Statistical inference on self-similar and increment stationary processes and random fields

This dissertation is about making statistical inference on self-similar and increment stationary processes/random fields. Self-similarity and, more generally, fractality are seen in many objects in nature which have similar features at different scales, for smaller scale and larger scale. The most well known statistical model that has self-similarity is fractional Brownian motion (fBm). It has been useful for its self-similarity, increment stationarity, and Gaussianity, and it naturally arises as the scaling limits of random walk, having many applications in hydrology, telecommunication network, finance, etc. Some extensions of fBm have been introduced including operator fractional Brownian motion (OFBM) and operator scaling Gaussian random field (OSGRF). OFBM and OSGRF are multivariate processes, random field, respectively, both have operator self-similarity/operator scaling property, Gaussianity, and increment stationarity.The first topic is about estimating Hurst parameter which is a measure for self-similarity in statistical model. Hurst estimation is examined/developed in OFBM and OSGRF using wavelet transform and discrete variation method, respectively. The asymptotics of the estimators are derived in continuous sample path, discrete sample, and discrete noisy sample in OFBM. In OSGRF, the asymptotics of the estimators are derived in different sampling methods, fixed domain/increasing domain, and/or samples on the exact directions/ samples on the grid lines. The performance of the estimators is examined through simulating OFBM and OSGRF, respectively, and the good choice for scale parameter in wavelet function and discrete variation method is recommended.The second topic is on measuring dependency between two random fields that are increment stationary. The dependency is measured in spectral domain by defining and estimating coherence in multivariate random field. The concept of coherence is originated from multivariate stationary time series, and it measures correlation between two time series in spectral domain. Recently, the definition is extended to multivariate random fields. In this dissertation, the concept of coherence is extended to multivariate random field with stationary increments, its properties are examined and its estimation method is developed. Especially, the concept and the estimator method are applied to OFBM.