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Register a new userSome Investigations about the Properties of Maximum Likelihood Estimations Based on Lower Record Values for a Sub-Family of the Exponential Family
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Added by
Saman Hosseini
Publication date
2017
fields
Mathematical Statistics
and research's language is
English
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Here, in this paper it has been considered a sub family of exponential family. Maximum likelihood estimations (MLE) for the parameter of this family, probability density function, and cumulative density function based on a sample and based on lower record values have been obtained. It has been considered Mean Square Error (MSE) as a criterion for determining which is better in different situations. Additionally, it has been proved some theories about the relations between MLE based on lower record values and based on a random sample. Also, some interesting asymptotically properties for these estimations have been shown during some theories.
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Here in this paper, it is tried to obtain and compare the ML estimations based on upper record values and a random sample. In continue, some theorems have been proven about the behavior of these estimations asymptotically.
This article presents a differential geometrical method for analyzing sequential test procedures. It is based on the primal result on the conformal geometry of statistical manifold developed in Kumon, Takemura and Takeuchi (2011). By introducing curvature-type random variables, the condition is first clarified for a statistical manifold to be an exponential family under an appropriate sequential test procedure. This result is further elaborated for investigating the efficient sequential test in a multidimensional curved exponential family. The theoretical results are numerically examined by using von Mises-Fisher and hyperboloid models.
We present theoretical properties of the log-concave maximum likelihood estimator of a density based on an independent and identically distributed sample in $mathbb{R}^d$. Our study covers both the case where the true underlying density is log-concave, and where this model is misspecified. We begin by showing that for a sequence of log-concave densities, convergence in distribution implies much stronger types of convergence -- in particular, it implies convergence in Hellinger distance and even in certain exponentially weighted total variation norms. In our main result, we prove the existence and uniqueness of a log-concave density that minimises the Kullback--Leibler divergence from the true density over the class all log-concave densities, and also show that the log-concave maximum likelihood estimator converges almost surely in these exponentially weighted total variation norms to this minimiser. In the case of a correctly specified model, this demonstrates a strong type of consistency for the estimator; in a misspecified model, it shows that the estimator converges to the log-concave density that is closest in the Kullback--Leibler sense to the true density.
In this article, inferences about the multicomponent stress strength reliability are drawn under the assumption that strength and stress follow independent Pareto distribution with different shapes $(alpha_1,alpha_2)$ and common scale parameter $theta$. The maximum likelihood estimator, Bayes estimator under squared error and Linear exponential loss function, of multicomponent stress-strength reliability are constructed with corresponding highest posterior density interval for unknown $theta.$ For known $theta,$ uniformly minimum variance unbiased estimator and asymptotic distribution of multicomponent stress-strength reliability with asymptotic confidence interval is discussed. Also, various Bootstrap confidence intervals are constructed. A simulation study is conducted to numerically compare the performances of various estimators of multicomponent stress-strength reliability. Finally, a real life example is presented to show the applications of derived results in real life scenarios.
A function $rho:[0,infty)to(0,1]$ is a completely monotonic function if and only if $rho(Vertmathbf{x}Vert^2)$ is positive definite on $mathbb{R}^d$ for all $d$ and thus it represents the correlation function of a weakly stationary and isotropic Gaussian random field. Radial positive definite functions are also of importance as they represent characteristic functions of spherically symmetric probability distributions. In this paper, we analyze the function [rho(beta ,gamma)(x)=1-biggl(frac{x^{beta}}{1+x^{beta}}biggr )^{gamma},qquad xge 0, beta,gamma>0,] called the Dagum function, and show those ranges for which this function is completely monotonic, that is, positive definite, on any $d$-dimensional Euclidean space. Important relations arise with other families of completely monotonic and logarithmically completely monotonic functions.
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