Appendix G: The Likelihood Function Calculation

The non-parametric likelihood function calculation procedure in the MAD software is just one of many possible ways to calculate the likelihood function, but it is a very generic method.  Note that the MAD software solves many different formulations of the likelihood function, as appropriate for the application configured in the pre-processing module:  ,  , , , , or .  Note that bold-face indicates a vector or matrix.

The MAD software uses kernels to regress the likelihood function joint PDF using the np package for R [Hayfield & Racine, 2008].  The generic formulation of multivariate kernel regression is

where  is a set of independent and identically distributed set of n samples, , h is a smoothing parameter (often called the bandwidth), and  that is non-negative, centered at 0, and integrates to 1 (often called the kernel) [Terrell & Scott, 1992; Scott 2009].

The extension to the likelihood function for MAD applications is straightforward after we define notation for the simulation data.  Let us introduce  as an ensemble of simulations, where  is a generic representation of any parameters on which the simulations are conditional, i.e.  or  etc.  Finally, knowing that function is evaluated at the measured Type-B location yields

Therefore, the two necessary pre-cursors to performing this calculation for any application are to acquire measurements of the Type-B data and generating ensembles of simulated data.  In the MAD software the kernel is Gaussian.

See the work of Scott [2009] for a much more in depth discussion of the properties of kernels and multivariate density estimation.

Note that extension of the application to fit any of the six likelihood function formulations listed above does not require any change to the second equation, but only requires that the  is a function of the proper .

Works Cited:

Hayfield, T. and J. S. Racine (2008) Nonparametric econometrics: The np package, Journal of Statistical Software, 27(5). http://www.jstatsoft.org/v27/i05  

Scott, D. W. (2009), Multivariate density estimation: theory, practice, and visualization, John Wiley & Sons, Inc., New York, NY.

Terrell, G. R. & D. W. Scott (1992), Variable kernel density estimation, The Annals of Statistics, 20(3), 1236-1265, http://www.jstor.org/stable/2242011

Last edited Sep 20, 2013 at 10:38 PM by frystacka, version 5

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