empirical characteristic function

Let $(X\_1,...,X\_n)$ be independent, identically distributed real-valued random variables with common characteristic function $\backslash varphi(t)$. The empirical characteristic function (ECF) defined as

:$$

\varphi_{n}(t)= \frac{1}{n} \sum_{j=1}^{n} e^{i tX_j},

is an unbiased and consistent estimator of the corresponding population characteristic function $\backslash varphi(t)$, for each $t\backslash in\backslash mathbb\; R$. The ECF apparently made its debut in page 342 of the classical textbook of Cramér (1946),Cramér H (1946) Mathematical Methods of Statistics. Princeton University Press, Princeton, New Jersey and then as part of the auxiliary tools for density estimation in Parzen (1962).Parzen E (1962) On estimation of a probability density function and mode. Annals of Mathematical Statistics. 33:1065–1076 Nearly a decade later the ECF features as the main object of research in two separate lines of application: In Press (1972)Press SJ (1972) Estimation in univariate and multivariate stable distributions. Journal of the American Statistical Association. 67:842–846 for parameter estimation and in Heathcote (1972)Heathcote CR (1972) A test for goodness of fit for symmetric random variables. Australian Journal of Statistics. 14:172-181 for goodness-of-fit testing.

Since that time there has subsequently been a vast expansion of statistical inference methods based on the ECF. For reviews of estimation methods based on the ECF the reader is referred to Csörgő (1984a),Csörgő S (1984a) Adaptive estimation of the parameters of stable laws. In P. Revesz (ed) Colloquia Mathematica Societatis Janos Bolyai 36. Limit Theorems in Probability and Statistics. North-Holland, Amsterdam: pp. 305-368 Rémillard and Theodorescu (2001),Rémillard B, Theodorescu R (2001) Estimation based on the empirical characteristic function. In: Balakrishnan, Ibragimov and Nevzorov (eds) Asymptotic Methods in Probability and Statistics with Applications. Birkhäuser, Boston: pp 435-449 Yu (2004),Yu J (2004) Empirical characteristic function estimation and its applications. Econometric Reviews. 23:93-123 and Carrasco and Kotchoni (2017),Carrasco M, Kotchoni R (2017) Efficient estimation using the characteristic function. Econometric Theory. 33:479-526 while testing procedures are surveyed by Csörgő (1984b),Csörgő S (1984b) Testing by the empirical characteristic function: A survey. In P Mandl, M Hušková (eds) Asymptotic Statistics. Elsevier, Amsterdam: pp. 45-56 Hušková and Meintanis (2008a),Hušková M, Meintanis SG (2008a) Testing procedures based on the empirical characteristic function I: Goodness-of-fit, testing for symmetry and independence. Tatra Mountains Mathematical Publications. 39:225-233 Hušková and Meintanis (2008b),Hušková M, Meintanis SG (2008b) Testing procedures based on the empirical characteristic function II: k-sample problem, change-point problem. Tatra Mountains Mathematical Publications. 39:235-243 and Meintanis (2016).Meintanis SG (2016) A review of testing procedures based on the empirical characteristic function (with discussion and rejoinder). South African Statistical Journal. 50:1-41 Ushakov (1999)Ushakov N (1999) Selected Topics in Characteristic Functions. VSP, Utrecht. and Prakasa Rao (1987)Prakasa Rao BLS (1987) Asymptotic Theory of Statistical Inference. Wiley, New York. (chapter 8) are also good sources of information on the limit properties of the ECF process, as well as on estimation and goodness-of-fit testing via the ECF.

One of the lines of research that deserves special mention is ECF testing for independence by means of distance correlation as originally suggested by Székely et al. (2007).Székely GJ, Rizzo M, Bakirov NK (2007) Measuring and testing independence by correlation of distances. The Annals of Statistics. 35 (6): 2769–2794 This approach has become extremely popular and is currently under vigorous development. We refer to Edelmann et al. (2019)Edelmann D, Fokianos K, Pitsillou M (2019) An updated literature review of distance correlation and its applications to time series. International Statistical Review. 87:237-262 and

Székely and Rizzo (2023)Székely GJ, Rizzo M (2023) The Energy of Data and Distance Correlation. CRC Press. Florida

for a recent survey on distance correlation methods.

Another popular line of research is goodness-of-fit testing for multivariate distributions, with special emphasis on testing for multivariate normality; for more information on this topic the reader is referred to the review by Ebner and Henze (2020). Ebner, B, Henze N (2020) Tests for multivariate normality - a critical review with emphasis on weighted L2-statistics. TEST 29: 845–892. This approach of ECF-based goodness-of-fit testing has been generalized by Chen et al. (2022) Chen, F, Jiménez-Gamero, MD, Meintanis, SG, Zhu, LX (2022) A general Monte Carlo method for multivariate goodness–of–fit testing applied to elliptical families. Computational Statistics and Data Analysis. 175: 107548. to testing fit for arbitrary elliptical distributions by means of modification of the two-sample test initially suggested by Meintanis (2005). Meintanis, SG (2005) Permutation tests for homogeneity based on the empirical characteristic function. Journal of Nonparametric Statistics. 17: 583-592.

A recent account of the basic ECF-based goodness-of-fit methods for testing symmetry, homogeneity and independence may be found in Chen et al. (2019). Chen F, Meintanis SG, Zhu, LX (2019) On some characterizations and multidimensional criteria for

testing homogeneity, symmetry and independence. Journal of Multivariate Analysis. 173: 125-144