## UNLV Theses, Dissertations, Professional Papers, and Capstones

12-1-2022

Thesis

#### Degree Name

Master of Science (MS)

#### Department

Mathematical Sciences

Hokwon Cho

#### Third Committee Member

Dieudonne Phanord

Ashok Singh

#### Abstract

A pivotal quantity is a random variable that is a function of both the random data and the unknown population parameters and whose probability distribution does not depend on any of the unknown parameters. The population parameters here may include nuisance parameters. Historically, pivotal quantities have been used for the construction of test statistics for hypothesis testing of some of these unknown parameters. They have also been used for the construction of confidence intervals for some of these parameters.Generalized pivotal quantities (GPQ) were introduced by Tsui and Weerahandi (1989) and Weerahandi (1993). A GPQ is a function, not only of the random data and the unknown parameters, but also of an independent copy of the random data. In addition, an observed GPQ does not depend on the nuisance parameters (but may depend on the parameters of interest). These GPQ’s can be used to construct generalized confidence intervals and to perform hypothesis tests on a single unknown parameter in cases where the traditional method fails. In this MS thesis, we ﬁrst estimate the parameters of a mixture of two normal distributions using a modified EM algorithm proposed by Ghojogh, Ghojogh, Crowley, and Karray (2020). We then calculate asymptotic confidence intervals after proposing a method for ﬁnding asymptotic standard errors for the estimates of these parameters. Next, we review the theory of classical pivotal quantities and we give some examples. In these examples, using pivotal quantities, we construct conﬁdence sets for single unknown parameters. We next review the theory of generalized pivotal quantities (GPQ’s) introduced by Tsui and Weerahandi (1989) and Weerahandi (1993). Using this generalized pivotal quantities, we compute generalized conﬁdence intervals for the parameters of a log-normal distribution. Finally, in this thesis, we use the modiﬁcations of Nkurunziza and Chen (2011) to the theory of Tsui and Weerahandi (1989) in order to deﬁne modiﬁed generalized pivotal quantities. Using these modiﬁed generalized pivotal quantities, we show how to calculate generalized p-values for tests of hypotheses on single unknown parameters. We hope that, with our work in this thesis, we lay the foundation for constructing generalized conﬁdence intervals and performing generalized tests of hypotheses for the ﬁve parameters of a mixture of two normal distributions using the generalized pivotal quantities method. We leave this subject as a future research topic.

#### Keywords

Asymptotic standard errors; EM algorithm; Generalized pivotal quantity method; Mixture of two univariate normal distributions; Parameter estimation in a mixture distribution

#### Disciplines

Statistics and Probability

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