Chesneau, Christophe Wavelet estimation of a density in a GARCH-type model. (English) Zbl 1298.62061 Commun. Stat., Theory Methods 42, No. 1, 98-117 (2013). Summary: We consider the GARCH-type model: \(S=\sigma^2Z\), where \(\sigma^2\) and \(Z\) are independent random variables. The density of \(\sigma^2\) is unknown whereas the one of \(Z\) is known. We want to estimate the density of \(\sigma^2\) from \(n\) observations of \(S\) under some dependence assumption (the exponentially strongly mixing dependence). Adopting the wavelet methodology, we construct a nonadaptive estimator based on projections and an adaptive estimator based on the hard thresholding rule. Taking the mean integrated squared error over Besov balls, we prove that the adaptive one attains a sharp rate of convergence. Cited in 10 Documents MSC: 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) Keywords:adaptive density estimation; GARCH models; hard thresholding; rate of convergence; wavelets PDFBibTeX XMLCite \textit{C. Chesneau}, Commun. Stat., Theory Methods 42, No. 1, 98--117 (2013; Zbl 1298.62061) Full Text: DOI HAL References: [1] Abbaszadeh , M. , Doosti , H. , Gachpazan , M. ( 2010 ). Density estimation by wavelets under multiplicative censoring.PreprintFerdowsi University . [2] Andersen K., J. Statist. Plann. and Infer. 98 pp 137– (2001) · Zbl 0977.62037 · doi:10.1016/S0378-3758(00)00237-8 [3] Butucea C., Bernoulli 11 (2) pp 309– (2005) · Zbl 1063.62044 · doi:10.3150/bj/1116340297 [4] Cai T., Ann. Statist. 27 pp 898– (1999) · Zbl 0954.62047 · doi:10.1214/aos/1018031262 [5] Cai T., J. Statist. Plann. Infer. 108 pp 329– (2002) · Zbl 1016.62025 · doi:10.1016/S0378-3758(02)00316-6 [6] Carrasco M., Econometric Theor. 18 pp 17– (2002) · Zbl 1181.62125 · doi:10.1017/S0266466602181023 [7] Caroll R. J., J. Amer. Statist. Assoc. 83 pp 1184– (1988) · doi:10.1080/01621459.1988.10478718 [8] Chaubey Y. P., J. Ind. Soc. Agricult. Statist. 65 pp 169– (2011) [9] Chaubey , Y. P. , Chesneau , C. , Doosti , H. ( 2010 ).Adaptive Wavelet Estimation of a Density from Mixtures Under Multiplicative Censoring.Preprint, Concordia University . · Zbl 1367.62102 [10] Chesneau C., Appl. Computat. Harmonic Anal. 28 (1) pp 67– (2010) · Zbl 1180.94005 · doi:10.1016/j.acha.2009.07.003 [11] Cohen A., Appl. Computat. Harmonic Anal. 24 (1) pp 54– (1993) · Zbl 0795.42018 · doi:10.1006/acha.1993.1005 [12] Comte F., Can. J. Statist. 34 pp 431– (2006) · Zbl 1104.62033 · doi:10.1002/cjs.5550340305 [13] Comte F., Econometric Theor. 24 (6) pp 1628– (2008) · Zbl 1277.62103 · doi:10.1017/S026646660808064X [14] Davydov Y., Theor. Probab. Appl. 15 (3) pp 498– (1970) [15] Delaigle A., Statistica Sinica 16 pp 773– (2006) [16] Delyon B., Appl. Computat. Harmonic Anal. 3 pp 215– (1996) · Zbl 0865.62023 · doi:10.1006/acha.1996.0017 [17] Devroye L., Can. J. Statist. 17 pp 235– (1989) · Zbl 0679.62029 · doi:10.2307/3314852 [18] Donoho D. L., Ann. Statist. 24 pp 508– (1996) · Zbl 0860.62032 · doi:10.1214/aos/1032894451 [19] Doukhan P., Mixing. Properties and Examples (1994) [20] Fan J., Ann. Statist. 19 pp 1257– (1991) · Zbl 0729.62033 · doi:10.1214/aos/1176348248 [21] Fan J., IEEE Trans. Inform. Theory 48 pp 734– (2002) · Zbl 1071.94511 · doi:10.1109/18.986021 [22] Härdle W., Wavelet, Approximation and Statistical Applications (1998) · doi:10.1007/978-1-4612-2222-4 [23] Lacour C., C. Roy. Acad. Sci. Paris Ser. I Math. 342 (11) pp 877– (2006) · Zbl 1095.62056 · doi:10.1016/j.crma.2006.04.006 [24] Masry E., IEEE Infor. Theor. 37 pp 1105– (1991) · Zbl 0732.60045 · doi:10.1109/18.87002 [25] Meyer Y., Wavelets and Operators (1992) [26] Modha D., IEEE Trans. Inform. Theor. 42 pp 2133– (1996) · Zbl 0868.62015 · doi:10.1109/18.556602 [27] Pensky M., Ann. Statist. 27 pp 2033– (1999) · Zbl 0962.62030 · doi:10.1214/aos/1017939249 [28] Van Zanten H., Statist. Infer. Stoch. Process. 11 pp 207– (2008) · Zbl 1204.62051 · doi:10.1007/s11203-007-9013-0 [29] Vardi Y., Biometrika 76 pp 751– (1989) · Zbl 0678.62051 · doi:10.1093/biomet/76.4.751 [30] Vardi Y., Ann. Statist. 20 pp 1022– (1992) · Zbl 0761.62056 · doi:10.1214/aos/1176348668 [31] Withers C. S., Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 57 pp 477– (1981) · Zbl 0465.60032 · doi:10.1007/BF01025869 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.