The Confidence Distribution Method to the Behrens-Fisher Problem

We use the methods of “The Welch-Satterthwaite test”, “The Cochran-Cox test”, “The Generalized p-value test”, “Computational Approach test” to structure different Confidence Distributions, and use the Confidence Distributions to give an new solution the confidence interval of the difference between two population means where the populations are assumed to be normal with unknown and unequal variances. Finally, we find the most effective solution through the numerical simulation.

The Combining Confidence Distribution Method to the Behrens-Fisher Problem

Based on the Confidence Distribution method to the Behrens-Fisher problem, we consider two approaches of combining Confidence Distributions: P Combination and AN Combination to solve the Behrens-Fisher problem. Firstly, we provide some Confidence Distributions to the Behrens-Fisher problem, and then we give the Confidence Distribution method to the Behrens-Fisher problem. Finally, we compare the “combination” and the “single” through the numerical simulation.

DISTRIBUTED MONITORING SYSTEM RELIABILITY ESTIMATION WITH CONSIDERATION OF STATISTICAL UNCERTAINTY

Taking into account the whole system structure and the component reliability estimation uncertainty, a system reliability estimation method based on probability and statistical theory for distributed monitoring systems is presented. The variance and confidence intervals of the system reliability estimation are obtained by expressing system reliability as a linear sum of products of higher order moments of component reliability estimates when the number of component or system survivals obeys binomial distribution. The eigenfunction of binomial distribution is used to determine the moments of component reliability estimates, and a symbolic matrix which can facilitate the search of explicit system reliability estimates is proposed. Furthermore, a case of application is used to illustrate the procedure, and with the help of this example, various issues such as the applicability of this estimation model, and measures to improve system reliability of monitoring systems are discussed.

Conditionally Optimal Weights of Evidence

A weight of evidence is a calibrated statistic whose values in [0, 1]indicate the degree of agreement between the data and either of two hypothesis, one being treated asthe null (H_0) and the other as the alternative (H_1). A value of zero means perfect agreement withthe null, whereas a value of one means perfect agreement with the alternative. The optimality weconsider is minimal mean squared error (MSE) under the alternative while keeping the MSE under thenull below a fixed bound. This paper studies such statistics from a conditional point of view, inparticular for location and scale models.

Randomized statistical inference:A unified statistical inference frame of frequentist,fiducial,and Bayesian inference

We propose randomized inference(RI),a new statistical inference approach.RI may be realized through a randomized estimate(RE)of a parameter vector,which is a random vector that takes values in the parameter space with a probability density function(PDF)that depends on the sample or sufficient statistics,such as the posterior distributions in Bayesian inference.Based on the PDF of an RE of an unknown parameter,we propose a framework for both the vertical density representation(VDR)test and the construction of a confidence region.This approach is explained with the aid of examples.For the equality hypothesis of multiple normal means without the condition of variance homogeneity,we present an exact VDR test,which is shown as an extension of one-way analysis of variance(ANOVA).In the case of two populations,the PDF of the Welch statistics is given by using the RE.Furthermore,through simulations,we show that the empirical distribution function,the approximated t,and the RE distribution function of Welch statistics are almost equal.The VDR test of the homogeneity of variance is shown to be more efficient than both the Bartlett test and the revised Bartlett test.Finally,we discuss the prospects of RI.

Linear expectile regression under massive data

In this paper,we study the large-scale inference for a linear expectile regression model.To mitigate the computational challenges in the classical asymmetric least squares(ALS)estimation under massive data,we propose a communication-efficient divide and conquer algorithm to combine the information from sub-machines through confidence distributions.The resulting pooled estimator has a closed-form expression,and its consistency and asymptotic normality are established under mild conditions.Moreover,we derive the Bahadur representation of the ALS estimator,which serves as an important tool to study the relationship between the number of submachines K and the sample size.Numerical studies including both synthetic and real data examples are presented to illustrate the finite-sample performance of our method and support the theoretical results.