36 <strong>np</strong>udensedatdatanewdataa p-variate data frame of density evaluation points. By default, evaluation takesplace on the data provided by tdat.an optional data frame, list or environment (or object coercible to a data frame byas.data.frame) containing the variables in the model. If not found in data,the variables are taken from environment(bws), typically the environmentfrom which <strong>np</strong>udensbw was called.An optional data frame in which to look for evaluation data. If omitted, thetraining data are used.DetailsValue<strong>np</strong>udens and <strong>np</strong>udist implement a variety of methods for estimating multivariate distributions(p-variate) defined over a set of possibly continuous and/or discrete (unordered, ordered) data. <strong>The</strong>approach is based on Li and Racine (2003) who employ ‘generalized product kernels’ that admit amix of continuous and discrete datatypes.Three classes of kernel estimators for the continuous datatypes are available: fixed, adaptive nearestneighbor,and generalized nearest-neighbor. Adaptive nearest-neighbor bandwidths change witheach sample realization in the set, x i , when estimating the density at the point x. Generalizednearest-neighbor bandwidths change with the point at which the density is estimated, x. Fixedbandwidths are constant over the support of x.Data contained in the data frame tdat (and also edat) may be a mix of continuous (default),unordered discrete (to be specified in the data frame tdat using the factor command), andordered discrete (to be specified in the data frame tdat using the ordered command). Data canbe entered in an arbitrary order and data types will be detected automatically by the routine (see <strong>np</strong>for details).A variety of kernels may be specified by the user. Kernels implemented for continuous datatypesinclude the second, fourth, sixth, and eighth order Gaussian and Epanechnikov kernels, and theuniform kernel. Unordered discrete datatypes use a variation on Aitchison and Aitken’s (1976)kernel, while ordered datatypes use a variation of the Wang and van Ryzin (1981) kernel.<strong>np</strong>udens returns a <strong>np</strong>density object, similarly <strong>np</strong>udist returns a <strong>np</strong>distribution object.<strong>The</strong> generic accessor functions fitted, and se, extract estimated values and asymptotic standarderrors on estimates, respectively, from the returned object. Furthermore, the functions summaryand plot support objects of both classes. <strong>The</strong> returned objects have the following components:evalthe evaluation points.dens or dist estimation of the density (cumulative distribution) at the evaluation pointsderrstandard errors of the density (cumulative distribution) estimateslog_likelihoodlog likelihood of the density estimatesUsage IssuesIf you are using data of mixed types, then it is advisable to use the data.frame function toconstruct your i<strong>np</strong>ut data and not cbind, since cbind will typically not work as intended onmixed data types and will coerce the data to the same type.
<strong>np</strong>udens 37Author(s)Tristen Hayfield 〈hayfield@phys.ethz.ch〉, Jeffrey S. Racine 〈racinej@mcmaster.ca〉ReferencesAitchison, J. and C.G.G. Aitken (1976), “ Multivariate binary discrimination by the kernel method,”Biometrika, 63, 413-420.Li, Q. and J.S. Racine (2007), No<strong>np</strong>arametric Econometrics: <strong>The</strong>ory and Practice, Princeton UniversityPress.Li, Q. and J.S. Racine (2003), “No<strong>np</strong>arametric estimation of distributions with categorical andcontinuous data,” Journal of Multivariate Analysis, 86, 266-292.Ouyang, D. and Q. Li and J.S. Racine (2006), “Cross-validation and the estimation of probabilitydistributions with categorical data,” Journal of No<strong>np</strong>arametric Statistics, 18, 69-100.Pagan, A. and A. Ullah (1999), No<strong>np</strong>arametric Econometrics, Cambridge University Press.Scott, D.W. (1992), Multivariate Density Estimation. <strong>The</strong>ory, Practice and Visualization, NewYork: Wiley.Silverman, B.W. (1986), Density Estimation, London: Chapman and Hall.Wang, M.C. and J. van Ryzin (1981), “A class of smooth estimators for discrete distributions,”Biometrika, 68, 301-309.See Also<strong>np</strong>udensbw , densityExamples# EXAMPLE 1 (INTERFACE=FORMULA): For this example, we load Giovanni# Baiocchi's Italian GDP panel (see Italy for details), then create a# data frame in which year is an ordered factor, GDP is continuous,# compute bandwidths using likelihood cross-validation, then create a# grid of data on which the density will be evaluated for plotting# purposes.data("Italy")attach(Italy)# Compute bandwidths using likelihood cross-validation (default), but we# override the default search tolerances to speed things up as this is a# well-behaved dataset (don't of course do this in general).bw
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npqreg 87ftol = 1.19209e-07,tol = 1
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npqreg 89Li, Q. and J.S. Racine (20
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npreg 91Usagenpreg(bws, ...)## S3 m
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npreg 93residR2MSEMAEMAPECORRSIGNif
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npreg 95summary(model)# Use npplot(
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npreg 97# - this may take a few min
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npreg 99# then a noisy samplen
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npregbw 101## S3 method for class '
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npregbw 103ckerorderukertypeokertyp
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npregbw 105ReferencesAitchison, J.
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npsigtest 107bw
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npsigtest 109Author(s)Tristen Hayfi
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npindex 111Usagenpindex(bws, ...)##
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npindex 113MAEMAPECORRSIGNif method
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npindex 115x2
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npindex 117# x1 is chi-squared havi
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npindexbw 119# plotting via persp()
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npindexbw 121methodnmultithe single
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npindexbw 123allows one to deploy t
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npscoef 125x1
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npscoef 127Valueeydatezdaterrorsres
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npscoef 129# We could manually plot
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npscoefbw 131optim.abstol,optim.max
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npscoefbw 133optim.maxattemptsmaxim
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npscoefbw 135ReferencesAitchison, J
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uocquantile 137uocquantileCompute Q
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INDEX 139npconmode, 29npindex, 110n