Estimación de la densidad utilizando el cuantil bootstrap varianza y covarianza media cuantil
Palabras clave:
estimación de densidades, varianza de cuantiles, covarianza entre media y cuantiles, bootstrap
Resumen
Evaluamos dos estimadores de densidades basados en la varianza y la covarianza entre media y varianza estimados por bootstrap. Revisamos otros desarrollos de estimadores de densidad relacionados con cuantiles. Las simulaciones de Monte Carlo para distintos procesos generadores de datos, tamaños de muestra, y otros parámetros muestran que los estimadores tienen buena performance en comparación con el estimador no paramétrico de kernel. Algunas de las técnicas de suavizamiento tienen menor error cuadrático medio integrado y sesgo, lo que indica que los estimadores propuestos son una estrategia promisoria.
Citas
Alejo, J., Bera, A., Montes-Rojas, G., Galvao, A., and Xiao, Z. (2016). Tests for normality based on the quantile-mean covariance. Stata Journal, 16(4):1039– 1057.
Alin, A., Martin, M. A., Beyaztas, U., and Pathak, P. K. (2017). Sufficient mout-of-n (m/n) bootstrap. Journal of Statistical Computation and Simulation, 87(9):1742–1753.
Babu, G. J. (1986). A note on bootstrapping the variance of sample quantile. Annals of the Institute of Statistical Mathematics, 38(3):439–443.
Bera, A. K., Galvao, A. F., and Wang, L. (2014). On testing the equality of mean and quantile effects. Journal of Econometric Methods, 3(1).
Bera, A. K., Galvao, A. F., Wang, L., and Xiao, Z. (2016). A new characterization of the normal distribution and test for normality. Econometric Theory, 32(5):12161252.
Cheng, C. (1998). A Berry-Esseen-type theorem of quantile density estimators. Statistics & Probability Letters, 39:255–262.
Chesneau, C., Dewan, I., and Doosti, H. (2016). Nonparametric estimation of a quantile density function by wavelet methods. Computational Statistics & Data Analysis, 94(C):161–174.
Cheung, K. Y. and Lee, S. M. S. (2005). Variance estimation for sample quantiles using the m out of n bootstrap. Annals of the Institute of Statistical Mathematics, 57(2):279–290.
Csörgő, M., Horváth, L., and Deheuvels, P. (1991). Estimating the QuantileDensity Function, pages 213–223. Springer Netherlands, Dordrecht.
David, H. A. and Nagaraja, H. (2003). Order Statistics. Wiley Series in Probability and Statistics. Wiley, New Jersey, third edition. doi:10.1002/0471722162.
Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist., 7(1):1–26.
Falk, M. (1986). ON THE ESTIMATION OF THE QUANTILE DENSITY FUNCTION. Statistics & Probability Letters, 4(March):69–73.
Ferguson, T. S. (1999). Asymptotic joint distribution of sample mean and a sample quantile. UCLA Unpublished Manuscript, pages 1–5. https://www.math.ucla. edu/~tom/papers/unpublished/meanmed.pdf.
Galton, F. (1889). Natural Inheritance. Macmillan, London and New York.
Gilchrist, W. (1980). Approximation Theorems of Mathematical Statistics. Wiley Series in Probability and Statistics. Wiley & Sons, Inc., New Jersey, first edition.
Hald, A. A. (1998). A history of mathematical statistics from 1750 to 1930. New York : Wiley. ”A Wiley-Interscience publication.”.
Hall, P. and Martin, M. A. (1988). Exact convergence rate of bootstrap quantile variance estimator. Probability Theory and Related Fields, 80(2):261–268.
Ho, Y. H. S. and Lee, S. M. S. (2005). Iterated smoothed bootstrap confidence intervals for population quantiles. Ann. Statist., 33(1):437–462.
Hodrick, R. and Prescott, E. (1997). Postwar u.s. business cycles: An empirical investigation. Journal of Money, Credit and Banking, 29(1):1–16.
Huang, J. (1991). Estimating the variance of the sample median, discrete case. Statistics & Probability Letters, 11(4):291 – 298.
Hutson, A. D. and Ernst, M. D. (2000). The Exact Bootstrap Mean and Variance of an L-Estimator. Journal of the Royal Statistical Society, 62:89–94.
Janas, D. (1993). A smoothed bootstrap estimator for a studentized sample quantile. Annals of the Institute of Statistical Mathematics, 45(2):317–329.
Jones, M. C. (1992). Estimating densities, quantiles, quantile densities and density quantiles. Annals of the Institute of Statistical Mathematics, 44(4):721–727.
Koenker, R. (1994). Confidence Intervals for Regression Quantiles. In Mandl, P. and Husková, M., editors, Asymptotic Statistics. Contributions to Statistics. Physica, Heidelberg.
Koenker, R. W. and Bassett, G. (1978). Regression quantiles. Econometrica, 46(1):33–50.
Maritz, J. S. and Jarrett, R. G. (1978). A Note on Estimating the Variance of the Sample Median. Journal of the American Statistical Association, 73(361):194– 196.
Miller, R. G. (1974). The jackknife–a review. Biometrika, 61(1):1–15.
Mnatsakanov, R. M. and Sborshchikovi, A. (2018). Recovery of quantile and quantile density function using the frequency moments. Statistics and Probability Letters, 10(xxxx):1–10.
Moore, D. S. (1969). Limiting distributions for sample quantiles. American Mathematical Monthly, 76(8):927–929.
Mosteller, F. et al. (1946). On some useful ”inefficient” statistics. Annals of Mathematical Statistics, 17(4):377–408.
Pagan, A. and Ullah, A. (1999). Nonparametric Econometrics. Cambridge University Press, New York, first edition.
Parzen, E. (1979). Nonparametric Statistical Data Modeling. Journal of the American Statistical Association, 74(365):105–121.
Parzen, E. (2004). Quantile Probability and Statistical Data Modeling. Statistical Science, 19(4):652–662.
Pyke, R. (1965). Spacings. Journal of the Royal Statistical Society, 27(3):395–449.
Rao, C., Pathak, P., and Koltchinskii, V. (1997). Bootstrap by sequential resampling. Journal of Statistical Planning and Inference, 64(2):257 – 281.
Saadi, N., Adjabi, S., and Djerroud, L. (2019). On the estimation of the quantile density function by orthogonal series. Communications in Statistics - Theory and Methods, pages 1–25.
Sheater, S. J. (1986). A finite sample estimate of the variance of the sample median. Statistics & Probability Letters, 4:337–342.
Siddiqui, M. M. (1960). Distribution of quantiles in samples from a bivariate population. Journal of Research of the National Institute of Standards and Technology, 64(3):145–151.
Soni, P., Dewan, I., and Jain, K. (2012). Nonparametric estimation of quantile density function. Computational Statistics and Data Analysis, 56(12):3876– 3886.
Stigler, S. M. (1973). Studies in the history of probability and statistics. xxxii: Laplace, fisher and the discovery of the concept of sufficiency. Biometrika, 60(3):439–445.
Tukey, J. W. (1965). Whic part of the sample contains the information? Proceedings of the National Academy of Sciences of the United States of America, 53(1):127–134.
Wang, D., Miecznikowski, J. C., and Hutson, A. D. (2012). Direct density estimation of L -estimates via characteristic functions with applications. Journal of Statistical Planning and Inference, 142(2):567–578.
Alin, A., Martin, M. A., Beyaztas, U., and Pathak, P. K. (2017). Sufficient mout-of-n (m/n) bootstrap. Journal of Statistical Computation and Simulation, 87(9):1742–1753.
Babu, G. J. (1986). A note on bootstrapping the variance of sample quantile. Annals of the Institute of Statistical Mathematics, 38(3):439–443.
Bera, A. K., Galvao, A. F., and Wang, L. (2014). On testing the equality of mean and quantile effects. Journal of Econometric Methods, 3(1).
Bera, A. K., Galvao, A. F., Wang, L., and Xiao, Z. (2016). A new characterization of the normal distribution and test for normality. Econometric Theory, 32(5):12161252.
Cheng, C. (1998). A Berry-Esseen-type theorem of quantile density estimators. Statistics & Probability Letters, 39:255–262.
Chesneau, C., Dewan, I., and Doosti, H. (2016). Nonparametric estimation of a quantile density function by wavelet methods. Computational Statistics & Data Analysis, 94(C):161–174.
Cheung, K. Y. and Lee, S. M. S. (2005). Variance estimation for sample quantiles using the m out of n bootstrap. Annals of the Institute of Statistical Mathematics, 57(2):279–290.
Csörgő, M., Horváth, L., and Deheuvels, P. (1991). Estimating the QuantileDensity Function, pages 213–223. Springer Netherlands, Dordrecht.
David, H. A. and Nagaraja, H. (2003). Order Statistics. Wiley Series in Probability and Statistics. Wiley, New Jersey, third edition. doi:10.1002/0471722162.
Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist., 7(1):1–26.
Falk, M. (1986). ON THE ESTIMATION OF THE QUANTILE DENSITY FUNCTION. Statistics & Probability Letters, 4(March):69–73.
Ferguson, T. S. (1999). Asymptotic joint distribution of sample mean and a sample quantile. UCLA Unpublished Manuscript, pages 1–5. https://www.math.ucla. edu/~tom/papers/unpublished/meanmed.pdf.
Galton, F. (1889). Natural Inheritance. Macmillan, London and New York.
Gilchrist, W. (1980). Approximation Theorems of Mathematical Statistics. Wiley Series in Probability and Statistics. Wiley & Sons, Inc., New Jersey, first edition.
Hald, A. A. (1998). A history of mathematical statistics from 1750 to 1930. New York : Wiley. ”A Wiley-Interscience publication.”.
Hall, P. and Martin, M. A. (1988). Exact convergence rate of bootstrap quantile variance estimator. Probability Theory and Related Fields, 80(2):261–268.
Ho, Y. H. S. and Lee, S. M. S. (2005). Iterated smoothed bootstrap confidence intervals for population quantiles. Ann. Statist., 33(1):437–462.
Hodrick, R. and Prescott, E. (1997). Postwar u.s. business cycles: An empirical investigation. Journal of Money, Credit and Banking, 29(1):1–16.
Huang, J. (1991). Estimating the variance of the sample median, discrete case. Statistics & Probability Letters, 11(4):291 – 298.
Hutson, A. D. and Ernst, M. D. (2000). The Exact Bootstrap Mean and Variance of an L-Estimator. Journal of the Royal Statistical Society, 62:89–94.
Janas, D. (1993). A smoothed bootstrap estimator for a studentized sample quantile. Annals of the Institute of Statistical Mathematics, 45(2):317–329.
Jones, M. C. (1992). Estimating densities, quantiles, quantile densities and density quantiles. Annals of the Institute of Statistical Mathematics, 44(4):721–727.
Koenker, R. (1994). Confidence Intervals for Regression Quantiles. In Mandl, P. and Husková, M., editors, Asymptotic Statistics. Contributions to Statistics. Physica, Heidelberg.
Koenker, R. W. and Bassett, G. (1978). Regression quantiles. Econometrica, 46(1):33–50.
Maritz, J. S. and Jarrett, R. G. (1978). A Note on Estimating the Variance of the Sample Median. Journal of the American Statistical Association, 73(361):194– 196.
Miller, R. G. (1974). The jackknife–a review. Biometrika, 61(1):1–15.
Mnatsakanov, R. M. and Sborshchikovi, A. (2018). Recovery of quantile and quantile density function using the frequency moments. Statistics and Probability Letters, 10(xxxx):1–10.
Moore, D. S. (1969). Limiting distributions for sample quantiles. American Mathematical Monthly, 76(8):927–929.
Mosteller, F. et al. (1946). On some useful ”inefficient” statistics. Annals of Mathematical Statistics, 17(4):377–408.
Pagan, A. and Ullah, A. (1999). Nonparametric Econometrics. Cambridge University Press, New York, first edition.
Parzen, E. (1979). Nonparametric Statistical Data Modeling. Journal of the American Statistical Association, 74(365):105–121.
Parzen, E. (2004). Quantile Probability and Statistical Data Modeling. Statistical Science, 19(4):652–662.
Pyke, R. (1965). Spacings. Journal of the Royal Statistical Society, 27(3):395–449.
Rao, C., Pathak, P., and Koltchinskii, V. (1997). Bootstrap by sequential resampling. Journal of Statistical Planning and Inference, 64(2):257 – 281.
Saadi, N., Adjabi, S., and Djerroud, L. (2019). On the estimation of the quantile density function by orthogonal series. Communications in Statistics - Theory and Methods, pages 1–25.
Sheater, S. J. (1986). A finite sample estimate of the variance of the sample median. Statistics & Probability Letters, 4:337–342.
Siddiqui, M. M. (1960). Distribution of quantiles in samples from a bivariate population. Journal of Research of the National Institute of Standards and Technology, 64(3):145–151.
Soni, P., Dewan, I., and Jain, K. (2012). Nonparametric estimation of quantile density function. Computational Statistics and Data Analysis, 56(12):3876– 3886.
Stigler, S. M. (1973). Studies in the history of probability and statistics. xxxii: Laplace, fisher and the discovery of the concept of sufficiency. Biometrika, 60(3):439–445.
Tukey, J. W. (1965). Whic part of the sample contains the information? Proceedings of the National Academy of Sciences of the United States of America, 53(1):127–134.
Wang, D., Miecznikowski, J. C., and Hutson, A. D. (2012). Direct density estimation of L -estimates via characteristic functions with applications. Journal of Statistical Planning and Inference, 142(2):567–578.
Publicado
2022-11-28
Cómo citar
Montes Rojas, G., & Mena, A. (2022). Estimación de la densidad utilizando el cuantil bootstrap varianza y covarianza media cuantil. Documentos De Trabajo Del Instituto Interdisciplinario De Economía Política, (50), 28. Recuperado a partir de https://ojs.econ.uba.ar/index.php/DT-IIEP/article/view/2449
Sección
Documento de trabajo
