×

Nonparametric density estimation for symmetric distributions by contaminated data. (English) Zbl 1241.62045

Summary: A semiparametric two-component mixture model is considered, in which the distribution of one (primary) component is unknown and assumed to be symmetric. The distribution of the other component (admixture) is known. We consider three estimates for the pdf of the primary component: a naive one, a symmetrized naive estimate and a symmetrized estimate with adaptive weights. Asymptotic behavior and small sample performance of the estimates are investigated. Some rules of thumb for bandwidth selection are discussed.

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
65C60 Computational problems in statistics (MSC2010)

Software:

KernSmooth; mixdist
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bordes L, Delmas C, Vandekerkhove P (2006) Semiparametric estimation of a two-component mixture model where one component is known. Scand J Stat 33: 733–752 · Zbl 1164.62331 · doi:10.1111/j.1467-9469.2006.00515.x
[2] Borovkov AA (1998) Mathematical statistics. Gordon and Breach Science Publishers, Amsterdam
[3] Bowman AW (1984) An alternative method of cross-validation for the smoothing of density estimates. Biometrika 71: 353–360 · doi:10.1093/biomet/71.2.353
[4] Chacón JE, Montanero J, Nogales AG (2008) Bootstrap bandwidth selection using an h-dependent pilot bandwidth. Scand J Stat 35: 139–157 · Zbl 1164.62006 · doi:10.1111/j.1467-9469.2007.00565.x
[5] Devroye L, Györfi L, Lugosi G (1996) A probabilistic theory of pattern recognition. Springer, New York · Zbl 0853.68150
[6] Devroye L (1997) Universal smoothing factor selection in density estimation: theory and practice (with discussion). Test 6: 223–320 · Zbl 0949.62026 · doi:10.1007/BF02564701
[7] Efron B, Tibshirani R, Storey JD, Tusher V (2001) Empirical Bayes analysis of a microarray experiment. J Am Stat Assoc 96: 1151–1160 · Zbl 1073.62511 · doi:10.1198/016214501753382129
[8] Hall P (1990) Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems. J Multivar Anal 32: 177–203 · Zbl 0722.62030 · doi:10.1016/0047-259X(90)90080-2
[9] Hall P (1992) Effect of bias estimation on coverage accuracy of bootstrap confidence intervals for a probability density. Ann Stat 20: 675–694 · Zbl 0748.62028 · doi:10.1214/aos/1176348651
[10] Hall P, Marron JS, Park BU (1992) Smoothed cross-validation. Probab Theor Relat Fields 92: 1–20 · Zbl 0742.62042 · doi:10.1007/BF01205233
[11] Hardle W, Muller M, Sperlich S, Werwatz A (2004) Nonparametric and semiparametric models. Springer, Berlin
[12] Hedenfalk I, Duggan D, Chen YD, Radmacher M, Bittner M, Simon R, Meltzer P, Gusterson B, Esteller M, Kallioniemi OP et al (2001) Gene-expression profiles in hereditary breast cancer. N Engl J Med 344: 539–548 · doi:10.1056/NEJM200102223440801
[13] Ho YHS, Lee SMS (2008) Iterated bootstrap-t confidence intervals for density functions. Scand J Stat 35: 295–308 · Zbl 1164.62009 · doi:10.1111/j.1467-9469.2007.00577.x
[14] Ibragimov IA, Khasminsky RZ (1979) Asymptotic estimation theory. Nauka, Moscow (in Russian)
[15] Kraft CH, Lepage Y, van Eeden C (1985) Estimation of a symmetric density function. Commun Stat A Theor Methods 14: 273–288 · Zbl 0565.62025 · doi:10.1080/03610928508828911
[16] Liu K, Tsokos CP (2001) Kernel estimates of symmetric density function. Int J Appl Math 6: 23–34 · Zbl 1023.62044
[17] Macdonald P, Du J (2008) mixdist: finite mixture distribution models. R package version 0.5-2. http://www.math.mcmaster.ca/peter/mix/mix.html
[18] Maiboroda R, Sugakova O (2010) Generalized estimating equations for symmetric distributions observed with admixture. Commun Stat Theor Method (to appear) · Zbl 1208.62059
[19] Meloche J (1991) Estimation of a symmetric density. Canad J Stat 19: 151–164 · Zbl 04502817 · doi:10.2307/3315794.o
[20] Pearson K (1894) Contribution to the mathematical theory of evolution. Phil Trans Roy Soc A 185: 71–110 · JFM 25.0347.02 · doi:10.1098/rsta.1894.0003
[21] Robin S, Bar-Hen A, Daudin J, Pierre L (2007) A semi-parametric approach for mixture models: application to local false discovery rate estimation. Comput Statist Data Anal 51: 5483–5493 · Zbl 1445.62075 · doi:10.1016/j.csda.2007.02.028
[22] Rudemo M (1982) Empirical choice of histograms and kernel density estimators. Scand J Stat 9: 65–78 · Zbl 0501.62028
[23] Shao J (1998) Mathematical statistics. Springer, New York
[24] Silverman BW (1986) Density estimation for statistics and data analysis. Chapman & Hall, London
[25] Storey JD, Tibshirani R (2003) Statistical significance for genome-wide studies. Proc Natl Acad Sci 100: 9440–9445 · Zbl 1130.62385 · doi:10.1073/pnas.1530509100
[26] Sugakova O (2009) Estimation of location parameter by observations with admixture. Teorija Imovirnosti ta Matematychna Statystyka 80: 81–91
[27] Sugakova O (2010) Density estimation by observations with admixture. Theory of Stochastic Processes (to appear) · Zbl 1223.62036
[28] Wand MP, Jones MC (1995) Kernel smoothing. Chapman & Hall, London
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.