Topic
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: In this paper, the exponential function type of learning model is treated and the four methods estimating parameters A, B and in C ) 1 0 ( ) ( < < + = B C AB x y x which is suited to realities are proposed.
Abstract: The production system which one operative worker performs all the processes of completing a product from an assembly to an inspection and packing is currently called the single-worker booth production system. It takes most time to complete a product at an early stage, but production time decreases gradually with the increase in the number of work and settles down about the stable time. The time is used as job standard time in work study. This phenomenon is called learning effect or experience effect. A power law function in the learning functions is well known as a learning curve model and the other function is an exponential function proposed by Pegels. In this study, the exponential function type of learning model is treated and the four methods estimating parameters A , B and in C ) 1 0 ( ) ( < < + = B C AB x y x which is suited to realities are proposed. Further, the validity for the proposed methods and the method of Pegels is confirmed by using the numerical example of the peg-board inspection and we discuss the fitness of the function in comparison with the proposed methods by using the residual square sum.
01 Jan 1998
TL;DR: In this article, it has been shown that in the case where the probability distributions live on a finite set, most exponential families which occur in applications are actually solution sets of binomial polynomials, which can be identified with the nonnegative real part of projective toric varieties.
Abstract: Exponential families are an important class of statistical models, i.e., parameterized families of probability distributions. It has been noted that in the case where the probability distributions live on a nite set most exponential families which occur in applications are actually solution sets of binomial polynomials. In fact they can be identied with the nonnegative real part of projective toric varieties. These toric varieties are not necessarily normal. This talk will explain this connection and give some examples. If time permits I will comment on how the generators of the binomial ideal dening the toric variety can be used in statistical testing.
TL;DR: For any given random variable Y with infinitely divisible distribution in a quadratic natural exponential family, this paper obtained a polynomial expansion of the power mixture density of Y.
Abstract: For any given random variable Y with infinitely divisible distribution in a quadratic natural exponential family we obtain a polynomial expansion of the power mixture density of Y. We approach the problem generally, and then consider certain distributions in greater detail. Various applications are indicated and the results are also applied to obtain approximations and their error bounds. Estimation of density and goodness-of-fit test are derived.
TL;DR: In this article, it is shown that one can construct set-compound estimators £ such that t ̂ times its risk less a minimum Bayes risk converges to zero with a rate as t -*•• •> provided that the components of θ are in a compact set and 2 ~ 2 matrix = σ I with σ unknown under squared error loss.
Abstract: Abstract. This paper is concerned with the problem of the estimation of the mean £ of a t variate normal distribution with covariance 2 ~ 2 matrix = σ I with σ unknown under squared error loss. It is shown that one can construct set-compound estimators £ such that t ̂ times its risk less a minimum Bayes risk converges to zero with a rate as t -*• •> provided that the components of θ are in a compact set and 2 ~
TL;DR: In this article, it was shown that for cylinder symmetric measures in dimension d 3, there are seven classes of measures with the tail property, corresponding to five symmetry groups, including the Pareto distribution, the exponential distribution, and the uniform distribution.
Abstract: A Pareto distribution has the property that any tail of the dis- tribution has the same shape as the original distribution. The exponential distribution and the uniform distribution have the tail property too. The tail property characterizes the univariate generalized Pareto distributions. There are three classes of univariate GPDs: Pareto distributions, power laws, and the exponential distribution. All these distributions extend to infinite mea- sures. The tail property translates into a group of symmetries for these infinite measures: translations for the exponential law; multiplications for the Pareto and power laws. In the multivariate case, for cylinder symmetric measures in dimension d 3, there are seven classes of measures with the tail property, corresponding to five symmetry groups. The second part of this paper estab- lishes this classification. The first part introduces the probabilistic setting, and discusses the associated geometric theory of multiparameter regular variation. We prove a remarkable result about a class of multiparameter slowly varying functions introduced in Ostrogorski (1995).