Working Title: Novel Methods Toward Independent and Identically Distributed Test

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Description: In probability theory, a collection of random variables is independent and identically distributed (IID) if all random variables follow the same probability distribution and are mutually independent. The assumption viewingobservations generated by IID random variables is called IID assumption or randomness assumption [5, 7, 10-12], which is ubiquitous and often serves as the foundation in statistics [5], machine learning [14], entropy source estimations[9] and so on. Nevertheless, the IID assumption is commonly violated in many practical environments [3, 4, 13] and it is therefore valuable to test the correctness of the assumption, called IID test or randomness test.This IID test problem has been discussed for at least 100 years since 1919 [8] and is still being actively researched [1, 5, 6, 9-12]. Our aims for the project is to investigate IID test and develop novel methods of it. This project fallswithin the EPSRC statistics and applied probability research area. More specifically, our motivations and objectives of proposing novel IID test methods are at least the following two-fold. First, we find that most existingIID test methods are useful only under certain cases and can be useless in many others, i.e. cannot reject the assumptions of IID under many non-IID cases when the IID assumption is violated. Therefore, it would be valuable todevelop a test method being able to reject the IID assumption under more non-IID cases. Secondly, nearly all IID test methods are based on hypothesis test framework, which is often criticized as no conclusion can be drawn if the IIDassumption is not rejected [2]. One of our future work is thus to propose a more informative test method than hypothesis test. For example, this method could be a measure of the strength of the IID property of observations, potentiallybetween 0 and 1 with closing to 0 indicating severe violation of IID and closing to 1 high confidence of IID. This measure not only indicate whether to reject the IID assumption as hypothesis test but also shows the strength of the IID ofobservations and therefore is more informative and could potentially be more useful in practice. For our aims of developing novel IID test methods, at least the following two potential methodologies could be carried out and deserve more research. It would be possible to develop new IID test methods using (1): the marginal like-1lihood ratio framework, in which marginal likelihoods reflecting the probability of different models (both models assumed observations IID and non-IID) generating observations are compared. Subsequently, the ratios of the probability of generating of observations of the models assuming IID and that of models assuming non-IID can indicate the strength of the IID of observations. (2):permutation, where we can compare the observations and permuted observations. Roughly speaking, if we regard the permuted observations as generated by some (unknown) IID random variables, we can use certain distance measurements to measure the distances of observations and permuted observations. Intuitively, the smaller the distance is, the more similar the observations are to the permuted observations and therefore the more likely the observations are IID. Two potential methodologies for developing our methods have been briefly discussed and needed more research. Once completed, our methods can be valuable for many statistic and machine learning models which make the IIDassumption of observations and our methods can be used to test this assumption. Compared with most existing methods, which may not be able to reject many non-IID cases and could potentially require a large number of observations toreject, our methods of IID test could be valuable in practice by rejecting IID assumptions under more non-IID cases using fewer observations and produce more informative test results.