A nonparametric method for estimating asymmetric densities based on skewed Birnbaum–Saunders distributions applied to environmental data

In this paper, we introduce a new nonparametric kernel method for estimating asymmetric densities based on generalized skew-Birnbaum–Saunders distributions. Kernels based on these distributions have the advantage of providing flexibility in the asymmetry and kurtosis levels. In addition, the generalized skew-Birnbaum–Saunders kernel density estimators are boundary bias free and achieve the optimal rate of convergence for the mean integrated squared error of the nonnegative asymmetric kernel estimators. We carry out a data analysis consisting of two parts. First, we conduct a Monte Carlo simulation study for evaluating the performance of the proposed method. Second, we use this method for estimating the density of three real air pollutant concentration data sets. These numerical results favor the proposed nonparametric estimators.

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

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

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

[4]  S. Nadarajah,et al.  Skewed distributions generated by the normal kernel , 2003 .

[5]  Gilberto A. Paula,et al.  Generalized Birnbaum‐Saunders distributions applied to air pollutant concentration , 2008 .

[6]  Víctor Leiva,et al.  On an extreme value version of the Birnbaum-Saunders distribution , 2012 .

[7]  Víctor Leiva,et al.  Fatigue Statistical Distributions Useful for Modeling Diameter and Mortality of Trees , 2012 .

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

[9]  Víctor Leiva,et al.  Modeling wind energy flux by a Birnbaum–Saunders distribution with an unknown shift parameter , 2011 .

[10]  Heleno Bolfarine,et al.  Skew‐symmetric distributions generated by the distribution function of the normal distribution , 2007 .

[11]  P. Haan,et al.  On the use of density kernels for concentration estimations within particle and puff dispersion models , 1999 .

[12]  G. Pagnini.  The kernel method to compute the intensity of segregation for reactive pollutants: Mathematical formulation , 2009 .

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

[14]  T. Chang,et al.  Comparison of a new kernel method and a sampling volume method for estimating indoor particulate matter concentration with Lagrangian modeling , 2012 .

[15]  M. Genton,et al.  The multivariate skew-slash distribution , 2006 .

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

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

[18]  A. Azzalini,et al.  Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution , 2003, 0911.2342.

[19]  Saralees Nadarajah,et al.  A truncated inverted beta distribution with application to air pollution data , 2008 .

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

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

[22]  José M. Angulo,et al.  A length-biased version of the Birnbaum-Saunders distribution with application in water quality , 2009 .

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

[24]  C. Loader.  Bandwidth selection: classical or plug-in? , 1999 .

[25]  George Christakos,et al.  An extended Birnbaum–Saunders model and its application in the study of environmental quality in Santiago, Chile , 2010 .

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

[27]  José A. Díaz-García,et al.  Erratum to “A new family of life distributions based on the elliptically contoured distributions”: [J. Statist. Plann. Inference 128(2) (2005) 445–457] , 2007 .

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

[29]  Magne Vollan Aarset,et al.  How to Identify a Bathtub Hazard Rate , 1987, IEEE Transactions on Reliability.

[30]  Filidor Vilca-Labra,et al.  A New Fatigue Life Model Based on the Family of Skew-Elliptical Distributions , 2006 .

[31]  A. Bowman.  An alternative method of cross-validation for the smoothing of density estimates , 1984 .

[32]  Graeme S. Lorimer.  The kernel method for air quality modelling—I. Mathematical foundation , 1986 .

[33]  A. Azzalini.  A class of distributions which includes the normal ones , 1985 .

[34]  Víctor Leiva,et al.  Air contaminant statistical distributions with application to PM10 in Santiago, Chile. , 2013, Reviews of environmental contamination and toxicology.

[35]  M. Rudemo.  Empirical Choice of Histograms and Kernel Density Estimators , 1982 .

[36]  Helton Saulo,et al.  Generalized Birnbaum-Saunders kernel density estimators and an analysis of financial data , 2013, Comput. Stat. Data Anal..

[37]  A. Azzalini,et al.  Statistical applications of the multivariate skew normal distribution , 2009, 0911.2093.

[38]  Narayanaswamy Balakrishnan,et al.  Mixture inverse Gaussian distributions and its transformations, moments and applications , 2009 .

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

[40]  Narayanaswamy Balakrishnan,et al.  A Skewed Sinh-Normal Distribution and Its Properties and Application to Air Pollution , 2010 .

[41]  R. Beaver,et al.  The skew-Cauchy distribution , 2000 .