watson estimators for finite samples. An illustration to a real data set is also provided and enables to investigate the sensitivity of the p − value to the bandwidth selection.

Adaptive directional estimator of the density in [formula omitted] for independent and mixing sequences

A new multivariate density estimator for stationary sequences is obtained by Fourier inversion of the thresholded empirical characteristic function. This estimator does not depend on the choice of parameters related to the smoothness of the density; it is directly adaptive. We establish oracle inequalities valid for independent, α-mixing and τ-mixing sequences, which allows us to derive optimal convergence rates, up to a logarithmic loss. On general anisotropic Sobolev classes, the estimator adapts to the regularity of the unknown density but also achieves directional adaptivity. More precisely, the estimator is able to reach the convergence rate induced by the best Sobolev regularity of the density of AX, where A belongs to a class of invertible matrices describing all the possible directions. The estimator is easy to implement and numerically efficient. It depends on the calibration of a parameter for which we propose an innovative numerical selection procedure, using the Euler characteristic of the thresholded areas.

Estimating spot volatility under infinite variation jumps with dependent market microstructure noise

Summary Jumps and market microstructure noise are stylized features of high-frequency financial data. It is well known that they introduce bias in the estimation of volatility (including integrated and spot volatilities) of assets, and many methods have been proposed to deal with this problem. When the jumps are intensive with infinite variation, the efficient estimation of spot volatility under serially dependent noise is not available and is thus in need. For this purpose, we propose a novel estimator of spot volatility with a hybrid use of the pre-averaging technique and the empirical characteristic function. Under mild assumptions, the results of consistency and asymptotic normality of our estimator are established. Furthermore, we show that our estimator achieves an almost efficient convergence rate with optimal variance when the jumps are either less active or active with symmetric structure. Simulation studies verify our theoretical conclusions. We apply our proposed estimator to empirical analyses, such as estimating the weekly volatility curve using second-by-second transaction price data.

Nonparametric Tests of Independence for Circular Data Based on Trigonometric Moments

We introduce nonparametric tests of independence for bivariate circular data based on trigonometric moments. Our contributions lie in (i) proposing nonparametric tests that are locally and asymptotically optimal against bivariate cosine von Mises alternatives and (ii) extending these tests, via the empirical characteristic function, to obtain consistent tests against broader sets of alternatives, eventually being omnibus. We thus provide a collection of trigonometric-based tests of varying generality and known optimalities. The large-sample behaviours of the tests under the null and alternative hypotheses are obtained, while simulations show that the new tests are competitive against previous proposals. Two data applications in astronomy and forest science illustrate the usage of the tests.

Specification procedures for multivariate stable-Paretian laws for independent and for conditionally heteroskedastic data

We consider goodness-of-fit methods for multivariate symmetric and asymmetric stable Paretian random vectors in arbitrary dimension. The methods are based on the empirical characteristic function and are implemented both in the i.i.d. context as well as for innovations in GARCH models. Asymptotic properties of the proposed procedures are discussed, while the finite-sample properties are illustrated by means of an extensive Monte Carlo study. The procedures are also applied to real data from the financial markets.

Goodness‐of‐fit tests for the multivariate Student‐t distribution based on i.i.d. data, and for GARCH observations

We consider goodness‐of‐fit tests for the multivariate Student's t‐distribution with i.i.d. data and for the innovation distribution in a generalized autoregressive conditional heteroskedasticity model. The methods are based on the empirical characteristic function and are relatively easy to implement, invariant under linear transformations, and globally consistent. Asymptotic properties of the proposed procedures are investigated, while the finite‐sample properties are illustrated by means of a Monte Carlo study. The procedures are also applied to real data from the financial markets.

DOA estimation of coherently and incoherently distributed sources using a characteristic function based ESPRIT algorithm with heavy-tailed signals and noises

Numerous methods exist for estimating the direction of arrival (DOA) of distributed sources in the presence of the Gaussian noises. However, they are inefficient when impulsive signals and noises are present. A novel empirical characteristic function (ECF)-based ESPRIT approach for localizing coherently and incoherently distributed sources is proposed in this study. This method makes use of the interesting properties of characteristic functions like simplicity, information preservation, analytic form, and high precision to build a technique that really is accurate and robust against impulsive signals and noises. The array characteristic function (CF) is modeled using the α-stable distribution to represent the impulsive nature of the signals and sources. This function generates a covariance matrix that can be utilized directly in ESPRIT algorithm. Due to the interesting properties of the ECF, an accurate algorithm is developed that is highly effective in suppressing the effect of impulsive noises, especially when the signal and noise have extremely heavy tails, the signal-to-noise ratio is low, and only a small number of snapshots are available. The results of Monte-Carlo simulation indicate that the suggested method beats existing methods in terms of resolution probability and estimation error. Therefore, this strategy is strongly recommended for localizing coherently and incoherently distributed sources in environments with strong impulses.

Robust Matched Field Processing Using an Empirical Characteristic Function Approach Under Impulsive Noise Environments

Robust Matched Field Processing Using an Empirical Characteristic Function Approach Under Impulsive Noise Environments

On the estimation of the jump activity index in the case of random observation times

We propose a nonparametric estimator of the jump activity index beta of a pure-jump semimartingale X driven by a beta -stable process when the underlying observations are coming from a high-frequency setting at irregular times. The proposed estimator is based on an empirical characteristic function using rescaled increments of X, with a limit that depends in a complicated way on beta and the distribution of the sampling scheme. Utilising an asymptotic expansion we derive a consistent estimator for beta and prove an associated central limit theorem.

On a new class of tests for the Pareto distribution using Fourier methods

We propose new classes of tests for the Pareto type I distribution using the empirical characteristic function. These tests are and statistics based on a characterization of the Pareto distribution involving the distribution of the sample minimum. In addition to deriving simple computational forms for the proposed test statistics, we prove consistency against a wide range of fixed alternatives. A Monte Carlo study is included in which the newly proposed tests are shown to produce high powers. These powers include results relating to fixed alternatives as well as local powers against mixture distributions. The use of the proposed tests is illustrated using an observed data set.

Homogeneity of marginal distributions for a large number of populations

SummaryAssume that a random vector is observed in populations and independent samples of that random vector are available at each population. Assume that and have the same dimension. Our purpose is to test the equality of the marginal distributions of and in the populations when is large compared to the sample sizes. With this aim, we propose and study a test statistic that compares the empirical characteristic functions of the marginal distributions. Under the null, the test statistic is asymptotically free‐distributed. An expression of the asymptotic power is also derived, which allows to study the consistency of the test. No assumption is made on the distribution of and , which can be continuous, discrete or mixed; moreover, no assumption is made about moments. A simulation study investigates the finite sample performance of the new test. The proposal is applied to study air pollution levels that are directly related to environmental health, in all countries where observations are available.

Bahadur efficiency for certain goodness-of-fit tests based on the empirical characteristic function

We study the Bahadur efficiency of several weighted L2-type goodness-of-fit tests based on the empirical characteristic function. The methods considered are for normality and exponentiality testing, and for testing goodness-of-fit to the logistic distribution. Our results are helpful in deciding which specific test a potential practitioner should apply. For the celebrated BHEP and energy tests for normality we obtain novel efficiency results, with some of them in the multivariate case, while in the case of the logistic distribution this is the first time that efficiencies are computed for any composite goodness-of-fit test.

Multivariate [formula omitted]-stable distributions: VAR(1) processes, measures of dependence and their estimations

Measuring the dependency between lagged random vectors in time series is extremely important to detect appropriate models for fitting real data. Vector autoregressive processes (VAR), in particular, are among the most popular multivariate time series that allow modeling schemes for two or more random variables. However, if the series of innovations process does not have finite second-order moments, the classical analysis that uses the empirical auto-covariance function is not adequate as its theoretical counterpart is not defined. After perceiving the lack of a suitable multivariate version of the codifference function, we propose an adaptation of this tool to measure interdependence in multivariate α-stable processes. We establish such a function based on notions presented in the existing literature. We also show some of its properties and give an estimator based on the empirical characteristic function. Subsequently, we evaluate another measure of dependence with different properties but similar usage for comparison reasons. These two measures are computed for VAR(1) processes with α-stable innovations, proving to be useful in the identification of associated patterns related to them. Finally, we demonstrate how these measures can be applied with real data sets by analyzing the monthly number of patients with community-acquired pneumonia and the registered values of PM10 in the city of São Paulo, Brazil, from January 2008 to December 2021.

Intraday cross-sectional distributions of systematic risk

We develop a test for the detection of intraday changes in the cross-sectional distribution of assets’ exposure to observable factors. The test is constructed for a panel of high-frequency asset returns, with the size of the cross-section and the sampling frequency increasing simultaneously. It is based on a comparison of the empirical characteristic functions of estimates of the assets’ factor loadings at different parts of the trading day, formed from local blocks of asset returns and the corresponding factor realizations. The limiting behavior of the test statistic is governed by unobservable latent factors in the asset prices. The critical values of the test are constructed on the basis of a novel simulation-based procedure. Empirical implementation of the test to stocks in the S&P 500 index and the five Fama–French factors, as well as the momentum factor, reveals different intraday behavior of the factor loadings: assets’ exposure to size, market and value risks vary systematically over the trading day while the three remaining factors do not exhibit statistically significant intraday variation. Moreover, we find diverse, and for some factors large, reactions in the assets’ factor loadings to major economic or firm specific news releases. Finally, we document that time-varying correlations between the observable risk factors drive a wedge between the time-of-day pattern of market betas, estimated with and without control for the other observable risk factors.

TESTING FOR STRICT STATIONARITY VIA THE DISCRETE FOURIER TRANSFORM

This paper proposes a model-free test for the strict stationarity of a potentially vector-valued time series using the discrete Fourier transform (DFT) approach. We show that the DFT of a residual process based on the empirical characteristic function weakly converges to a zero spectrum in the frequency domain for a strictly stationary time series and a nonzero spectrum otherwise. The proposed test is powerful against various types of nonstationarity including deterministic trends and smooth or abrupt structural changes. It does not require smoothed nonparametric estimation and, thus, can detect the Pitman sequence of local alternatives at the parametric rate $T^{-1/2}$ , faster than all existing nonparametric tests. We also design a class of derivative tests based on the characteristic function to test the stationarity in various moments. Monte Carlo studies demonstrate that our test has reasonarble size and excellent power. Our empirical application of exchange rates strongly suggests that both nominal and real exchange rate returns are nonstationary, which the augmented Dickey–Fuller and Kwiatkowski–Phillips–Schmidt–Shin tests may overlook.

A test for normality and independence based on characteristic function

In this article we prove a generalization of the Ejsmont characterization (Ejsmont in Stat Probab Lett 114:1–5, 2016) of the multivariate normal distribution. Based on it, we propose a new test for independence and normality. The test uses an integral of the squared modulus of the difference between the product of empirical characteristic functions and some constant. Special attention is given to the case of testing for univariate normality in which we derive the test statistic explicitly in terms of Bessel function and explore asymptotic properties. The simulation study also includes the cases of testing for bivariate and trivariate normality and independence, as well as multivariate normality. We show the quality performance of our test in comparison to some popular powerful competitors. The practical application of the proposed normality and independence test is discussed and illustrated using a real dataset.

Performance estimation when the distribution of inefficiency is unknown

We show how to compute inefficiency or performance scores when the distribution of the one-sided error component in Stochastic Frontier Models (SFMs) is unknown; and we do the same with Data Envelopment Analysis (DEA). Our procedure, which is based on the Fast Fourier Transform (FFT), utilizes the empirical characteristic function of the residuals in SFMs or efficiency scores in DEA. The new techniques perform well in Monte Carlo experiments and deliver reasonable results in an empirical application to large U.S. banks. In both cases, deconvolution of DEA scores with the FFT brings the results much closer to the inefficiency estimates from SFM.

ON MULTIPLE STRUCTURAL BREAKS IN DISTRIBUTION: AN EMPIRICAL CHARACTERISTIC FUNCTION APPROACH

We estimate and test for multiple structural breaks in distribution via an empirical characteristic function approach. By minimizing the sum of squared generalized residuals, we can consistently estimate the break fractions. We propose a sup-F type test for structural breaks in distribution as well as an information criterion and a sequential testing procedure to determine the number of breaks. We further construct a class of derivative tests to gauge possible sources of structural breaks, which is asymptotically more powerful than the smoothed nonparametric tests for structural breaks. Simulation studies show that our method performs well in determining the appropriate number of breaks and estimating the unknown breaks. Furthermore, the proposed tests have reasonable size and excellent power in finite samples. In an application to exchange rate returns, our tests are able to detect structural breaks in distribution and locate the break dates. Our tests also indicate that the documented breaks appear to occur in variance and higher-order moments, but not so often in mean.

ECF-MUSIC: An empirical characteristic function based direction of arrival (DOA) estimation in the presence of impulsive noise

DOA estimation in the presence of impulsive signals and noises is a challenging issue. Conventional and fractional lower order based methods do not work properly in the presence of very heavy-tailed noise. Interesting properties of empirical characteristic function (ECF) such as great simplicity and high accuracy leads to develop new methods, which are presented in this paper. It is analytically shown that by using the characteristic function of the array signal, a new sample covariance matrix can be extracted to use in algorithms like MUSIC. Numerical methods and the optimum set of parameters are also presented to estimate this matrix from the ECF of the array signal. Monte-Carlo simulation results are also presented which demonstrate the effectiveness and improved performance of the algorithms in highly impulsive environments. It is particularly effective in comparison to available methods in terms of probability of resolution and estimation error. The results indicate that the proposed method has a superior performance under the condition of heavy-tailed noise.