An Empirical Characteristic Function Approach to Selecting a Transformation to Normality

We consider goodness-of-fit tests of the Cauchy distribution based on weighted integrals of the squared distance between the empirical characteristic function of the standardized data and the characteristic function of the standard Cauchy distribution. For standardization of data Gurtler and Henze (2000,Annals of the Institute of Statistical Mathematics,52, 267–286) used the median and the interquartile range. In this paper we use the maximum likelihood estimator (MLE) and an equivariant integrated squared error estimator (EISE), which minimizes the weighted integral. We derive an explicit form of the asymptotic covariance function of the characteristic function process with parameters estimated by the MLE or the EISE. The eigenvalues of the covariance function are numerically evaluated and the asymptotic distributions of the test statistics are obtained by the residue theorem. A simulation study shows that the proposed tests compare well to tests proposed by Gurtler and Henze and more traditional tests based on the empirical distribution function.

This paper examines the estimation of the Stochastic Conditional Duration model by the empirical characteristic function and the generalized method of moments when maximum likelihood is unavailable. The joint characteristic function for the durations along with general expressions for the moments are derived, leading naturally to estimation via the empirical characteristic function and generalized method of moments. In a Monte Carlo study as well as an empirical application, these alternative methods are compared with quasi maximum likelihood. These experiments reveal that the empirical characteristic function approach outperforms the quasi maximum likelihood and generalized method of moments in terms of both bias and root mean square error. Copyright The Author(s). Journal compilation Royal Economic Society 2008

What Jump Process to Use to Model S&P 500 Returns?

A stochastic model for intermittent fluctuations in the scrape-off layer of magnetically confined plasmas has been constructed based on a super-position of uncorrelated pulses arriving according to a Poisson process. In the most common applications of the model, the pulse amplitudes are assumed exponentially distributed, supported by conditional averaging of large-amplitude fluctuations in experimental measurement data. This basic assumption has two potential limitations. First, statistical analysis of measurement data using conditional averaging only reveals the tail of the amplitude distribution to be exponentially distributed. Second, exponentially distributed amplitudes leads to a positive definite signal which cannot capture fluctuations in for example electric potential and radial velocity. Assuming pulse amplitudes which are not positive definite often make finding a closed form for the probability density function (PDF) difficult, even if the characteristic function remains relatively simple. Thus estimating model parameters requires an approach based on the characteristic function, not the PDF. In this contribution, the effect of changing the amplitude distribution on the moments, PDF and characteristic function of the process is investigated and a parameter estimation method using the empirical characteristic function is presented and tested on synthetically generated data. This proves valuable for describing intermittent fluctuations of all plasma parameters in the boundary region of magnetized plasmas.

SUMMARY An omnibus test of normality is proposed, which has high power against many alternative hypotheses. The test uses a weighted integral of the squared modulus of the difference between the characteristic functions of the sample and of the normal distribution. together with the one-to-one correspondence between +(t) and F(x), suggest utilizing kn(t) in statistical inference. Here we use /n(t) to test the composite hypothesis that F(x) is normal. Let 00(t) = exp (it1 - 1t2 a2) be the characteristic function under the null hypothesis, with 1u and U2 the unspecified mean and variance. The test introduced here is based on a weighted integral over t of the squared modulus of kn(t) - $O(t), where $0(t) depends on sample estimates of ju and a2. We compare the power of the test with that of prominent tests based on order statistics and on sample moments. Heathcote (1972) and Feigin & Heathcote (1977) have previously considered using either the real or imaginary part alone of 4n(t) in tests of simple hypotheses. Their tests relied on the fact that, for given t, either component of 4n(t) is asymptotically normal with mean given by its population counterpart. If the alternative hypothesis is also simple, it may be possible to find a value of t that maximizes the power of the test in large samples. Epps, Singleton & Pulley (1982) showed how the sample moment generating function may be employed to test composite hypotheses that the data come from one or the other of two separate families of distributions. Murota & Takeuchi (1981) have recently proposed a location and scale invariant test based on the statistic -an(t) = I n(t/S) 12, where S is the sample standard deviation. Applied to the problem of testing normality, an(t) was found to have high power in the vicinity of t = 10 against members of six families of symmetric alternatives, but its

The purpose of this paper is to show that the empirical characteristic function, when suitably normalised, converges weakly to a stationary Gaussian process whose autocovariance function is the theoretical characteristic function.

Extended Abstract. Suppose n i.i.d. observations, X1, …, Xn, are available from the unknown distribution F(.), goodness-of-fit tests refer to tests such as

A new technique is presented for the characterization of impulsive interference with /spl alpha/-stable processes. The proposed model is constructed by estimating the /spl alpha/-stable distribution parameters using the empirical characteristic function. Characteristic function based methods have been shown to yield the best performance but are plagued by the amount of computation required. The new parameter estimation algorithm is shown to achieve the same level of performance as other characteristic function based methods while greatly reducing the amount of computation required.

We propose an estimator for the mean of random variables in separable real Banach spaces using the empirical characteristic function. Assuming that the covariance operator of the random variable is bounded in a precise sense, we show that the proposed estimator achieves the optimal sub-Gaussian rate, except for a faster decaying mean-dependent term. Under a mild condition, an iterative refinement procedure can essentially eliminate the mean-dependent term and provide a shift-equivariant estimate. Our results particularly suggests that a certain Gaussian width that appears in the best known rate in the literature might not be necessary. Furthermore, using the boundedness of the characteristic functions, we also show that, except possibly at high signal-to-noise ratios, the proposed estimator is gracefully robust against adversarial "contamination". Our analysis is overall concise and transparent, thanks to the tractability of the characteristic functions.

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

This paper has to do with a Cramer-von Mises test for symmetry of the error distribution in a class of absolutely regular and non-necessarily stationary heteroscedastic models. The test statistic is based on the empirical characteristic function. Its convergence, as well as that of the residual-based empirical distribution function are established. From these results, the null cumulative distribution function of the test statistic is approximated. A simulation experiment shows that the test performs well on the examples tested.

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.

SUMMARY The iterative method for the parameter estimation of stable laws proposed by Koutrouvelis (1981) is modified to handle the case where the distribution is assumed to belong to a Cauchy family with location and/or scale parameter unknown. It is shown that the determination of optimum values at which the empirical characteristic function needs to be evaluated reduces in this case to the determination of the asymptotically optimum quantiles for the parameter estimation of an exponential distribution by linear functions of order statistics. The estimators of the scale and location parameters are asymptotically independent and normally distributed, and achieve asymptotic efficiencies close to unity. The proposed procedure is compared to a method of estimation using linear functions of order statistics.

The estimation of characteristic exponent /spl alpha/ is the most important step for the parameter estimation of symmetric /spl alpha/ stable distribution. For lack of closed form of probability density function (pdf), classical methods are no longer available. In this paper improved algorithms based on the empirical characteristic function are proposed to solve this problem under the conditions of short data samples. Simulations show that the proposed algorithms are robust and they are more efficient and applicable than the conventional ones.

Tests for the multivariate k-sample problem are considered. The tests are based on the weighted L2 distance between empirical characteristic functions, and afford an interesting interpretation in terms of a corresponding test statistic based on the L2 distance of pairs of non-parametric density estimators. Depending on the choice of weighting, a corresponding Dirac-type weight function reduces the test to a normalised version of the L2 distance between the sample means of the k populations. Theoretical and computational issues are considered, while the finite-sample implementation based on the permutation distribution of the test statistic shows that the new test performs well in comparison with alternative procedures of the change-point type.