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.

A note on the exponential distribution

Abstract: The exponential distribution is an example of a continuous distribution. A random variable X is said to follow the exponential distribution with parameter λ if its distribution function F is given by: F (x) = 1 − e−λ x for x > 0. Recall that the distribution function F (x) = P (X ≤ x) by definition and is an increasing function of x. Since F (0) = 0, it follows that X is bigger than 0 with probability 1.

Estimation in an Exponential Distribution with Common Location and Scale Parameters

Abstract: We consider several point estimators including the jackknife estimator for the estimation of the common location and scale parameters and the right tail probability in an exponential distribution . Several interval estimators of the common location and scale parameters arc also obtained.

Necessary and sufficient conditions for consistency of approximate M-estimators in nonlinear models

Abstract: The concept of M-estimator η n = arg min Σ ρ(Y i -ρ(η)) i=1 is widely used in mathematical statistics to construct robust estimators. We apply this concept to a special class of nonlinear models with non random regressors x i . The model consists of a one parametric parent family of distributions {F η } and a function of f(θ, α) which links the parameter θ and the regressors to the distribution. Hence Y i ∼ F f(θ0 , α i ) with some θ 0 ∈ Θ which is to be estimated. Special attention is paid to the pseudolinear models where the function f has the special structure f(θ,α) = g(θ'α). The M-estimator of θ is constructed by the minimization n of M n (θ) = Σ ρ(Y i - φ(f(θ, α i ))) where φ is a suitably i=1 chosen monotone function. In this paper we study the larger class of approximate M-estimators θ n which attain the minimum of M n (θ) only approximately for large n. Sufficient conditions for the consistency of approximate M-estimators are derived, motivated by sufficient conditions for the simpler parent estimator η n . In the class of approximate M-estimators which is larger than the class of ordinary M-estimators we are able to show that the sufficient conditions are necessary for the consistency of all approximate M-estimators. If {F η } is a natural exponential family and f(θ,α) = g(θ'α) has a pseudolinear structure our model reduces to the well known generalized linear model. Another special case is the class of nonlinear regression models.

A note on accelerated sequential estimation of the mean of NEF-PVF distributions

Abstract: The minimum risk point estimation for the mean is addressed for a natural exponential family (NEF) that also has a power variance function (PVF) under a loss function given by the squared error plus linear cost An appropriate accelerated version of the full purely sequential methodology of Bose and Boukai (1993b, submitted) is proposed along the lines of Mukhopadhyay (1993a, Tech Report, No 93-27, Department of Statistics, University of Connecticut) in order to achieve operational savings The main result provides the asymptotic second-order expansion of the regret function associated with the accelerated sequential estimator of the population mean