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SKEW-NORMAL AND SKEW-T DISTRIBUTIONS BASED ON A UNIFIED SKEW-NORMAL DISTRIBUTION

Name: SKEW-NORMAL AND SKEW-T DISTRIBUTIONS BASED ON A UNIFIED SKEW-NORMAL DISTRIBUTION
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Ph.D. DISSERTATION IN

MATHEMATICAL STATISTICS

SKEW-NORMAL AND SKEW-T DISTRIBUTIONS BASED ON A UNIFIED SKEW-NORMAL DISTRIBUTION

In this thesis, a skew and unimodal or bimodal (uni/bi-modal) extension of the student-t distribution is considered. This model is more flexible and has wider ranges of skewness and kurtosis than that of the other asymmetric distributions available in literature. As an special case, a skew and uni/bi-modal extension of the cauchy distribution is considered. Some representation for the proposed models are given. With a simulation and real data study, applicability of the proposed models are illustrated. An extended multivariate skew-normal(SN) model is introduced by extending the univariate normal-skew-normal distributions, which is proposed by Gómez et al. (2013), to the multivariate case. This new class of distributions is presented as a shape mixture of the multivariate extended skew-normal distribution. The extend technique leads to deriving stochastic representations for the proposed distribution. Some basic properties of the new family of distributions are given. Computational techniques using EM-type algorithms are employed for iteratively computing maximum likelihood estimates. An application of the new distribution is illustrated using some real data sets. Further the truncated version of the multivariate unified skew-normal (SUN) distributions is introduced. This method of truncation is applied to derive the joint distribution of consecutive order statistics and some conditional distributions from an exchangeable -dimensional normal random vector. These results are used to determine some measures in the reliability theory such as the mean past life (MPL) and mean residual life (MRL) functions.

Keywords: Skew-normal distribution; Stochastic representation; Unified skew-normal distribution; Order statistics.

دسته: زبان خارجه, زبان خارجه برچسب: Keywords: Skew-normal distribution; Stochastic representation; Unified skew-normal distribution; Order statistics.

توضیحات

Contents Page

List of Figures……………………………………………………………………………………………………….. xi

Chapter 1: Introduction…………………………………………………………………………………………. 2

1.1 Literature review…………………………………………………………………………………………….. 2

1.2 Preliminaries…………………………………………………………………………………………………… 7

2.1 Skew-symmetric distributions…………………………………………………………………………. 7

2.2 Univariate skew-normal distribution……………………………………………………………….. 8

2.3 Some extensions of the skew-symmetric distributions………………………………………. 9

2.3.1 Normal-skew-normal distribution……………………………………………………….. 10

2.3.2 Skew-flexible distribution…………………………………………………………………… 11

2.4 Multivariate skew-normal distribution…………………………………………………………… 13

2.4.1 Mardia’s measures of skewness and kurtosis……………………………………….. 14

2.5 Multivariate extended skew-normal distribution…………………………………………….. 15

2.6 Multivariate Unified skew-normal distribution……………………………………………….. 16

2.6.1 Marginal and conditional distributions………………………………………………… 17

2.6.2 Multivariate singular unified skew-normal distribution…………………………. 18

1.3 Aims……………………………………………………………………………………………………………… 21

Chapter 2: An uni/bi-modal class of skew-t distributions………………………………………. 24

2.1 Introduction………………………………………………………………………………………………….. 24

2.2 Skew-flexible-t-normal distribution……………………………………………………………….. 25

2.1 Skew-flexible-cauchy-normal distribution………………………………………………………. 27

2.2 Uni/bi-modality property……………………………………………………………………………… 27

2.3 Stochastic Representation and data generation……………………………………………… 28

2.3 Moments………………………………………………………………………………………………………. 29

2.4 Inference………………………………………………………………………………………………………. 31

4.1 Fisher information matrix…………………………………………………………………………….. 33

2.5 Numerical illustration……………………………………………………………………………………. 38

2.6 Illustrations with real data sets………………………………………………………………………. 41

Chapter 3: A new multivariate skew-normal distribution………………………………………. 48

3.1 Introduction………………………………………………………………………………………………….. 48

3.2 The univariate case………………………………………………………………………………………. 52

3.3 The multivariate case……………………………………………………………………………………. 53

3.1 Stochastic representations……………………………………………………………………………. 58

3.4 Some features of the multivariate case………………………………………………………….. 59

4.1 Marginals…………………………………………………………………………………………………… 60

4.2 Linear transforms……………………………………………………………………………………….. 61

4.3 Independence……………………………………………………………………………………………… 62

4.4 Random data generation……………………………………………………………………………… 64

4.5 An extension……………………………………………………………………………………………….. 65

4.6 Conditional distributions………………………………………………………………………………. 65

4.7 Quadratic forms………………………………………………………………………………………….. 67

3.5 Inference……………………………………………………………………………………………………… 69

5.1 EM-type algorithm………………………………………………………………………………………. 70

5.2 Large sample properties of the estimators……………………………………………………. 75

5.3 Two numerical examples…………………………………………………………………………….. 76

5.4 Real data study………………………………………………………………………………………….. 77

Chapter 4: Truncated SUN distributions and their application………………………………. 83

4.1 Introduction………………………………………………………………………………………………….. 83

4.2 Truncated SUN distributions………………………………………………………………………….. 84

2.1 Special univariate cases……………………………………………………………………………….. 85

2.2 Special multivariate cases……………………………………………………………………………. 88

4.3 Application to order statistics………………………………………………………………………… 93

3.1 Exchangeable case………………………………………………………………………………………. 95

4.4 Application to reliability……………………………………………………………………………… 101

References…………………………………………………………………………………………………………. 107

SKEW-NORMAL AND SKEW-T DISTRIBUTIONS BASED ON A UNIFIED SKEW-NORMAL DISTRIBUTION

تعدا135 صفحه در فایل word

Ph.D. DISSERTATION IN

MATHEMATICAL STATISTICS

SKEW-NORMAL AND SKEW-T DISTRIBUTIONS BASED ON A UNIFIED SKEW-NORMAL DISTRIBUTION

Keywords: Skew-normal distribution; Stochastic representation; Unified skew-normal distribution; Order statistics.

Contents Page

List of Figures……………………………………………………………………………………………………….. xi

Chapter 1: Introduction…………………………………………………………………………………………. 2

1.1 Literature review…………………………………………………………………………………………….. 2

1.2 Preliminaries…………………………………………………………………………………………………… 7

2.1 Skew-symmetric distributions…………………………………………………………………………. 7

2.2 Univariate skew-normal distribution……………………………………………………………….. 8

2.3 Some extensions of the skew-symmetric distributions………………………………………. 9

2.3.1 Normal-skew-normal distribution……………………………………………………….. 10

2.3.2 Skew-flexible distribution…………………………………………………………………… 11

2.4 Multivariate skew-normal distribution…………………………………………………………… 13

2.4.1 Mardia’s measures of skewness and kurtosis……………………………………….. 14

2.5 Multivariate extended skew-normal distribution…………………………………………….. 15

2.6 Multivariate Unified skew-normal distribution……………………………………………….. 16

2.6.1 Marginal and conditional distributions………………………………………………… 17

2.6.2 Multivariate singular unified skew-normal distribution…………………………. 18

1.3 Aims……………………………………………………………………………………………………………… 21

Chapter 2: An uni/bi-modal class of skew-t distributions………………………………………. 24

2.1 Introduction………………………………………………………………………………………………….. 24

2.2 Skew-flexible-t-normal distribution……………………………………………………………….. 25

2.1 Skew-flexible-cauchy-normal distribution………………………………………………………. 27

2.2 Uni/bi-modality property……………………………………………………………………………… 27

2.3 Stochastic Representation and data generation……………………………………………… 28

2.3 Moments………………………………………………………………………………………………………. 29

2.4 Inference………………………………………………………………………………………………………. 31

4.1 Fisher information matrix…………………………………………………………………………….. 33

2.5 Numerical illustration……………………………………………………………………………………. 38

2.6 Illustrations with real data sets………………………………………………………………………. 41

Chapter 3: A new multivariate skew-normal distribution………………………………………. 48

3.1 Introduction………………………………………………………………………………………………….. 48

3.2 The univariate case………………………………………………………………………………………. 52

3.3 The multivariate case……………………………………………………………………………………. 53

3.1 Stochastic representations……………………………………………………………………………. 58

3.4 Some features of the multivariate case………………………………………………………….. 59

4.1 Marginals…………………………………………………………………………………………………… 60

4.2 Linear transforms……………………………………………………………………………………….. 61

4.3 Independence……………………………………………………………………………………………… 62

4.4 Random data generation……………………………………………………………………………… 64

4.5 An extension……………………………………………………………………………………………….. 65

4.6 Conditional distributions………………………………………………………………………………. 65

4.7 Quadratic forms………………………………………………………………………………………….. 67

3.5 Inference……………………………………………………………………………………………………… 69

5.1 EM-type algorithm………………………………………………………………………………………. 70

5.2 Large sample properties of the estimators……………………………………………………. 75

5.3 Two numerical examples…………………………………………………………………………….. 76

5.4 Real data study………………………………………………………………………………………….. 77

Chapter 4: Truncated SUN distributions and their application………………………………. 83

4.1 Introduction………………………………………………………………………………………………….. 83

4.2 Truncated SUN distributions………………………………………………………………………….. 84

2.1 Special univariate cases……………………………………………………………………………….. 85

2.2 Special multivariate cases……………………………………………………………………………. 88

4.3 Application to order statistics………………………………………………………………………… 93

3.1 Exchangeable case………………………………………………………………………………………. 95

4.4 Application to reliability……………………………………………………………………………… 101

References…………………………………………………………………………………………………………. 107

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