Empirical characteristic function

Let be independent, identically distributed real-valued random variables with common characteristic function . The empirical characteristic function (ECF) defined as

is an unbiased and consistent estimator of the corresponding population characteristic function , for each . The ECF apparently made its debut in page 342 of the classical textbook of Cramér (1946),[1] and then as part of the auxiliary tools for density estimation in Parzen (1962).[2] Nearly a decade later the ECF features as the main object of research in two separate lines of application: In Press (1972)[3] for parameter estimation and in Heathcote (1972)[4] 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),[5] Rémillard and Theodorescu (2001),[6] Yu (2004),[7] and Carrasco and Kotchoni (2017),[8] while testing procedures are surveyed by Csörgő (1984b),[9] Hušková and Meintanis (2008a),[10] Hušková and Meintanis (2008b),[11] and Meintanis (2016).[12] Ushakov (1999)[13] is also a good source of information regarding the limit properties of the ECF process, as well as on estimation and goodness-of-fit testing via the ECF. A line of research that deserves special mention is ECF testing for independence by means of distance correlation as originally suggested by Székely et al. (2007).[14] This approach has become extremely popular and is currently under vigorous development. We refer to Edelmann et al. (2019)[15] for a recent survey on distance correlation methods.

References

  1. Cramér H (1946) Mathematical Methods of Statistics. Princeton University Press, Princeton, New Jersey
  2. Parzen E (1962) On estimation of a probability density function and mode. Annals of Mathematical Statistics. 33:1065–1076
  3. Press SJ (1972) Estimation in univariate and multivariate stable distributions. Journal of the American Statistical Association. 67:842–846
  4. Heathcote CR (1972) A test for goodness of fit for symmetric random variables. Australian Journal of Statistics. 14:172-181
  5. 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
  6. 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
  7. Yu J (2004) Empirical characteristic function estimation and its applications. Econometric Reviews. 23:93-123
  8. Carrasco M, Kotchoni R (2017) Efficient estimation using the characteristic function. Econometric Theory. 33:479-526
  9. Csörgő S (1984b) Testing by the empirical characteristic function: A survey. In P Mandl, M Hušková (eds) Asymptotic Statistics. Elsevier, Amsterdam: pp. 45-56
  10. 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
  11. 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
  12. 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
  13. Ushakov N (1999) Selected Topics in Characteristic Functions. VSP, Utrecht.
  14. Székely GJ, Rizzo M, Bakirov NK (2007) Measuring and testing independence by correlation of distances. The Annals of Statistics. 35 (6): 2769–2794
  15. 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
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