Higher Order variance and Gauss Jacobi Quadrature

René Blacher (1)

1 IPS - Inférence Processus Stochastiques

In this report, we study in a detailed way higher order variances and quadrature Gauss Jacobi. Recall that the variance of order j measures the concentration of a probability close to j points $ x_{j,s} $ with weight $ \lambda_{j,s} $ which are determined by the parameters of the quadrature Gauss Jacobi. We shall study many example in which these measures specify adequately the distribution of probabilities. We shall also study their estimation and their asymptotic distributions under very wide assumptions. In particular we look what happens when the probabilities are a mixture of points with measures nonzero and of continuous densities. We will see that the Gauss Jacobi Quadrature can be used in order to detect these points of nonzero measures. We apply these results to the decomposition of Gaussian mixtures. Moreover, in the case of regression we can apply these results to estimate higher order regression.

Mots clés: Gaussian mixtures; Higher order variance; Gauss Jacobi quadrature; Central limit theorem; Higher order regression

Domaines: Probabilités [math.PR]; Statistiques [math.ST]; Théorie [stat.TH]

René Blacher. Higher Order variance and Gauss Jacobi Quadrature. [Research Report] LJK. 2012. ⟨hal-00708152⟩