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1 Kernel smoothed CDF estimation with akdensity Philippe Van Kerm CEPS/INSTEAD, Luxembourg Abstract This note describes formulas for estimation of kernel smoothed CDF in the Stata user-written package akdensity, with usage example. July 200 Keywords akdensity ; Stata ; smoothed CDF Introduction akdensity is a user-written Stata command for univariate density estimation using adaptive kernel estimation methods (Van Kerm, 2003). The command is freely available online for installation in net-aware Stata. At the Stata command prompt, type or ssc install akdensity net install st0037_2, from( A new functionality has been added in a software update in July 200 (version 4.0). A new option cdf(newvarname) is now available to supplement estimation of the density function with estimation of the kernel smoothed cumulative distribution function of the data. Both the PDF and CDF function estimates produced are based on identical (adaptive) bandwidth and kernel function specifications set by the other command options. This note describes syntax, formulas and usage examples for this new option which was not available at the time of publication of Van Kerm (2003). 2 Methods and formulas The adaptive kernel density estimate as computed by akdensity is given by ˆf(x) = n i= w i n ( ) w i x xi K i= where the x i s are the data points (associated with sample weights w i ), K is a kernel function, and = h λ i where s a global bandwidth parameter and λ i is a bandwidth adaptation factor proportional to the square root of the density of the data at each sample point (Van Kerm, 2003). Centre d Etudes de Populations, de Pauvreté et de Politiques Socio-Economiques/International Networks for Studies in Technology, Environment, Alternatives, Development. B.P. 48, L-450 Differdange, Luxembourg. philippe.vankerm@ceps.lu. The latest version of the akdensity package is 4.0 (of ). Stata 7.0 or later is required.

2 The corresponding kernel smoothed CDF estimate is given by ˆF(x) = x where IK is the integral of the kernel function K: = ˆf(v)dv n i= w i IK(x) = x n ( ) x xi w i IK i= K(v)dv. (See, e.g., Reiss (98), Kulczycki and Dawidowicz (997), Bowman et al. (998) or Li and Racine (2006).) For a Gaussian kernel, K(z) = φ(z) and IK(z) = Φ(z) where φ and Φ are the Gaussian PDF and CDF, respectively. For Epanechnikov kernel functions, we have or K(z) = IK(z) = K(z) = IK(z) = { ( 5 z2) if z < 5 0 otherwise 0 if z < ( z 5 z3) if z < 5 if z > 5 { 3 4 ( z 2 ) if z < 0 otherwise 0 if z < 2 + ( 3 4 z 3 z3) if z < if z > 3 Syntax The syntax for akdensity follows the official kdensity syntax: akdensity varname [ weight ][ if ][ in ][, nograph noadaptive generate(newvar x newvar density) n(#) width(#) [ epan gauss ] cdf(string) normal student(#) at(var x) stdbands(#) symbol(...) connect(...) title(string) graph options ] The syntax for the companion command akdensity0 is: akdensity0 varname [ weight ][ if ][ in ], width(# varname) at(var x) generate(newvar density) [ stdbands(#) lambda(string) [ epan gauss ] double ] cdf(string) requests estimation of the kernel smoothed cumulative distribution function in addition to the density function. Both function estimates are based on identical (adaptive) bandwidth and kernel function specifications. CDF estimates for each point on the grid specified by at or n options are stored in the newly created variable newvarname. With the exception of cdf(newvarname), all options are described in Van Kerm (2003) or in [R] kdensity (for the options in common). 2

3 4 Example The following example illustrates usage of akdensity s cdf option using the coral trout length data distribution used in Van Kerm (2003).. use trocolen.dta, clear First, cdf is used with the default bandwidth selected by akdensity. The first plot (top panel below) illustrates the resulting estimates of both the PDF and CDF estimates along with the classic empirical CDF estimate.. range x (26 missing values generated). label var x "Length". akdensity length, nograph gen(fx) at(x) cdf(fx) Two-stage adaptive kernel density estimation Step : Pilot density and local bandwidth factors estimation Step 2: Adaptive kernel density estimation. cumul length, gen(ecdf). tw (line fx x) (line Fx x, yaxis(2)) (line ecdf length, connect(stairstep) > sort yaxis(2) ) `gropts Note that preference over bandwidth size may differ according to focus on smoothing the PDF or the CDF. The second plot (bottom panel) illustrates PDF and CDF estimates with a smaller bandwidth. For consistency, however, akdensity will estimate both PDF and CDF using identical bandwidths.. akdensity length, nograph gen(fx2) at(x) cdf(fx2) w(0) Two-stage adaptive kernel density estimation Step : Pilot density and local bandwidth factors estimation Step 2: Adaptive kernel density estimation. tw (line fx2 x) (line Fx2 x, yaxis(2)) (line ecdf length, connect(stairstep > ) sort yaxis(2) ) `gropts References Bowman, A., Hall, P. and Prvan, T. (998), Bandwidth selection for the smoothing of distribution functions, Biometrika 85(4), Kulczycki, P. and Dawidowicz, A. L. (997), Kernel estimator of quantile, Universitatis Iagellonicae Acta Mathematica 37, Li, Q. and Racine, J. S. (2006), Nonparametric Econometrics: Theory and Practice, Princeton University Press. Reiss, R.-D. (98), Nonparametric estimation of smooth distribution functions, Scandinavian Journal of Statistics 8(2), 6 9. Van Kerm, P. (2003), Adaptive kernel density estimation, Stata Journal 3(2),

4 (a) Default bandwidth Adaptive kernel PDF estimate Adaptive kernel CDF estimate Coral trout length (in mm.) (b) Bandwidth set to 0 mm. Adaptive kernel PDF estimate Adaptive kernel CDF estimate Coral trout length (in mm.) Figure. Adaptive kernel PDF and CDF estimates and empirical CDF 4

5 Acknowledgements Implementation of the new cdf() option and preparation of this note was financially supported by the World Bank Knowledge for Change Program (KCP II - TF094570). Suggested citation Van Kerm, P. (200), Kernel smoothed CDF estimation with akdensity, CEPS/INSTEAD, Differdange, Luxembourg. 5

Self-consistent density estimation

Self-consistent density estimation Joerg Luedicke Alberto Bernacchia Manuscript currently under review by The Stata Journal 5 April 2013 Contact: joerg.luedicke@ufl.edu The Stata Journal (yyyy) vv, Number