الفهرس 
introduction and motivationOnly 14 pages are availabe for public view
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Abstract
The statistical literature contains many new G families of continuous distributions which have been generated either by merging (compounding) common G families of continuous distributions or by adding one or more parameters to the G family. These novel G families have been employed for modeling real-life datasets in many applied studies such as insurance, engineering, econometrics, biology, medicine, statistical forecasting, and environmental sciences.
Three new compound families of distributions are proposed in this thesis. These new families are:
The Poisson inverse generalized Rayleigh-G (PIGR-G) family
The quasi Poisson Topp Leone generated-G (QPTLG-G) family
The Poisson exponentiated Weibull-G (PEW-G) family.
These families extend some well-known families in the literature. Some of the statistical properties including the quantiles, moments, incomplete moments, moment of the residual life and moment generating function are presented.
Based on the exponential model as a baseline, the density PIGR-G family can be ”asymmetric right skewed with heavy tail”, ”symmetric” and bimodal density with different shapes. The hazard function of the PIGR-G family can be ”upside-down-constant”, ”increasing-constant”, ”upside-down-increasing”, ”increasing”, ”decreasing” and ”constant”. The skewness of the PIGR-G family ∈ (-0.18234,∞). Furthermore, the spread for its kurtosis is ranging from nearly 2.987858 to nearly ∞.
Based on the Lomax model as a baseline, the density QPTLG-G family can be ”asymmetric right skewed” with different shapes and ”symmetric” density. The hazard function of the QPTLG-G family can be ”upside-down-upside-down”, ”monotonically decreasing”, ”J shape”, ”upside-down ”. The skewness of the QPTLG-G family ∈ (-0.6276,∞). Furthermore, the spread for its kurtosis is ranging from nearly 3.7575 to nearly ∞.
Based on the Lomax model as a baseline, the density PEW-G family can be ”asymmetric and right skewed ” with different shapes, ”symmetric” and ”bimodal”. The hazard function of the PEW-G family can be ”upside-down-constant”, ”increasing”, ”decreasing”, ”upside-down-upside-down- increasing”, upside-down, ”increasing-constant”, ” bathtub (U shape)” and ”upside-down- bathtub (upside-down-U)”. The skewness of the PEW-G family ∈ (-0.997,2886.6). Furthermore, the spread for its kurtosis is ranging from nearly 3.036296 to nearly +∞.
The PIGR-G family is much better than the odd Lindley-G family, Marshall-Olkin-G family, the Burr-Hatke-G family, generalized Marshall-Olkin-G family, Beta-G family, Marshall-Olkin Kumaraswamy-G family, Kumaraswamy-G family, the Burr X -G family and Kumaraswamy Marshall-Olkin-G family, based on the exponential model as a baseline.
The QPTLG-G family is better than the odd log-logistic-G family, the generalized mixture-G family, the transmuted Topp-Leone-G family, the Gamma-G family, the Burr-Hatke-G family, the exponentiated-G family, the Kumaraswamy-G family and the proportional reversed hazard rate-G family, based on the Lomax model as a baseline.
The PEW-G family is better than Quasi Poisson Topp Leone generated-G family, Weibull generalized-G family, Marshall-Olkin Lehmann-G family, the generalized mixture-G family, Odd log-logistic-G family, Burr-Hatke -G family, Transmuted Topp-Leone-G family, Gamma-G, Kumaraswamy-G family, Mcdonald-G, Beta-G family, exponentiated-G family, proportional reversed hazard rate -G family, based on the Lomax model as a baseline.
A simulation study is performed to assess the performance of the maximum likelihood method for estimating the unknown parameters. The maximum likelihood estimates are assessed in terms of the average values, the bias and the mean square errors.
According to the simulation results, the average values of the maximum likelihood estimates tend to the initial values, the bias tends to zero and the mean square errors decreases and tend to zero as n→∞. It can be concluded the maximum likelihood estimates and their asymptotic results can be adopted for estimating the model parameters. Applications to real-life data sets are presented for illustrating the superiority of the new families.
Some new bivariate versions of these families are proposed using Farlie Gumbel Morgenstern copula, modified FGM copula, Clayton copula, Renyi’s entropy copula and Ali-Mikhail-Haq copula.
The statistical literature contains many new G families of continuous distributions which have been generated either by merging (compounding) common G families of continuous distributions or by adding one or more parameters to the G family. These novel G families have been employed for modeling real-life datasets in many applied studies such as insurance, engineering, econometrics, biology, medicine, statistical forecasting, and environmental sciences.
Three new compound families of distributions are proposed in this thesis. These new families are:
The Poisson inverse generalized Rayleigh-G (PIGR-G) family
The quasi Poisson Topp Leone generated-G (QPTLG-G) family
The Poisson exponentiated Weibull-G (PEW-G) family.
These families extend some well-known families in the literature. Some of the statistical properties including the quantiles, moments, incomplete moments, moment of the residual life and moment generating function are presented.
Based on the exponential model as a baseline, the density PIGR-G family can be ”asymmetric right skewed with heavy tail”, ”symmetric” and bimodal density with different shapes. The hazard function of the PIGR-G family can be ”upside-down-constant”, ”increasing-constant”, ”upside-down-increasing”, ”increasing”, ”decreasing” and ”constant”. The skewness of the PIGR-G family ∈ (-0.18234,∞). Furthermore, the spread for its kurtosis is ranging from nearly 2.987858 to nearly ∞.
Based on the Lomax model as a baseline, the density QPTLG-G family can be ”asymmetric right skewed” with different shapes and ”symmetric” density. The hazard function of the QPTLG-G family can be ”upside-down-upside-down”, ”monotonically decreasing”, ”J shape”, ”upside-down ”. The skewness of the QPTLG-G family ∈ (-0.6276,∞). Furthermore, the spread for its kurtosis is ranging from nearly 3.7575 to nearly ∞.
Based on the Lomax model as a baseline, the density PEW-G family can be ”asymmetric and right skewed ” with different shapes, ”symmetric” and ”bimodal”. The hazard function of the PEW-G family can be ”upside-down-constant”, ”increasing”, ”decreasing”, ”upside-down-upside-down- increasing”, upside-down, ”increasing-constant”, ” bathtub (U shape)” and ”upside-down- bathtub (upside-down-U)”. The skewness of the PEW-G family ∈ (-0.997,2886.6). Furthermore, the spread for its kurtosis is ranging from nearly 3.036296 to nearly +∞.
The PIGR-G family is much better than the odd Lindley-G family, Marshall-Olkin-G family, the Burr-Hatke-G family, generalized Marshall-Olkin-G family, Beta-G family, Marshall-Olkin Kumaraswamy-G family, Kumaraswamy-G family, the Burr X -G family and Kumaraswamy Marshall-Olkin-G family, based on the exponential model as a baseline.
The QPTLG-G family is better than the odd log-logistic-G family, the generalized mixture-G family, the transmuted Topp-Leone-G family, the Gamma-G family, the Burr-Hatke-G family, the exponentiated-G family, the Kumaraswamy-G family and the proportional reversed hazard rate-G family, based on the Lomax model as a baseline.
The PEW-G family is better than Quasi Poisson Topp Leone generated-G family, Weibull generalized-G family, Marshall-Olkin Lehmann-G family, the generalized mixture-G family, Odd log-logistic-G family, Burr-Hatke -G family, Transmuted Topp-Leone-G family, Gamma-G, Kumaraswamy-G family, Mcdonald-G, Beta-G family, exponentiated-G family, proportional reversed hazard rate -G family, based on the Lomax model as a baseline.
A simulation study is performed to assess the performance of the maximum likelihood method for estimating the unknown parameters. The maximum likelihood estimates are assessed in terms of the average values, the bias and the mean square errors.
According to the simulation results, the average values of the maximum likelihood estimates tend to the initial values, the bias tends to zero and the mean square errors decreases and tend to zero as n→∞. It can be concluded the maximum likelihood estimates and their asymptotic results can be adopted for estimating the model parameters. Applications to real-life data sets are presented for illustrating the superiority of the new families.
Some new bivariate versions of these families are proposed using Farlie Gumbel Morgenstern copula, modified FGM copula, Clayton copula, Renyi’s entropy copula and Ali-Mikhail-Haq copula.