  Modern datasets are characterized by a large number of features that may
conceal complex dependency structures. To deal with this type of data,
dimensionality reduction techniques are essential. Numerous dimensionality
reduction methods rely on the concept of intrinsic dimension, a measure of the
complexity of the dataset. In this article, we first review the TWO-NN model, a
likelihood-based intrinsic dimension estimator recently introduced in the
literature. The TWO-NN estimator is based on the statistical properties of the
ratio of the distances between a point and its first two nearest neighbors,
assuming that the points are a realization from an homogeneous Poisson point
process. We extend the TWO-NN theoretical framework by providing novel
distributional results of consecutive and generic ratios of distances. These
distributional results are then employed to derive intrinsic dimension
estimators, called Cride and Gride. These novel estimators are more robust to
noisy measurements than the TWO-NN and allow the study of the evolution of the
intrinsic dimension as a function of the scale used to analyze the dataset. We
discuss the properties of the different estimators with the help of simulation
scenarios.
