Incorporating support constraints into nonparametric estimators of densities

Abstract

L e t f * ( x ) be t h e u s u a l P a r z e n -R o s e n b l a t t k e r n e l e s t i m a t o r n o f t h e p d f f o f a random v a r i a b l e X based on a sample XI , . . . ,X n f r o m X. I n many p r a c t i c a l a p p l i c a t i o n s , i t i s known t h a t X > c a n d / o r X < d f o r g i v e n c o n s t a n t s c and d. A d d i t i o n a l l y , one m i g h t know t h e v a l u e s o f f ( c ) a n d / o r f ( d ) . "mi r r o r image" and " t i e d down" m o d i f i c a t i o n s i n c o r p o r a t e t h i s a d d i t i o n a l i n f o r m a t i o n i n t o f n w h i c h has s u p p o r t [c,d]. T h i s e s t i m a t o r T h i s p a p e r s t u d i e s * o f f n w h i c h an e s t i m a t o r i s i n t e r p r e t e d i n a manner w h i c h a l l o w s one t o use most o f t h e known convergence p r o p e r t i e s o f k e r n e l e s t i m a t e s i n s t u d y i n g t h e b e h a v i o r of fn.

Improving partial mutual information-based input variable selection by consideration of boundary issues associated with bandwidth estimation

“… Reflection correction (RC) (Schuster, 1985;Silverman, 1986), which 'reflects' the 266 data at the boundary and adds the density outside the support of ̂ back to the 267 boundary region; 268  Boundary kernel (BK) (Gasser and Müller, 1979;Marshall and Hazelton, 2010; 269 Zhang and Karunamuni, 2000), which replaces the conventional Gaussian kernel with 270 a more adaptive kernel that is able to capture any shape of the density, although 271 negative densities can be generated near the boundary; 272  Pseudo-data approach (PA) (Cowling and Hall, 1996), which generates additional data 273 based on the 'three-point-rule' and combines them with the original data before 274 implementing kernel estimation; 275  Kernel transformation (KT) (Marron and Ruppert, 1994), which requires (i) a 276 transformation function so that ( ) has a first derivative of 0 at the boundary; (ii) 277 a kernel estimator with reflection on ( ); and (iii) a back-conversion through the 278 change-of-variables formula to achieve ̂. As a result of applying the transformation 279 function , the impact of the boundary issue becomes insignificant because the non-280…”

“…In total, 525data 353 points are generated for each of the exogenous inputs for the three functions considered 354 (details given below) and the first 25 points are rejected in order to prevent initialisation 355 effects (May et al, 2008b), resulting in 500 data points to be used in the analysis. 356 The output data are generated by substituting the generated input data into three synthetic 367 (Schuster, 1985;396 Silverman, 1986), and the boundary kernel (BK) (Gasser and Müller, 1979;Marshall and 397 Hazelton, 2010; Zhang and Karunamuni, 2000) are applied in this study. The CK is selected 398 as a benchmark model against which the performance of the other approaches can be 399 compared; the RC is adopted because it can be extended into a bivariate setting with relative 400 ease; while the BK is implemented because it has theoretically amenable derivations and 401 successful applications to both univariate and bivariate cases.…”

Adaptive Smoothing of Seismicity in Time, Space, and Magnitude for Time-Dependent Earthquake Forecasts for California

“…Several methods have been proposed to deal with this problem (Müller, 1991). In this work, we use the reflection technique introduced by Schuster (1985) to correct for missing earthquakes with M ≤ M D : for each earthquake of magnitude M ≥ M D we add a mirror earthquake with magnitude equal (Müller, 1991). This method works correctly even if the density function is not uniform close to the boundary.…”

Why Entrepreneurs Choose Risky R&D Projects – But Still Not Risky Enough

“…Because the distributions of L and H have bounded support, the standard kernel density estimator is inconsistent at the boundary which invalidates its use in the current context. Instead, we employ the Schuster (1985) and Silverman (1986) reflection method yielding the following consistent density estimator:…”