Exponential family random graph models (ERGMs) are widely used to model networks by parameterizing probability in terms of a set user-selected sufficient statistics. Equivalently, ERGMs ca...

Descriptive statistics is the process of summarizing gathered raw data from a research and creating useful statistics, which help the better understanding of data. According to the types of variables, which consist of qualitative and quantitative variables, some descriptive statistics have been introduced. Frequency percentage is used in qualitative data, and mean, median, mode, standard deviat...

The aim of this paper is to present a framework for asymptotic analysis of likelihood ratio and minimum discrepancy test statistics. First order asymptotics are presented in a general framework under minimal regularity conditions and for not necessarily nested models. In particular, these asymptotics give sufficient and in a sense necessary conditions for asymptotic normality of test statistics...

We introduce photon-statistics excitation spectroscopy and exemplarily apply it to a quantum-dot micropillar laser. Both the intensity and the photon number statistics of the emission from the micropillar show a strong dependence on the photon statistics of the light used for excitation of the sample. The results under coherent and pseudothermal excitation reveal that a description of the laser...

We establish necessary and sufficient conditions for the shock statistics to approach self-similar form in Burgers turbulence with Lévy process initial data. The proof relies upon an elegant closure theorem of Bertoin and Carraro and Duchon that reduces the study of shock statistics to Smoluchowski’s coagulation equation with additive kernel, and upon our previous characterization of the domain...

Commutative Jordan algebras are used to express the structure of mixed orthogonal models and to derive complete sufficient statistics. From these statistics, UMVUE, (Uniformly Minimum Variance Unbiased Estimators), are derived for the relevant parameters, first of single models then of several such models. These models may correspond to experiments designed separately so our results may be seen...

This material is designed to introduces basic concepts and fundamental theory of mathematical statistics. A review of basic concepts will include likelihood functions, sufficient statistics, and exponential family of distributions. Then point estimation will be discussed, including minimum variance unbiased estimates, Cramér-Rao inequality, maximum likelihood estimates and asymptotic theory. To...

In this note we derive Γ-minimax estimators for the location parameter of the normal and uniform models. All distributions with uniformly bounded variances and means belonging to a fixed interval form a class of priors, Γ. We restrict ourselves to the rules which are linear combinations of order statistics. Optimal decision rules in such decisiontheoretic framework are, as expected, rules linea...

Under some mild regularity on the normalizing sequence, we obtain necessary and sufficient conditions for the Strong Law of Large Numbers for (symmetrized) U-statistics. We also obtain nasc’s for the a.s. convergence of series of an analogous form.