Gilchrist W. Statistical modelling with quantile functions

Boca Raton: Chapman & Hall/CRC, 2000. — 317 p.Galton used quantiles more than a hundred years ago in describing data. Tukey and Parzen used them in the 60s and 70s in describing populations. Since then, the authors of many papers, both theoretical and practical, have used various aspects of quantiles in their work. Until now, however, no one put all the ideas together to form what turns out to be a general approach to statistics.Statistical Modelling with Quantile Functions does just that. It systematically examines the entire process of statistical modelling, starting with using the quantile function to define continuous distributions. The author shows that by using this approach, it becomes possible to develop complex distributional models from simple components. A modelling kit can be developed that applies to the whole model - deterministic and stochastic components - and this kit operates by adding, multiplying, and transforming distributions rather than data.Statistical Modelling with Quantile Functions adds a new dimension to the practice of statistical modelling that will be of value to anyone faced with analyzing data. Not intended to replace classical approaches but to supplement them, it will make some of the traditional topics easier and clearer, and help readers build and investigate models for their own practical statistical problems.Statistical Modelling with Quantile FunctionsList of Figures
List of TablesSome Useful Mathematical Results
Further Studies in the Method of Maximum Likelihood
Bivariate TransformationsSample properties
The cumulative distribution function
The probability density function
The quantile function
The quantile density function
A modelling kit for distributions
Modelling with quantile functions
Simple properties of population quantile functions
Elementary model components
Choosing a model
Fitting a model
Applications
ConclusionsQuantiles and moments
The five-number summary and measures of spread
Measures of skewness
Other measures of shape
Bibliographic notes
Problems
Defining the population
The reflection rule
The intermediate rule
The standardization rule
The Q-transformation rule
The p-transformation rule
The addition rule for quantile density functions
Population moments
Quantile measures of distributional form
L-moments
Probability-weighted moments
Problems
The process of statistical modelling
Order statistics
The order statistics distribution rule
The median rankit rule
Transformation
Simulation
Approximation
Correlation
Tailweight
The TW(p) function
Limiting distributions
Quantile models and generating models
Smoothing
Evaluating linear moments
Problems
The uniform distribution
The reciprocal uniform distribution
The exponential distribution
The power distribution
The Pareto distribution
The extreme type distribution and the Cauchy distribution
The sine distribution
The normal and log-normal distributions
Problems
Position and scale change — generalizing
Using addition — linear and semi-linear models
Using multiplication
Using Q-transformations
Using p-transformations
Distributions of largest and smallest observations
Conditional probabilities
Truncated distributions
Conceptual model building
Problems
The logistic distributions
The lambda distributions
The three-parameter, symmetric, Tukey-lambda distribution
The four-parameter lambda
The generalized lambda
The five-parameter lambda
Extreme value distributions
The Burr family of distributions
Sampling distributionsThe geometric distribution
The binomial distribution
Problems
The context
Numerical summaries
Skewness
Interpretation
Starting points
Identification plots
Identification plots for common distributions
Using p-transformations
Identification involving component models
Sequential model building
Problems
Matching methods
Methods based on lack of fit criteria
The method of maximum likelihood
Discounted estimation
Intervals and regions
Initial estimates
ProblemsDensity probability plots
Residual plots
Unit exponential spacing control chart
Application validation
Testing the model
Testing using the uniform distribution
Tests based on the criteria of fit
Problems
Definitions
p-Hazards
Hydrology
Capability
Control charts
Problems
Approaches to regression modelling
Quantile autoregression models
Semi-linear and non-linear regression quantile functions
ProblemsThe circular distributions
The Weibull circular distribution
The generalized Pareto circular distribution
The elliptical family of distributions
Additive models
Marginal/conditional models
Estimation
Problems
A Postscript
Indefinite Integrals