Abstract
The density deconvolution problem is considered for random variables assumed to belong to the generalized skew-symmetric (GSS) family of distributions. The approach is semiparametric in that the symmetric component of the GSS distribution is assumed known, and the skewing function capturing deviation from the symmetric component is estimated using a deconvolution kernel approach. This requires the specification of a bandwidth parameter. The mean integrated square error (MISE) of the GSS deconvolution estimator is derived, and two bandwidth estimation methods based on approximating the MISE are also proposed. A generalized method of moments approach is also developed for estimation of the underlying GSS location and scale parameters. Simulation study results are presented including a comparing the GSS approach to the nonparametric deconvolution estimator. For most simulation settings considered, the GSS estimator is seen to have performance superior to the nonparametric estimator.
Highlights
The density deconvolution problem arises when it is of interest to estimate the probability density function fx(x) of a random variable X using observations contaminated by measurement error
This paper presents a semiparametric approach for estimating fx(x) that assumes X belongs to the class of generalized skew-symmetric (GSS) distributions
In this paper, the density deconvolution problem is considered for variables belonging to the family of generalized skew-symmetric (GSS) distributions
Summary
The density deconvolution problem arises when it is of interest to estimate the probability density function (pdf ) fx(x) of a random variable X using observations contaminated by measurement error. There is one instance where median ISE of the nonparametric estimator is smaller than that of the GSS estimator – skewing function π2(z) with NSR = 0.5, Laplace measurement error, and sample size n = 200. (The same holds true for sample sizes n = 50 and 100 in Table 6.) the equivalent scenario with sample size n = 500 has the GSS estimator with smaller median ISE This possibly indicates the effect of estimating the location and scale parameters in smaller samples and when large amounts of heavier-tailed-than-normal measurement error is present. The GSS deconvolution estimator for fx(x) is calculated assuming a normal symmetric component, f0(z) = φ(z), along with a Laplace distribution for the measurement error ε. It is reassuring that the GSS estimator is not dissimilar in appearance
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