Successive Approximation of Nonlinear Confidence Regions (SANCR)

This paper studies the existence and uniqueness solution of fractional integro-differential equation, by using some numerical graphs with successive approximation method of fractional integro –differential equation. The results of written new program in Mat-Lab show that the method is very interested and efficient. Also we extend the results of Butris [3].

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The Methodology of Wealth of Nations

10.1093/oso/9780190690120.003.0012 ◽

2017 ◽

Author(s):

Eric Schliesser

Keyword(s):

Successive Approximation ◽

David Hume ◽

Scientific Theory ◽

Research Tool ◽

Natural Course ◽

Market Prices ◽

Wealth Of Nations ◽

Natural Prices

This chapter aims to give an interpretation of Adam Smith’s main methods in Wealth of Nations. It does so by focusing on the crucial analytical distinction between natural prices and market prices. Smith postulates a “natural course” of events in order to stimulate research into institutions and other causes that cause actual events to deviate from it. For Smith, scientific theory is among other functions, a research tool that allows for a potentially open-ended process of successive approximation. Smith’s explanation of the introduction of commerce in Europe is contrasted with that of his friend and forerunner David Hume. Hume’s essay “Of the Populousness of Ancient Nations” may have inspired Smith to develop his method.

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Bootstrap Analysis

10.1093/oso/9780198505044.003.0004 ◽

2017 ◽

Author(s):

Russell Cheng

Keyword(s):

Goodness Of Fit ◽

Null Distribution ◽

Real Data ◽

Confidence Regions ◽

Confidence Bands ◽

Cumulative Distribution ◽

Attractive Alternative ◽

Bootstrap Analysis ◽

Parametric Bootstrapping ◽

Anderson Darling

Parametric bootstrapping (BS) provides an attractive alternative, both theoretically and numerically, to asymptotic theory for estimating sampling distributions. This chapter summarizes its use not only for calculating confidence intervals for estimated parameters and functions of parameters, but also to obtain log-likelihood-based confidence regions from which confidence bands for cumulative distribution and regression functions can be obtained. All such BS calculations are very easy to implement. Details are also given for calculating critical values of EDF statistics used in goodness-of-fit (GoF) tests, such as the Anderson-Darling A2 statistic whose null distribution is otherwise difficult to obtain, as it varies with different null hypotheses. A simple proof is given showing that the parametric BS is probabilistically exact for location-scale models. A formal regression lack-of-fit test employing parametric BS is given that can be used even when the regression data has no replications. Two real data examples are given.

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Mean Empirical Likelihood Inference for Response Mean with Data Missing at Random

Discrete Dynamics in Nature and Society ◽

10.1155/2020/8893594 ◽

2020 ◽

Vol 2020 ◽

pp. 1-12

Author(s):

Hanji He ◽

Guangming Deng

Keyword(s):

Empirical Likelihood ◽

Missing At Random ◽

Confidence Regions ◽

Likelihood Inference ◽

High Dimensional ◽

Finite Sample ◽

The Mean ◽

Data Missing ◽

Consistency And Asymptotic Normality ◽

The Impact

We extend the mean empirical likelihood inference for response mean with data missing at random. The empirical likelihood ratio confidence regions are poor when the response is missing at random, especially when the covariate is high-dimensional and the sample size is small. Hence, we develop three bias-corrected mean empirical likelihood approaches to obtain efficient inference for response mean. As to three bias-corrected estimating equations, we get a new set by producing a pairwise-mean dataset. The method can increase the size of the sample for estimation and reduce the impact of the dimensional curse. Consistency and asymptotic normality of the maximum mean empirical likelihood estimators are established. The finite sample performance of the proposed estimators is presented through simulation, and an application to the Boston Housing dataset is shown.

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