scholarly journals Asymptotic methods of testing statistical hypotheses

2021 ◽  
Author(s):  
◽  
Thuong Nguyen
Keyword(s):  
Hypothesis Testing ◽  
Null Hypothesis ◽  
Goodness Of Fit ◽  
Empirical Process ◽  
Chi Square ◽  
Distribution Free ◽  
Statistical Hypotheses ◽  
Regularly Varying ◽  
Asymptotically Distribution Free ◽  
Tail Distributions

<p>For a long time, the goodness of fit (GOF) tests have been one of the main objects of the theory of testing of statistical hypotheses. These tests possess two essential properties. Firstly, the asymptotic distribution of GOF test statistics under the null hypothesis is free from the underlying distribution within the hypothetical family. Secondly, they are of omnibus nature, which means that they are sensitive to every alternative to the null hypothesis.   GOF tests are typically based on non-linear functionals from the empirical process. The first idea to change the focus from particular functionals to the transformation of the empirical process itself into another process, which will be asymptotically distribution free, was first formulated and accomplished by {\bf Khmaladze} \cite{Estate1}. Recently, the same author in consecutive papers \cite{Estate} and \cite{Estate2} introduced another method, called here the {\bf Khmaladze-2} transformation, which is distinct from the first Khmaladze transformation and can be used for an even wider class of hypothesis testing problems and is simpler in implementation.   This thesis shows how the approach could be used to create the asymptotically distribution free empirical process in two well-known testing problems.   The first problem is the problem of testing independence of two discrete random variables/vectors in a contingency table context. Although this problem has a long history, the use of GOF tests for it has been restricted to only one possible choice -- the chi-square test and its several modifications. We start our approach by viewing the problem as one of parametric hypothesis testing and suggest looking at the marginal distributions as parameters. The crucial difficulty is that when the dimension of the table is large, the dimension of the vector of parameters is large as well. Nevertheless, we demonstrate the efficiency of our approach and confirm by simulations the distribution free property of the new empirical process and the GOF tests based on it. The number of parameters is as big as $30$. As an additional benefit, we point out some cases when the GOF tests based on the new process are more powerful than the traditional chi-square one.   The second problem is testing whether a distribution has a regularly varying tail. This problem is inspired mainly by the fact that regularly varying tail distributions play an essential role in characterization of the domain of attraction of extreme value distributions. While there are numerous studies on estimating the exponent of regular variation of the tail, using GOF tests for testing relevant distributions has appeared in few papers. We contribute to this latter aspect a construction of a class of GOF tests for testing regularly varying tail distributions.</p>

scholarly journals Asymptotic methods of testing statistical hypotheses

2021 ◽  
Author(s):  
◽  
Thuong Nguyen
Keyword(s):  
Hypothesis Testing ◽  
Null Hypothesis ◽  
Goodness Of Fit ◽  
Empirical Process ◽  
Chi Square ◽  
Distribution Free ◽  
Statistical Hypotheses ◽  
Regularly Varying ◽  
Asymptotically Distribution Free ◽  
Tail Distributions

<p>For a long time, the goodness of fit (GOF) tests have been one of the main objects of the theory of testing of statistical hypotheses. These tests possess two essential properties. Firstly, the asymptotic distribution of GOF test statistics under the null hypothesis is free from the underlying distribution within the hypothetical family. Secondly, they are of omnibus nature, which means that they are sensitive to every alternative to the null hypothesis.   GOF tests are typically based on non-linear functionals from the empirical process. The first idea to change the focus from particular functionals to the transformation of the empirical process itself into another process, which will be asymptotically distribution free, was first formulated and accomplished by {\bf Khmaladze} \cite{Estate1}. Recently, the same author in consecutive papers \cite{Estate} and \cite{Estate2} introduced another method, called here the {\bf Khmaladze-2} transformation, which is distinct from the first Khmaladze transformation and can be used for an even wider class of hypothesis testing problems and is simpler in implementation.   This thesis shows how the approach could be used to create the asymptotically distribution free empirical process in two well-known testing problems.   The first problem is the problem of testing independence of two discrete random variables/vectors in a contingency table context. Although this problem has a long history, the use of GOF tests for it has been restricted to only one possible choice -- the chi-square test and its several modifications. We start our approach by viewing the problem as one of parametric hypothesis testing and suggest looking at the marginal distributions as parameters. The crucial difficulty is that when the dimension of the table is large, the dimension of the vector of parameters is large as well. Nevertheless, we demonstrate the efficiency of our approach and confirm by simulations the distribution free property of the new empirical process and the GOF tests based on it. The number of parameters is as big as $30$. As an additional benefit, we point out some cases when the GOF tests based on the new process are more powerful than the traditional chi-square one.   The second problem is testing whether a distribution has a regularly varying tail. This problem is inspired mainly by the fact that regularly varying tail distributions play an essential role in characterization of the domain of attraction of extreme value distributions. While there are numerous studies on estimating the exponent of regular variation of the tail, using GOF tests for testing relevant distributions has appeared in few papers. We contribute to this latter aspect a construction of a class of GOF tests for testing regularly varying tail distributions.</p>

scholarly journals Raking-ratio empirical process with auxiliary information learning

2020 ◽  
Vol 24 ◽  
pp. 435-453
Author(s):  
Mickael Albertus
Keyword(s):  
Asymptotic Behavior ◽  
Goodness Of Fit ◽  
Empirical Process ◽  
Information Source ◽  
Auxiliary Information ◽  
Computational Method ◽  
Empirical Measure ◽  
Ratio Method ◽  
Goodness Of Fit Test ◽  
Chi Square

The raking-ratio method is a statistical and computational method which adjusts the empirical measure to match the true probability of sets of a finite partition. The asymptotic behavior of the raking-ratio empirical process indexed by a class of functions is studied when the auxiliary information is given by estimates. These estimates are supposed to result from the learning of the probability of sets of partitions from another sample larger than the sample of the statistician, as in the case of two-stage sampling surveys. Under some metric entropy hypothesis and conditions on the size of the information source sample, the strong approximation of this process and in particular the weak convergence are established. Under these conditions, the asymptotic behavior of the new process is the same as the classical raking-ratio empirical process. Some possible statistical applications of these results are also given, like the strengthening of the Z-test and the chi-square goodness of fit test.

scholarly journals THE EFFECT OF PRODUCT FASCINATION, SERVICE FASCINATION, AND COMPETITIVE POWER OF TARIFF ON CUSTOMER SATISFACTION AND IMPACT ON TRANSACTION DECISIONS (Study on Agent BRILink BRI Branch of Pati)

2019 ◽  
Vol 17 (2) ◽  
pp. 89
Author(s):  
Rizal Mahdi Kurniawan ◽  
Harry Soesanto ◽  
Sugiarto Sugiarto
Keyword(s):  
Customer Satisfaction ◽  
Hypothesis Testing ◽  
Goodness Of Fit ◽  
Sem Analysis ◽  
Competitive Power ◽  
Chi Square

Bank Rakyat Indonesia (BRI) innovates to launch BRILink Agent services as a collaborative partner between BRI Bank and its customers. The problem in this research is how to improve customer satisfaction which will influence the decisions of transactions at BRILink BRI Branch of Pati. Data on observed variables were obtained through interviews with questionnaires to 110 BRI Bank customers who transacted at BRILink Branch of Pati and tested using AMOS statistic software. The result of SEM analysis confirm the criteria of Goodness of fit that is Chi-Square is 154.616 with probability equal to 0.277; CMIN/DF (1.066); GFI (0.873); AGFI (0.833); TLI (0.991); CFI (0.992); and RMSEA (0.025). The result of hypothesis testing is: transaction decisions are statistically proven to be affected by customer satisfaction, customer satisfaction is statistically proven to be affected by service attractiveness and tariff competitiveness, while the attractiveness of unpaid products has a significant effect. 

scholarly journals Asymptotically distribution-free goodness-of-fit testing for tail copulas

2015 ◽  
Vol 43 (2) ◽  
pp. 878-902 ◽  
Author(s):  
Sami Umut Can ◽  
John H. J. Einmahl ◽  
Estate V. Khmaladze ◽  
Roger J. A. Laeven
Keyword(s):  
Goodness Of Fit ◽  
Distribution Free ◽  
Goodness Of Fit Testing ◽  
Fit Testing ◽  
Asymptotically Distribution Free

Answering Two Criticisms of Hypothesis Testing

1999 ◽  
Vol 85 (1) ◽  
pp. 3-18 ◽  
Author(s):  
Les Leventhal
Keyword(s):  
Hypothesis Testing ◽  
Null Hypothesis ◽  
Relative Frequency ◽  
Hypothesis Test ◽  
Hypothesis Tests ◽  
Two Generations ◽  
Statistical Hypotheses ◽  
Frequency Probability ◽  
The One

Two generations of methodologists have criticized hypothesis testing by claiming that most point null hypotheses are false and that hypothesis tests do not provide the probability that the null hypothesis is true. These criticisms are answered. (1) The point-null criticism, if correct, undermines only the traditional two-tailed test, not the one-tailed test or the little-known directional two-tailed test. The directional two-tailed test is the only hypothesis test that, properly used, provides for deciding the direction of a parameter, that is, deciding whether a parameter is positive or negative or whether it falls above or below some interesting nonzero value. The point-null criticism becomes unimportant if we replace traditional one- and two-tailed tests with the directional two-tailed test, a replacement already recommended for most purposes by previous writers. (2) If one interprets probability as a relative frequency, as most textbooks do, then the concept of probability cannot meaningfully be attached to the truth of an hypothesis; hence, it is meaningless to ask for the probability that the null is true. (3) Hypothesis tests provide the next best thing, namely, a relative frequency probability that the decision about the statistical hypotheses is correct. Two arguments are offered.

scholarly journals Asymptotically Distribution-Free Goodness-of-Fit Testing for Tail Copulas

2014 ◽  
Author(s):  
Sami Umut Can ◽  
John H. J. Einmahl ◽  
Estate V. Khmaladze ◽  
Roger J. A. Laeven
Keyword(s):  
Goodness Of Fit ◽  
Distribution Free ◽  
Goodness Of Fit Testing ◽  
Fit Testing ◽  
Asymptotically Distribution Free

scholarly journals Impact of Age, Gender and Job Satisfaction on Employee Engagement in the IT sector

2019 ◽  
Vol 9 (2) ◽  
pp. 4841-4845
Keyword(s):  
Job Satisfaction ◽  
Employee Engagement ◽  
Null Hypothesis ◽  
Goodness Of Fit ◽  
Positive Association ◽  
Chi Square ◽  
Age And Gender ◽  
Gender And Age ◽  
And Gender ◽  
The Impact

The present analysis has been conducted to understand the influence of job satisfaction, gender and age on the employee engagement levels in the Information and Technology sector. 196 bona fide questionnaire responses were received from two Information Technology (IT) firms in Odisha, India to perceive the impact of factors like job satisfaction, age and gender on the work commitment levels of employees. Correlation analysis was done to unravel the interrelationship linking gender, age and job satisfaction. The findings indicated that there was no effect of gender or age on the engagement of employees in the IT sector. Gender and age were independent of each other but influenced the Job satisfaction of the employees in the IT area. The goodness of fit was calculated for the dataset by doing the chi square test. Based on the values, for the first and the second hypothesis the null hypothesis was rejected and the alternative was selected. For the third hypothesis the null hypothesis was accepted. This showed that age and gender do not have a major impact on engagement in IT employees. But job satisfaction has a positive association with employee engagement. Regression analysis was also carried out to check the relationship between age and gender which are the independent variables with job satisfaction which is dependent. It was concluded that there is a 50% association between the independent and dependent variables. Thus organizations should make sure that the work culture is a healthy mix of the right elements so that a diverse taskforce is always driven to work and shares mutual goals with the organization.

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