Self-Similarity Parameter Estimation for K-Dimensional Processes

Efficient Parameter Estimation for Self - Similar Processes

In the paper [1] the author claimed to have established asymptotic normality and efficiency of the Gaussian maximum likelihood estimator (MLE) for long range dependent processes (with the increments of the self-similar fractional Brownian motion as a special case—hence, the title of the paper). The case considered in the paper was Gaussian stationary sequences with spectral densities fθ(x)∼ |x|...

Self-similarity parameter estimation and reproduction property for non-Gaussian Hermite processes

We consider the class of all the Hermite processes (Z t )t2[0;1] of order q 2 N and with Hurst index H 2 ( 1 2 ; 1). The process Z (q;H) is H-self-similar, it has stationary increments and it exhibits long-range dependence identical to that of fractional Brownian motion (fBm). For q = 1, Z is fBm, which is Gaussian; for q = 2, Z is the Rosenblatt process, which lives in the second Wiener chaos;...

Parameter Estimation for Multi-dimensional Filtered Poisson Processes

Statistical Self-Similarity of One-Dimensional Growth Processes

For one-dimensional growth processes we consider the distribution of the height above a given point of the substrate and study its scale invariance in the limit of large times. We argue that for self-similar growth from a single seed the universal distribution is the Tracy-Widom distribution from the theory of random matrices and that for growth from a flat substrate it is some other, only nume...

Self-Similarity: Part II-Optimal Estimation of Fractal Processes

In a companion paper we characterized the class of scale-invariant convolution operators: the generalized fractional derivatives of order γ. We used these operators to specify regularization functionals for a series of Tikhonov-like least squares data fitting problems and proved that the general solution is a fractional spline of twice the order. We investigated the deterministic properties of ...