THE AUSTRALIAN NATIONAL UNIVERSITY

WORKING PAPERS IN ECONOMICS AND ECONOMETRICS

Nonparametric Density Estimation for Stratified Samples

Robert Breunig The Australian National University∗

Working Paper No. 457

First version: February, 2001 Current version: November, 2005

ISBN: 086831 457 9

∗ Center for Economic Policy Research, Economics Program, Research School of Social Sciences, The Australian National University, Canberra ACT 0200, AUSTRALIA phone: +61 (0)2 6125-2148; fax: +61 (0)2 6125-0087, e-mail: Robert.Breunig@anu.edu.au

Nonparametric Density Estimation forStrati�ed Samples

Robert BreunigThe Australian National University�

First version: February, 2001Current version: November 1, 2005

Abstract

In this paper, we consider the non-parametric, kernel estimate of the density,f(x), for data drawn from strati�ed samples. Much of the data used by socialscientists is gathered in some type of complex survey violating the usual assump-tions of independently and identically distributed data. Such e¤ects induced bythe survey structure are rarely considered in the literature on non-parametricdensity estimation, yet they may have serious consequences for our analysis, asshown in this paper.

A weighted estimator is developed which provides asymptotically unbi-ased density estimation for strati�ed samples. A data-based method for choosingthe optimal bandwidth is suggested, using information on within-stratum vari-ances and means. The weighted estimator and proposed bandwidth are shownto give smaller mean squared error for strati�ed samples than an unweightedestimator and a commonly used method of choosing the bandwidth. Severalillustrations from simulation are provided. We also show that the optimalsampling scheme in this case is always strati�ed sampling proportional to size,irrespective of the stratum-speci�c densities.

Keywords: nonparametric density estimation, bandwidth selection, strati�ed sam-pling, optimal sampling

JEL Classi�cation: C14, C42

.

�Center for Economic Policy Research, Economics Program, Research School of Social Sciences,The Australian National University, Canberra ACT 0200, AUSTRALIAphone: +61 (02) 6125-2148; fax: +61 (02) 6125-0087, e-mail: Robert.Breunig@anu.edu.auI am grateful to Aman Ullah for his sugestions and comments on this paper. I have also bene-�tted from conversations with Chris Skeels and Nilanjana Roy and the comments of participantsin seminars at Georgia State University, the U.S. Bureau of Labor Statistics, University of Califor-nia, Riverside, the Econometric Society Australasian Meetings 1998, University of Sydney and theUniversity of New South Wales.

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Table 1Comparison of IMSE from weighted and unweighted estimation

Identical Standard deviations IMSE(hst)/

h*hst IMSE(h*) IMSE(hst) IMSE(h*)

Proportional 0.00 0.26626 0.26607 0.00133 0.00133 1.00000Sampling: 0.50 0.28290 0.27455 0.00129 0.00128 0.99815

1.00 0.33283 0.30181 0.00119 0.00117 0.979221.50 0.41603 0.34725 0.00110 0.00102 0.926102.00 0.53252 0.36956 0.00135 0.00095 0.705492.50 0.68229 0.34059 0.00375 0.00104 0.276223.00 0.86535 0.31439 0.01320 0.00112 0.084963.50 1.08168 0.30216 0.03859 0.00117 0.030244.00 1.33130 0.29899 0.09292 0.00118 0.012694.50 1.61420 0.30016 0.19669 0.00118 0.005975.00 1.93039 0.30242 0.38729 0.00117 0.00301

Dis-proportional 0.00 0.26626 0.27243 0.00146 0.00146 0.99897

Sampling: 0.50 0.28290 0.28112 0.00141 0.00141 0.99992n2/n1=2 1.00 0.33283 0.30903 0.00130 0.00128 0.98825

1.50 0.41603 0.35556 0.00118 0.00112 0.944662.00 0.53252 0.37840 0.00142 0.00105 0.739152.50 0.68229 0.34874 0.00380 0.00114 0.299483.00 0.86535 0.32191 0.01324 0.00123 0.093103.50 1.08168 0.30939 0.03862 0.00128 0.033214.00 1.33130 0.30615 0.09295 0.00130 0.013954.50 1.61420 0.30735 0.19671 0.00129 0.006575.00 1.93039 0.30966 0.38731 0.00128 0.00331

Identical means IMSE(hst)/

h*hst IMSE(h*) IMSE(hst) IMSE(h*)

Proportional 1.00 0.26626 0.26607 0.00133 0.00133 1.00000Sampling: 1.50 0.43267 0.31478 0.00145 0.00112 0.77163

2.00 0.66565 0.33664 0.00363 0.00105 0.288852.50 0.96519 0.34506 0.01280 0.00102 0.079823.00 1.33130 0.34834 0.04341 0.00101 0.023323.50 1.76397 0.34971 0.13070 0.00101 0.007724.00 2.26321 0.35034 0.35068 0.00101 0.002874.50 2.82901 0.35066 0.85215 0.00101 0.001185.00 3.46138 0.35082 1.90508 0.00101 0.000535.50 4.16031 0.35092 3.97038 0.00101 0.000256.00 4.92581 0.35097 7.79642 0.00101 0.00013

Dis-proportional 1.00 0.26626 0.27243 0.00146 0.00146 0.99897Sampling: 1.50 0.43267 0.32231 0.00153 0.00123 0.80293n2/n1=2 2.00 0.66565 0.34470 0.00368 0.00115 0.31292

2.50 0.96519 0.35333 0.01284 0.00112 0.087493.00 1.33130 0.35668 0.04343 0.00111 0.025623.50 1.76397 0.35809 0.13072 0.00111 0.008484.00 2.26321 0.35873 0.35070 0.00111 0.003164.50 2.82901 0.35905 0.85216 0.00111 0.001305.00 3.46138 0.35922 1.90509 0.00111 0.000585.50 4.16031 0.35932 3.97039 0.00111 0.000286.00 4.92581 0.35937 7.79643 0.00110 0.00014

12 µµ −

12 /σσ

Table 2Results of simulation exercise

Weighted Non-Parametric Density Estimation for Stratified Samples

Average results of 1000 repetitions

Stratum 1 values fixed: n1=1000

n2 hst ha h* FigureProportional 0 1 1000 0.23159 0.19553 0.23138 4 No difference between stratasampling: 2 1 1000 0.32764 0.27818 0.32071 5 Strata differ only by mean

3 1 1000 0.41773 0.35468 0.27351 6 Strata differ only by mean0 3 1000 0.51800 0.31662 0.30318 7 Strata differ only by standard deviation3 3 1000 0.62402 0.49159 0.30584 8 Both means and standard deviations differ

Non-proportional 2 1 2000 0.29363 0.24931 0.30272 9 Strata differ only by meansampling: 3 1 2000 0.37006 0.31420 0.25824 10 Strata differ only by mean

0 3 2000 0.53760 0.36005 0.28605 11 Strata differ only by standard deviation3 3 2000 0.61692 0.52379 0.28921 12 Both means and standard deviations differ

11 =σ01 =µ

2µ 2σ