Nonparametric Kernel Density Estimation Near the Boundary

Standard fixed symmetric kernel-type density estimators are known to encounter problems for positive random variables with a large probability mass close to zero. It is shown that, in such settings, alternatives of asymmetric gamma kernel estimators are superior, but also differ in asymptotic and finite sample performance conditionally on the shape of the density near zero and the exact form of the chosen kernel. Therefore, a refined version of the gamma kernel with an additional tuning parameter adjusted according to the shape of the density close to the boundary is suggested. A data-driven method for the appropriate choice of the modified gamma kernel estimator is also provided. An extensive simulation study compares the performance of this refined estimator to those of standard gamma kernel estimates and standard boundary corrected and adjusted fixed kernels. It is found that the finite sample performance of the proposed new estimator is superior in all settings. Two empirical applications based on high-frequency stock trading volumes and realized volatility forecasts demonstrate the usefulness of the proposed methodology in practice.

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

[2]  A. Engelen,et al.  A Strategy Perspective on the Performance Relevance of the CFO , 2012 .

[3]  Francis X. Diebold,et al.  Parametric and Nonparametric Measurements of Volatility: Volume 1: Tools and Techniques , 2010 .

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

[5]  Nikolaus Hautsch,et al.  Modelling Irregularly Spaced Financial Data: Theory and Practice of Dynamic Duration Models , 2004 .

[6]  George Kapetanios,et al.  An Automatic Leading Indicator of Economic Activity: Forecasting GDP Growth for European Countries , 1999 .

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

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

[9]  Fabrizio Cipollini,et al.  Intra-Daily Volume Modeling and Prediction for Algorithmic Trading , 2010 .

[10]  R. C. Merton,et al.  AN INTERTEMPORAL CAPITAL ASSET PRICING MODEL , 1973 .

[11]  R. Engle.  New Frontiers for Arch Models , 2002 .

[12]  Ralph E. Steuer,et al.  Is socially responsible investing just screening? : Evidence from mutual funds , 2012 .

[13]  Ralf Sabiwalsky.  Does Basel II Pillar 3 Risk Exposure Data Help to Identify Risky Banks? , 2012 .

[14]  Z. Q. John Lu,et al.  Nonparametric Functional Data Analysis: Theory And Practice , 2007, Technometrics.

[15]  Matthias Ritter,et al.  Forecast based Pricing of Weather Derivatives , 2015 .

[16]  Jakob Söhl,et al.  Confidence sets in nonparametric calibration of exponential Lévy models , 2012, Finance Stochastics.

[17]  Thorsten Dickhaus,et al.  Simultaneous Statistical Inference in Dynamic Factor Models , 2012 .

[18]  Enzo Weber,et al.  The Signal of Volatility , 2012 .

[19]  Wolfgang Härdle,et al.  Copula dynamics in CDOs , 2014 .

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

[21]  Hanna Wielandt,et al.  The Polarization of Employment in German Local Labor Markets , 2012 .

[22]  B. Werker,et al.  Semiparametric Duration Models , 2004 .

[23]  P. Hansen,et al.  Realized Variance and Market Microstructure Noise , 2005 .

[24]  Ulf Brüggemann,et al.  Intended and Unintended Consequences of Mandatory IFRS Adoption: A Review of Extant Evidence and Suggestions for Future Research , 2012 .

[25]  C. Hafner,et al.  Volatility of price indices for heterogeneous goods , 2012 .

[26]  Franziska Lottmann.  Explaining regional unemployment differences in Germany: a spatial panel data analysis , 2012 .

[27]  Dieter Nautz,et al.  Correlated Trades and Herd Behavior in the Stock Market , 2012 .

[28]  N. Shephard,et al.  Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise , 2006 .

[29]  Mammen Enno,et al.  Generated Covariates in Nonparametric Estimation: A Short Review. In: Recent Developments in Modeling and Applications in Statistics , 2013 .

[30]  Stefan Sperlich,et al.  Simple and effective boundary correction for kernel densities and regression with an application to the world income and Engel curve estimation , 2010, Comput. Stat. Data Anal..

[31]  Wolfgang Karl Härdle,et al.  HMM in Dynamic HAC Models , 2012 .

[32]  M. Schienle,et al.  Nonparametric Kernel Density Estimation Near the Boundary , 2013 .

[33]  M. C. Jones,et al.  A SIMPLE NONNEGATIVE BOUNDARY CORRECTION METHOD FOR KERNEL DENSITY ESTIMATION , 1996 .

[34]  O. Scaillet,et al.  Local Multiplicative Bias Correction for Asymmetric Kernel Density Estimators , 2003 .

[35]  Joachim Gassen,et al.  Comparability Effects of Mandatory IFRS Adoption , 2012 .

[36]  Marc Henry,et al.  Higher-Order Kernel Semiparametric M-Estimation of Long Memory , 2002 .

[37]  J. Marron,et al.  Transformations to reduce boundary bias in kernel density estimation , 1994 .

[38]  Wolfgang Härdle,et al.  Dynamic activity analysis model-based win-win development forecasting under environment regulations in China , 2014, Comput. Stat..

[39]  Ruihong Huang,et al.  On the Dark Side of the Market: Identifying and Analyzing Hidden Order Placements , 2012 .

[40]  W. Härdle,et al.  Quantile Regression in Risk Calibration , 2012 .

[41]  Joachim Grammig,et al.  Nonparametric specification tests for conditional duration models , 2005 .

[42]  Jürgen Symanzik,et al.  Computational Statistics (Journal) , 2012 .

[43]  Philip B. Whyman,et al.  The Impact of the Euro , 2000 .

[44]  Chitru S. Fernando,et al.  Managerial Overconfidence and Corporate Risk Management , 2012 .

[45]  Sebastian Braun,et al.  Implementing Quotas in University Admissions: An Experimental Analysis , 2011, Games Econ. Behav..

[46]  Fulvio Corsi,et al.  A Simple Approximate Long-Memory Model of Realized Volatility , 2008 .

[47]  U. Horst,et al.  Hidden Liquidity: Determinants and Impact , 2012 .

[48]  Shunpu Zhang.  A note on the performance of the gamma kernel estimators at the boundary , 2010 .

[49]  Ostap Okhrin,et al.  Hierarchical Archimedean Copulae: The HAC Package , 2012 .

[50]  T. Dickhaus,et al.  Multiple point hypothesis test problems and effective numbers of tests , 2012 .

[51]  R. Nickl,et al.  A Donsker Theorem for Lévy Measures , 2012 .

[52]  Masayuki Hirukawa,et al.  Nonparametric multiplicative bias correction for kernel-type density estimation on the unit interval , 2010, Comput. Stat. Data Anal..

[53]  P. Michels.  Asymmetric kernel functions in non-parametric regression analysis and prediction , 1992 .

[54]  N. Shephard,et al.  Econometric analysis of realized volatility and its use in estimating stochastic volatility models , 2002 .

[55]  Predictive density estimators for daily volatility based on the use of realized measures , 2009 .

[56]  Neil Shephard,et al.  Designing Realised Kernels to Measure the Ex-Post Variation of Equity Prices in the Presence of Noise , 2008 .

[57]  R. Engle,et al.  A Multiple Indicators Model for Volatility Using Intra-Daily Data , 2003 .

[58]  Taras Bodnar,et al.  Copula-Based Dynamic Conditional Correlation Multiplicative Error Processes , 2012 .

[59]  O. Scaillet,et al.  CONSISTENCY OF ASYMMETRIC KERNEL DENSITY ESTIMATORS AND SMOOTHED HISTOGRAMS WITH APPLICATION TO INCOME DATA , 2005, Econometric Theory.

[60]  Frédéric Ferraty,et al.  Nonparametric Functional Data Analysis: Theory and Practice (Springer Series in Statistics) , 2006 .

[61]  Zografia Anastasiadou,et al.  Statistical Modelling of Temperature Risk , 2012 .

[62]  Alexandra Spitz-Oener,et al.  Bye Bye, G.I. - The Impact of the U.S. Military Drawdown on Local German Labor Markets , 2012 .

[63]  Peter N. C. Mohr,et al.  The Aging Investor: Insights from Neuroeconomics , 2012 .

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

[65]  N. Shephard,et al.  Realised Kernels in Practice: Trades and Quotes , 2008 .

[66]  E. Mammen,et al.  Additive Models: Extensions and Related Models. , 2012 .

[67]  P. Vieu,et al.  SPECIFICATION TEST FOR CONDITIONAL DISTRIBUTION WITH FUNCTIONAL DATA , 2011, Econometric Theory.

[68]  Mathias Trabs,et al.  Option calibration of exponential Lévy models: Implementation and empirical results , 2012 .

[69]  Dedy Dwi Prastyo,et al.  Support Vector Machines with Evolutionary Feature Selection for Default Prediction , 2012 .

[70]  Lars Winkelmann,et al.  Assessing the Anchoring of Inflation Expectations , 2015 .

[71]  Ulf Brüggemann,et al.  Fair Value Reclassifications of Financial Assets During the Financial Crisis , 2014 .

[72]  Hong Lan,et al.  Existence and Uniqueness of Perturbation Solutions to DSGE Models , 2012 .

[73]  Jesus M. Salas,et al.  Why Do Firms Engage in Selective Hedging , 2012 .

[74]  W. Härdle,et al.  Local Adaptive Multiplicative Error Models for High- Frequency Forecasts , 2012 .

[75]  Johanna Kappus.  Nonparametric adaptive estimation of linear functionals for low frequency observed L evy processes , 2012 .

[76]  Nikolaus Hautsch,et al.  Capturing the Zero: A New Class of Zero-Augmented Distributions and Multiplicative Error Processes , 2012 .

[77]  N. Hjort,et al.  Nonparametric Density Estimation with a Parametric Start , 1995 .

[78]  Mathias Trabs,et al.  A uniform central limit theorem and efficiency for deconvolution estimators , 2012, 1208.0687.

[79]  L. Bauwens,et al.  The Logarithmic Acd Model: An Application to the Bid-Ask Quote Process of Three NYSE Stocks , 2000 .

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

[81]  Rainer Schulz,et al.  Location, location, location: Extracting location value from house prices , 2012 .

[82]  Rainer Schulz,et al.  A Slab in the Face: Building Quality and Neighborhood Effects , 2012 .

[83]  Wenjuan Chen,et al.  Do Japanese Stock Prices Reflect Macro Fundamentals , 2012 .

[84]  H. Müller,et al.  Kernel estimation of regression functions , 1979 .

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

[86]  Taoufik Bouezmarni,et al.  Nonparametric density estimation for multivariate bounded data , 2007 .

[87]  D. Hamermesh,et al.  Total work and gender: facts and possible explanations , 2012 .

[88]  Tony Lancaster,et al.  The Econometric Analysis of Transition Data. , 1992 .

[89]  F. Diebold,et al.  The Distribution of Realized Exchange Rate Volatility , 2000 .

[90]  P. Schotman,et al.  Price Discovery in Fragmented Markets , 2003 .

[91]  N. Shephard,et al.  Realized Kernels in Practice: Trades and Quotes , 2009 .

[92]  P. Vieu,et al.  Analysis of Time of Occurrence of Earthquakes: A Functional Data Approach , 2011 .