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FAMILIES OF DISTRIBUTIONS
As part of the development of continuous univariate distributions, the construction of generalized families of probability distributions has featured prominently in the literature.
These distributional families include, amongst others, the Pearson family (Pearson, 1895), the Burr family (Burr, 1942, 1968, 1973; Burr & Cislak, 1968) and the family of S distributions (Voit, 1992), where each of these families is defined through a differential equation.
Examples of transformation-based families of distributions are the Johnson families, which include transformations to the normal distribution (Johnson, 1949), Laplace distribution (Johnson, 1954) and logistic distribution (Tadikamalla & Johnson, 1982), the kappa family of distributions (Mielke, 1973; Hosking, 1994), which include transformations from the exponential, Gumbel and logistic distributions, and Tukey’s lambda family of distributions (Hastings et al., 1947; Tukey, 1960, 1962; Ramberg & Schmeiser, 1972, 1974; Freimer et al., 1988) obtained through transformation of the uniform distribution.
TUKEY’S LAMBDA FAMILY OF DISTRIBUTIONS
Among the distributional families listed above, Tukey’s lambda family of distributions is unique in two ways. Firstly, this family is defined exclusively through its quantile function, also known as the inverse cumulative distribution function, and is therefore a quantile-based distribution. No closed-form expressions exist for either its probability density function or cumulative distribution function. Consequently exploration and utilization of this family are in some ways more challenging. However, there are also distinctive opportunities available by modeling the family through its quantile function.
Secondly, all the members from any selected type from the lambda family possess a single functional form given by that type’s quantile function. In contrast, different members of the other listed families in Section 1.2 have different functional forms. A single functional form is beneficial in that one does not need to move from one function to another when exploring different distributional shapes.
Collectively the various generalizations of the lambda family are referred to as the generalized lambda distribution (GLD), where each generalization is a distinct type. Unfortunately in the literature each different type of the GLD is incorrectly referred to as a “parameterization” of the GLD. When a distribution has different parameterizations, the parameters of the one parameterization can be transformed to the parameters of the other parameterization and vice versa. For example, the uniform distribution can be parameterized by its minimum and maximum parameters, a and b, or by its location and scale parameters, a = a and b = b − a . With the GLD there exist no simple transformations between different “parameterizations”. Therefore in this thesis the term “type” is used instead of the term “parameterization”.
INITIAL DEVELOPMENT OF THE NEW TYPE OF THE GLD
In 2008, M.T. (Theodor) Loots completed his Honors Essay under my supervision, focusing on the theory and application of L-moments (Loots, 2008). To illustrate the interpretation of L-skewness, Loots utilized the skew logistic distribution (SLD) of Gilchrist (2000), while he used Tukey’s lambda distribution (Hastings et al., 1947; Tukey, 1960, 1962) to demonstrate the interpretation of L-kurtosis. The SLD has a single shape parameter controlling the Lskewness ratio and has a constant value for its L-kurtosis ratio. Tukey’s lambda distribution is symmetric and its shape parameter controls the L-kurtosis ratio. Upon noting furthermore that the expressions for the L-kurtosis ratios of Tukey’s lambda distribution and the GPD are exactly the same, we realized that a new type of the GLD could be constructed from the GPD. In the resulting GPD Type of the GLD, the shape properties of both the SLD and Tukey’s lambda distribution are incorporated. Specifically the L-kurtosis ratio of the GPD Type is explained by just one of its shape parameters. As a result, the GPD Type has closed-form expressions for method of L-moments estimation. Basic results for the GPD Type of the GLD were given in van Staden & Loots (2009a).
Subsequently, exploring the distributional properties of the GPD Type in more depth, it was found that the quantile-based measures of kurtosis of this type also exhibit skewnessinvariance. This led to the development of a general methodology for the construction of families of quantile-based distributions with skewness-invariant kurtosis measures. The GPD Type of the GLD is an example of such a quantile-based distribution.
1. INTRODUCTION
1.1 AIMS AND OBJECTIVES
1.2 FAMILIES OF DISTRIBUTIONS
1.3 TUKEY’S LAMBDA FAMILY OF DISTRIBUTIONS
1.4 INITIAL DEVELOPMENT OF THE NEW TYPE OF THE GLD
1.5 OUTLINE OF THESIS
1.6 CONTRIBUTIONS OF THESIS
2. QUANTILE MODELING
2.1 INTRODUCTION
2.2 CLASSICAL AND QUANTILE STATISTICAL UNIVERSES
2.3 CONSTRUCTION RULES FOR DISTRIBUTIONAL MODEL BUILDING
2.4 MOMENTS
2.5 L-MOMENTS
2.6 QUANTILE-BASED MEASURES OF LOCATION, SPREAD AND SHAPE
2.7 SKEWNESS-INVARIANT KURTOSIS MEASURES
2.8 PROPOSITION: QUANTILE-BASED DISTRIBUTIONS WITH SKEWNESS-INVARIANT KURTOSIS MEASURES
2.9 METHOD OF L-MOMENTS ESTIMATION
2.10 Q-Q PLOTS AND GOODNESS-OF-FIT
2.11 TAIL BEHAVIOR
2.12 CONCLUSION
2.13 DERIVATIONS
3. THE GENERALIZED LAMBDA DISTRIBUTION (GLD)
3.1 INTRODUCTION
3.2 TUKEY’S LAMBDA DISTRIBUTION
3.3 RAMBERG-SCHMEISER TYPE (GLDRS)
3.4 FREIMER-MUDHOLKAR-KOLLIA-LIN TYPE (GLDFMKL)
3.5 PARAMETER SPACE, SUPPORT AND SPECIAL CASES
3.6 MOMENTS
3.7 L-MOMENTS
3.8 QUANTILE-BASED MEASURES OF LOCATION, SPREAD AND SHAPE
3.9 LOCATION AND SPREAD
3.10 DISTRIBUTIONAL SHAPE
3.11 REGIONS OF THE GLDRS
3.12 CLASSES OF THE GLDFMKL
3.13 PARAMETER ESTIMATION
3.14 MONTE CARLO SIMULATION
3.15 APPLICATIONS
3.16 CONCLUSION
3.17 DERIVATIONS
4. A GLD TYPE WITH SKEWNESS-INVARIANT MEASURES OF KURTOSIS
4.1 INTRODUCTION
4.2 GENESIS AND SPECIAL CASES
4.3 PARAMETER SPACE AND SUPPORT
4.4 CLASSES OF THE GLDGPD
4.5 MOMENTS
4.6 L-MOMENTS
4.7 QUANTILE-BASED MEASURES OF LOCATION, SPREAD AND SHAPE
4.8 DISTRIBUTIONAL SHAPE
4.9 METHOD OF L-MOMENTS ESTIMATION
4.10 FITTING OF THE GLDGPD TO DATA
4.11 GLDGPD APPROXIMATION OF DISTRIBUTIONS
4.12 CONCLUSION 182
4.13 DERIVATIONS 182
5. CONCLUSION
5.1 CONSTRUCTION OF QUANTILE-BASED FAMILIES OF DISTRIBUTIONS
5.2 CONSTRUCTION OF THE GPD TYPE OF THE GLD
5.3 THEORETICAL DEVELOPMENT AND PRACTICAL UTILIZATION
5.4 FUTURE RESEARCH
REFERENCES
