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Natural exponential family
About: Natural exponential family is a research topic. Over the lifetime, 1973 publications have been published within this topic receiving 60189 citations. The topic is also known as: NEF.
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TL;DR: The question of the power-law exponent of exponential growth networks is studied in this paper, where the degree distribution of the pseudofractal scale-free web, an exponential growth network, is obtained.
Abstract: The question of the power-law exponent of exponential growth networks is studied here. In a discrete case, the degree distribution is defined as the probability distribution of the discrete variable. Based on this, the degree distribution of the pseudofractal scale-free web, an exponential growth network, is obtained. The power-law exponent ln3/ln2 is analyzed according to the maximum likelihood principle. It satisfies consistency and is good for small generations of the network. For many exponential growth networks, their power-law exponent needs to be tested. The work provides a new view on the power-law exponent of an exponential growth network.
1 citations
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TL;DR: The proposed approach modifications the exponential random graph model, commonly used for modeling small-size graph distributions, to address the well-known issue of model degeneracy, resulting in a non-parametric model similar to kernel density estimators.
Abstract: We propose a novel approach for density estimation with exponential families for the case when the true density may not fall within the chosen family. Our approach augments the sufficient statistics with features designed to accumulate probability mass in the neighborhood of the observed points, resulting in a non-parametric model similar to kernel density estimators. We show that under mild conditions, the resulting model uses only the sufficient statistics if the density is within the chosen exponential family, and asymptotically, it approximates densities outside of the chosen exponential family. Using the proposed approach, we modify the exponential random graph model, commonly used for modeling small-size graph distributions, to address the well-known issue of model degeneracy.
1 citations
01 Jan 1981
TL;DR: In this paper, Srivastava's characterisation of the Exponential Distribution based on record values is presented, as well as the relationship between the conditional and unconditional distribution of a random variable.
Abstract: Section I: Continuous Models.- Statistical Predictive Distributions.- Hyperbolic Distributions and Ramifications: Contributions to Theory and Application.- Multivariate Distributions of Hyperbolic Type.- The Multimodal Exponential Families of Statistical Catastrophe Theory.- Regression Models for the Inverse Gaussian Distribution.- A Note on the Inverse Gaussian Distribution.- Some Properties of the Log-Laplace Distribution.- Compound Distributions Relevant to Life Testing.- Distributions Associated with Neutrality Properties for Random Proportions.- The Independence of Size and Shape Before and After Scale Change.- Distributions on the Simplex for the Analysis of Neutrality.- Section II: Discrete Models.- Chance Mechanisms for the Univariate Generalized Waring Distribution and Related Characterizations.- On a New Family of Discrete Distributions.- On the Stirling Distribution of the First Kind.- On the Moments and Factorial Moments of a MPSD.- On Bivariate Discrete Distributions Generated By Compounding.- Bivariate Generalized Discrete Distributions and Bipartitional Polynomials.- A Bivariate Hyper-Poisson Distribution.- On the Multinomial Distributions Generated By Stochastic Matrices and Applications.- Section III: Structural Properties.- Distributions with Sufficient Statistics for Multivariate Location Parameter and Transformation Parameter.- Analytic Distribution Functions.- Some Recent Statistical Results for Infinitely Divisible Distributions.- An Alternate Simpler Method of Evaluating the Multivariate Beta Function and an Inverse Laplace Transform Connected with Wishart Distribution.- On a Theorem of Polya.- Asymptotic Distributions of Functions of Eigenvalues.- Section IV: Computer Generation.- A Rejection Technique for the Generation of Random Variables with the Beta Distribution.- Fast Methods for Generating Bivariate Discrete Random Variables.- Frugal Methods of Generating Bivariate Discrete Random Variables.- Section V: Characterizations.- A Characterization of the Negative Multinomial Distribution.- On the Rao-Rubin Characterization of the Poisson Distribution.- On Some Characterizations of the Geometric Distribution.- On Splitting Model and Related Characterizations of Some Statistical Distributions.- Rao-Rubin Condition for a Certain Class of Continuous Damage Models.- On Matrix-Variate Beta Type I Distribution and Related Characterization of Wishart Distribution.- On the Relationship Between the Conditional and Unconditional Distribution of a Random Variable.- Some Bivariate Distributions of (X,Y) Where the Conditional Distribution of Y, Given X, is Either Beta or Unit-Gamma.- Some Relationships Between the Logistic and the Exponential Distributions.- Some Characterizations of the Exponential Distribution Based on Record Values.- A Note on Srivastava's Characterization of the Exponential Distribution Based on Record Values.- On the Stochastic Equation X+Y=XY.- On the Stability of Characterizations of Non-Normal Stable Distributions.- Author Index.
1 citations
TL;DR: In this article, decision theory is applied to determine minimax policies of simultaneous control and estimation for stochastic system (1), where the binomial distribution with unknown parameter and the class of prior distributions of parameter is restricted by fixing the second moment.
Abstract: In the present paper methods of the decision theory are applied to determine minimax policies of simultaneous control and estimation for stochastic system (1). There are solved the following cases: a)when disturbances of the system have the binomial distribution with unknown parameter b)when disturbances of the system have distribution belonging to an exponential family dependent on natural unknown parameter and the class of prior distributions of parameter is restricted by fixing the second moment In both cases open analytical forms of minimax policies are given
1 citations
TL;DR: In this article, the authors considered the class of nature exponential families generated by stable distributions and obtained an explicit expression for the uniformly minimum variance unbiased estimator for the r-th cumlant, the density function, and the reliability function.
Abstract: The class of nature exponential families generated by stable distributions has been introduced in different contexts by several authors. Tweedie (1984) and Jorgensen (1987) studied this class in the context of generalized liner models and exponential dispersion models. Bar-Lev and Enis (1986) introduced this class in the context of the property of reproducibility in natural exponential families and Hougaard (1986) found the distributions in this class to be natural candidates for applications as survival distributions in life tables for heterogeneous populations. In this note, we consider such a class in the context of minimum variance unbiased estimation. For each family in this class, we obtain an explicit expression for the uniformly minimum variance unbiased estimator for the r-th cumlant, the density function, and the reliability function.
1 citations