Generalized Birnbaum-Saunders kernel density estimators and an analysis of financial data

The kernel method is a nonparametric procedure used to estimate densities with support in R. When nonnegative data are modeled, the classical kernel density estimator presents a bias problem in the neighborhood of zero. Several methods have been developed to reduce this bias, which include the boundary kernel, data transformation and reflection methods. An alternative proposal is to use kernel estimators based on distributions with nonnegative support, as is the case of the Birnbaum-Saunders (BS), gamma, inverse Gaussian and lognormal models. Generalized BS (GBS) distributions have received considerable attention, due to their properties and their flexibility in modeling different types of data. In this paper, we propose, characterize and implement the kernel method based on GBS distributions to estimate densities with nonnegative support. In addition, we provide a simple method to choose the corresponding bandwidth. In order to evaluate the performance of these new estimators, we conduct a Monte Carlo simulation study. The obtained results are illustrated by analyzing financial real data.

[1]  J. Johannes,et al.  Nonparametric estimation for dependent data , 2011 .

[2]  H. Müller,et al.  Hazard rate estimation under random censoring with varying kernels and bandwidths. , 1994, Biometrics.

[3]  Olivier Scaillet,et al.  Density estimation using inverse and reciprocal inverse Gaussian kernels , 2004 .

[4]  D. W. Scott,et al.  Biased and Unbiased Cross-Validation in Density Estimation , 1987 .

[5]  Peter Hall,et al.  On Pseudodata Methods for Removing Boundary Effects in Kernel Density Estimation , 1996 .

[6]  M. C. Jones,et al.  An Improved Estimator of the Density Function at the Boundary , 1999 .

[7]  Mia Hubert,et al.  An adjusted boxplot for skewed distributions , 2008, Comput. Stat. Data Anal..

[8]  N. Klutchnikoff,et al.  Minimax properties of beta kernel estimators , 2011 .

[9]  Narayanaswamy Balakrishnan,et al.  Estimation of extreme percentiles in Birnbaum-Saunders distributions , 2011, Comput. Stat. Data Anal..

[10]  A. Ullah,et al.  Nonparametric Econometrics , 1999 .

[11]  A. Goldenshluger,et al.  Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality , 2010, 1009.1016.

[12]  Jianqing Fan,et al.  On automatic boundary corrections , 1997 .

[13]  Robert F. Engle,et al.  The Econometrics of Ultra-High Frequency Data , 1996 .

[14]  Boundary Aware Estimators of Integrated Density Derivative Products , 1997 .

[15]  Taoufik Bouezmarni,et al.  Nonparametric Density Estimation for Positive Time Series , 2006, Comput. Stat. Data Anal..

[16]  Xiaodong Jin,et al.  Birnbaum-Saunders and Lognormal Kernel Estimators for Modelling Durations in High Frequency Financial Data , 2003 .

[17]  M. Lejeune,et al.  Smooth estimators of distribution and density functions , 1992 .

[18]  Narayanaswamy Balakrishnan,et al.  Shape and change point analyses of the Birnbaum-Saunders-t hazard rate and associated estimation , 2012, Comput. Stat. Data Anal..

[19]  H. Müller,et al.  Kernels for Nonparametric Curve Estimation , 1985 .

[20]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[21]  G. Kerkyacharian,et al.  Nonlinear estimation in anisotropic multi-index denoising , 2001 .

[22]  José A. Díaz-García,et al.  A new family of life distributions based on the elliptically contoured distributions , 2005 .

[23]  A. Ullah,et al.  Nonparametric Econometrics: Semiparametric and Nonparametric Estimation of Simultaneous Equation Models , 1999 .

[24]  Ruey S. Tsay,et al.  A nonlinear autoregressive conditional duration model with applications to financial transaction data , 2001 .

[25]  Chad R. Bhatti,et al.  The Birnbaum-Saunders autoregressive conditional duration model , 2010, Math. Comput. Simul..

[26]  Gilberto A. Paula,et al.  An R implementation for generalized Birnbaum-Saunders distributions , 2009, Comput. Stat. Data Anal..

[27]  M. Rosenblatt.  Remarks on Some Nonparametric Estimates of a Density Function , 1956 .

[28]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .

[29]  Hans-Georg Müller,et al.  Smooth optimum kernel estimators near endpoints , 1991 .

[30]  James Stephen Marron,et al.  Transformations in Density Estimation , 1991 .

[31]  J. D. Wilson,et al.  A smoothed EM approach to indirect estimation problems, with particular reference to stereology and emission tomography , 1990 .

[32]  Gilberto A. Paula,et al.  Influence diagnostics in log-Birnbaum-Saunders regression models with censored data , 2007, Comput. Stat. Data Anal..

[33]  Narayanaswamy Balakrishnan,et al.  Lifetime analysis based on the generalized Birnbaum-Saunders distribution , 2008, Comput. Stat. Data Anal..

[34]  B. Park,et al.  New methods for bias correction at endpoints and boundaries , 2002 .

[35]  Víctor Leiva,et al.  Nuevas cartas de control basadas en la distribución Birnbaum-Saunders y su implementación , 2011 .

[36]  Peter Hall,et al.  A Geometrical Method for Removing Edge Effects from Kernel-Type Nonparametric Regression Estimators , 1991 .

[37]  Fabienne Comte,et al.  Convolution power kernels for density estimation , 2012 .

[38]  Marcelo Fernandes,et al.  Central limit theorem for asymmetric kernel functionals , 2005 .

[39]  Eugene F. Schuster,et al.  Incorporating support constraints into nonparametric estimators of densities , 1985 .

[40]  Daren B. H. Cline,et al.  Kernel Estimation of Densities with Discontinuities or Discontinuous Derivatives , 1991 .

[41]  K. Abadir,et al.  Optimal Asymmetric Kernels , 2004 .

[42]  Rohana J. Karunamuni,et al.  A locally adaptive transformation method of boundary correction in kernel density estimation , 2006 .

[43]  Young K. Truong,et al.  On bandwidth choice for density estimation with dependent data , 1995 .

[44]  M. C. Jones,et al.  Simple boundary correction for kernel density estimation , 1993 .

[45]  Rohana J. Karunamuni,et al.  On kernel density estimation near endpoints , 1998 .

[46]  Narayanaswamy Balakrishnan,et al.  The Generalized Birnbaum–Saunders Distribution and Its Theory, Methodology, and Application , 2008 .

[47]  Thomas H. McInish,et al.  An Analysis of Intraday Patterns in Bid/Ask Spreads for NYSE Stocks , 1992 .

[48]  Alexandre B. Tsybakov,et al.  Introduction to Nonparametric Estimation , 2008, Springer series in statistics.

[49]  Song-xi Chen,et al.  Probability Density Function Estimation Using Gamma Kernels , 2000 .

[50]  Gilberto A. Paula,et al.  Robust statistical modeling using the Birnbaum-Saunders- t distribution applied to insurance , 2012 .

[51]  Rohana J. Karunamuni,et al.  On boundary correction in kernel density estimation , 2005 .

[52]  I. N. Volodin,et al.  On limit distributions emerging in the generalized Birnbaum-Saunders model , 2000 .

[53]  R. Berk,et al.  Continuous Univariate Distributions, Volume 2 , 1995 .

[54]  P. Diggle.  A Kernel Method for Smoothing Point Process Data , 1985 .

[55]  Thomas H. McInish,et al.  HOURLY RETURNS, VOLUME, TRADE SIZE, AND NUMBER OF TRADES , 1991 .

[56]  W. Härdle.  Applied Nonparametric Regression , 1991 .

[57]  Song-xi Chen,et al.  Beta kernel estimators for density functions , 1999 .

[58]  Z. Birnbaum,et al.  A new family of life distributions , 1969 .

[59]  J. Kiefer,et al.  Asymptotic Minimax Character of the Sample Distribution Function and of the Classical Multinomial Estimator , 1956 .

[60]  Rohana J. Karunamuni,et al.  On nonparametric density estimation at the boundary , 2000 .

[61]  E. Parzen.  On Estimation of a Probability Density Function and Mode , 1962 .