By Stefano Cabras (auth.), Nicola Torelli, Fortunato Pesarin, Avner Bar-Hen (eds.)
This quantity contains contributions chosen after a double blind evaluate procedure and provided as a initial model on the forty fifth assembly of the Italian Statistical Society. The papers supply major and cutting edge unique contributions and canopy a vast variety of subject matters together with: statistical concept; tools for time sequence and spatial info; statistical modeling and knowledge research; survey method and reputable information; research of social, demographic and well-being info; and fiscal facts and econometrics.
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I / . e. inverse of the probability of exceeding a ﬁxed value xT /. If the parent distribution is known it is trivial to observe that exact unbiased plotting positions P. Erto ( ) A. it N. Torelli et al. 1007/978-3-642-35588-2 3, © Springer-Verlag Berlin Heidelberg 2013 23 3 24 P. Erto and A. Lepore can be obtained via order statistics-theory. Unfortunately, they are usually complex to determine, thus invalidating the practical aim of the graphical estimation. e. the parent distribution) and on the use that has to be made of the results.
Let X 0:5N. 1 D 2; 1 D 3=2/ C 0:5N. 2 D 2; 1 D 1/. The simplicial depth function is increasing for x < x0:5 D 0:4 and decreasing for x > x0:5 . 2/ two local maxima Â ' 1:994 and Â ' 1:962, near to the modes. The plot of the (normalized) depth functions is shown in the left panel of Fig. 3. 0 1 0 −1 −2 −6 −4 −2 0 x 2 4 6 0 2 4 6 8 τ Fig. 3 Mixture of normal distributions. Left: global and local simplicial depth (functions are normalized with respect to maximum value). Right: local maxima versus parameter; horizontal line marks the median results do depend on the value of , both the number and the locations of the local maxima stay almost constant for a wide range of the smoothing parameter, up to 52% quantile order (see the right panel of Fig.
2:5; 2:5/T . 1 shows the estimated density contours and simplicial depth contours. 1:25; 1:25/T and monotonically decreases along any ray from this point. On the other hand, the local simplicial depth contours (see Fig. 3 −2 −2 −2 0 2 −2 4 0 2 4 x x[,1] Fig. 1 Normal mixture distribution. 8 2 y y 0. 9 0. 1 −2 −2 −2 0 2 x 4 −2 0 2 4 x Fig. 2 Normal mixture distribution. Estimated local simplicial depth contours 1:758, corresponding to the 60% quantile) are no longer nested and they show the partial centers, provided that is sufﬁciently low.
Advances in Theoretical and Applied Statistics by Stefano Cabras (auth.), Nicola Torelli, Fortunato Pesarin, Avner Bar-Hen (eds.)