scholarly journals On J M Keynes’s Rejection of the Moscow School of Probability’s Limiting Frequency Approach to Probability and Kolmogorov’s Axiom of Additivity (Countable Additivity ): Non –Additivity was the fundamental, basic axiom upon which all of the Economics of Ke

2015 ◽  
Vol 2 (11) ◽  
pp. 7 ◽  
Author(s):  
Michael Emmett Brady
Keyword(s):  
Goodness Of Fit ◽  
Countable Additivity ◽  
Goodness Of Fit Tests ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Exploratory Data ◽  
Probability And Statistics ◽  
Basic Axiom ◽  
Moscow School ◽  
Gaussian Normal Distribution

<p>J M Keynes was an acknowledged, world renown, and internationally recognized expert in probability and statistics in the 1930’s based on his A Treatise on Probability (1921). . Keynes had been selected by statistics journals to serve as a referee during the 1930’s. It is, therefore, no surprise that he was selected as the referee by the League of Nations to review Jan Tinbergen’s work on business cycles that used an econometrics approach based on The Law of Large Numbers, the Central Limit Theorem, and the Gaussian (Normal) Distribution .The fundamental axiom used by Tinbergen was additivity . Kolmogorov and the Moscow School of Probability’s main innovation was to go from the axiom of additivity to the axiom of countable additivity. However, Keynes rejected additivity except in the special case that the weight of the evidence, w, which measured the relative completeness of the evidence ,defined on the closed unit interval [0,1],equaled 1 , approached 1,or approximated 1. Keynes also accepted goodness of fit tests, such as the Lexis –Q test, and exploratory data analysis as evidence that could be used to support using a particular probability distribution.</p>

scholarly journals On the ubiquitous notion of mean in probability and statistics

2019 ◽  
Vol 2 (2) ◽  
pp. 35-41
Author(s):  
Aboubakar MAITOURNAM
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Law Of Large Numbers ◽  
Probability Distributions ◽  
Basic Notion ◽  
The Core ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Essential Parameter ◽  
Probability And Statistics

In probability and statistics, the basic notion of probability of an event can be expressed as a mathematical expectation. The latter is a theoretical mean and is an essential parameter of most probability distributions, in particular of the Gaussian distribution. Last but not least, the notion of mean is at the core of two main theorems of probabilities and statistics, that is : the law of large numbers and the central limit theorem. Whether it is a theoretical or empirical version, the concept of mean is omnipresent in probability and statistics, is consubstantial to these two disciplines and is a bridge between randomness and determinism.

scholarly journals Dependence Modeling in Energy Markets using Sibuya-type Copulas

2017 ◽  
Vol 6 (3) ◽  
pp. 43
Author(s):  
Nikolai Kolev ◽  
Jayme Pinto
Keyword(s):  
Goodness Of Fit ◽  
Basic Assumption ◽  
Energy Markets ◽  
Dependence Structure ◽  
Dependence Function ◽  
Dependence Modeling ◽  
Goodness Of Fit Tests ◽  
Oil And Natural Gas ◽  
Error Terms ◽  
Time Invariant

The dependence structure between 756 prices for futures on crude oil and natural gas traded on NYMEX is analyzed  using  a combination of novel time-series and copula tools.  We model the log-returns on each commodity individually by Generalized Autoregressive Score models and account for dependence between them by fitting various copulas to corresponding  error terms. Our basic assumption is that the dependence structure may vary over time, but the ratio between the joint distribution of error terms and the product of marginal distributions (e.g., Sibuya's dependence function) remains the same, being time-invariant.  By performing conventional goodness-of-fit tests, we select the best copula, being member of the currently  introduced class of  Sibuya-type copulas.

scholarly journals Goodness-of-Fit Tests for Bivariate Time Series of Counts

Econometrics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 10
Author(s):  
Šárka Hudecová ◽  
Marie Hušková ◽  
Simos G. Meintanis
Keyword(s):  
Goodness Of Fit ◽  
Probability Generating Function ◽  
Parametric Bootstrap ◽  
Real Data ◽  
Data Sets ◽  
Test Statistics ◽  
Finite Sample ◽  
Generalized Poisson ◽  
Goodness Of Fit Tests ◽  
Monte Carlo Experiments

This article considers goodness-of-fit tests for bivariate INAR and bivariate Poisson autoregression models. The test statistics are based on an L2-type distance between two estimators of the probability generating function of the observations: one being entirely nonparametric and the second one being semiparametric computed under the corresponding null hypothesis. The asymptotic distribution of the proposed tests statistics both under the null hypotheses as well as under alternatives is derived and consistency is proved. The case of testing bivariate generalized Poisson autoregression and extension of the methods to dimension higher than two are also discussed. The finite-sample performance of a parametric bootstrap version of the tests is illustrated via a series of Monte Carlo experiments. The article concludes with applications on real data sets and discussion.

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