Trimmed regions induced by parameters of a probability

Consider any kind of parameter for a probability distribution and a fixed distribution. We study the subsets of the parameter space constituted by all the parameters of the probabilities in the @a-trimming of the fixed distribution. These sets will be referred to as parameter trimmed regions. They are composed of all parameter candidates whose degree of suitability as such a parameter for the distribution is, at least, a specific value @a. In particular, we analyze location, scale, and location-scale parameters and study the properties of the trimmed regions induced by them. Several specific examples of parameter trimmed regions are studied. Among them, we should mention the zonoid trimmed regions obtained when the chosen parameter is the mean value and the location-scale regions of a univariate distribution obtained when the parameter is the pair given by the mean and the standard deviation.

[1]  Integral trimmed regions , 2005 .

[2]  I. Mizera.  On depth and deep points: a calculus , 2002 .

[3]  Regina Y. Liu,et al.  Multivariate analysis by data depth: descriptive statistics, graphics and inference, (with discussion and a rejoinder by Liu and Singh) , 1999 .

[4]  I. Molchanov.  Theory of Random Sets , 2005 .

[5]  Karl Mosler,et al.  Weighted-mean trimming of multivariate data , 2011, J. Multivar. Anal..

[6]  R. Dyckerhoff.  Computing zonoid trimmed regions of bivariate data sets , 2000 .

[7]  R. Serfling,et al.  Structural properties and convergence results for contours of sample statistical depth functions , 2000 .

[8]  Consistency of the α-trimming of a probability. Applications to central regions , 2008, 0805.2219.

[9]  J. A. Cuesta-Albertos,et al.  Uniqueness and approximate computation of optimal incomplete transportation plans , 2011 .

[10]  C. Müller,et al.  Location–Scale Depth , 2004 .

[11]  C. Small.  A Survey of Multidimensional Medians , 1990 .

[12]  Regina Y. Liu.  On a Notion of Data Depth Based on Random Simplices , 1990 .

[13]  Regina Y. Liu,et al.  Regression depth. Commentaries. Rejoinder , 1999 .

[14]  R. Serfling,et al.  General notions of statistical depth function , 2000 .

[15]  Ignacio Cascos,et al.  The expected convex hull trimmed regions of a sample , 2007, Comput. Stat..

[16]  K. Mosler.  "Multivariate Dispersion, Central Regions, and Depth": The Lift Zonoid Approach , 2002 .

[17]  J. A. Cuesta-Albertos,et al.  Trimmed Comparison of Distributions , 2008 .

[18]  A. Gordaliza.  Best approximations to random variables based on trimming procedures , 1991 .

[19]  K. Mosler,et al.  Zonoid trimming for multivariate distributions , 1997 .