The Distribution of Spacings between Quadratic Residues, Ii

The Distribution of Spacings between Quadratic Residues

We study the distribution of spacings between squares modulo q, where q is square-free and highly composite, in the limit as the number of prime factors of q goes to infinity. We show that all correlation functions are Poissonian, which among other things, implies that the spacings between nearest neighbors, normalized to have unit mean, have an exponential distribution. Date: Dec 14, 1998. Sup...

On Multivariate Likelihood Ratio Ordering among Generalized Order Statistics and their Spacings

The most of the results obtained about stochastic properties of generalized order statistics and their spacings in the literature are based on equal model parameters. In this paper, with less restrictive conditions on the model parameters, we prove some new multivariate likelihood ratio ordering results between two sub-vectors of GOS's as well as two sub-vectors of $p$-spacings based on two con...

The Distribution of Spacings between Quadratic

Values at Integers of Homogeneous Polynomials

1. Introduction. Berry and Tabor conjecture in [1] that the distribution of local spacings between the eigenvalues of the quantization of the Hamiltonian for generic, completely integrable systems follow those of random numbers. (Such a distribution is often called " Poissonian. ") In [8], Sarnak considers the specific case of the geodesic flow on a flat two-dimensional torus, for which the Ber...

Evaluation of estimation methods for parameters of the probability functions in tree diameter distribution modeling

One of the most commonly used statistical models for characterizing the variations of tree diameter at breast height is Weibull distribution. The usual approach for estimating parameters of a statistical model is the maximum likelihood estimation (likelihood method). Usually, this works based on iterative algorithms such as Newton-Raphson. However, the efficiency of the likelihood method is not...