Abstract
AbstractOlkin and Spiegelman introduced a semiparametric estimator of the density defined as a mixture between the maximum likelihood estimator and the kernel density estimator. Due to the absence of any leave‐one‐out strategy and the hardness of estimating the Kullback–Leibler loss of kernel density estimate, their approach produces unsatisfactory results. This article investigates an alternative approach in which only the kernel density estimate is modified. From a theoretical perspective, the estimated mixture parameter is shown to converge in probability to one if the parametric model is true and to zero otherwise. From a practical perspective, the utility of the approach is illustrated on real and simulated data sets.
Highlights
There exist two general approaches to density estimation
The semiparametric density estimator is the combination between the maximum likelihood estimator and the kernel density estimator where the mixing coefficient can be either the fitness coefficient (LR method) or the OS coefficient (OS method)
For each of the simulated data sets, the copula parameter ξ was estimated as mentioned above and the marginals were estimated under three scenarios: using a kernel density estimator; a maximum likelihood estimate based on the exponential distribution for both margins; and using the semiparametric combination based on the LR method
Summary
There exist two general approaches to density estimation. On the one hand, parametric methods are known to be precise but their results depend on the assumed statistical model. Nonparametric methods often require more data but are free from model specification In this context, Olkin and Spiegelman (1987), abbreviated OS in the present article, proposed to combine the two approaches by forming a convex combination αfθn + (1 − α)fn between a. The estimator αn introduced in (1) is called the fitness coefficient because it may be interpreted as how well the model fits the data (see Section 5). Following the idea of combining parametric and nonparametric estimator, some authors (Lee & Soleymani, 2015; Rahman et al, 1997; Soleymani & Lee, 2014) investigate different strategies based on the mean squared error between the combination αfθn + (1 − α)fn and the true density, but the solution depends on the unknown distribution and heavy bootstrap methods need to be employed.
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