Statistical modelling with quantile functions

Warren Gilchrist1584881747, 9781584881742, 9781420035919

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.

---

Table of contents :
Statistical Modelling with Quantile Functions……Page 2
Contents……Page 4
List of Figures……Page 10
List of Tables……Page 13
Preface……Page 16
APPENDIX 1: Some Useful Mathematical Results……Page 302
APPENDIX 2: Further Studies in the Method of Maximum Likelihood……Page 304
APPENDIX 3: Bivariate Transformations……Page 308
References……Page 310
1.1 Introduction……Page 18
1.3 Sample properties……Page 20
The cumulative distribution function……Page 26
The probability density function……Page 28
The quantile function……Page 29
The quantile density function……Page 31
1.5 A modelling kit for distributions……Page 32
1.6 Modelling with quantile functions……Page 34
1.7 Simple properties of population quantile functions……Page 41
1.8 Elementary model components……Page 45
1.9 Choosing a model……Page 48
1.10 Fitting a model……Page 51
1.12 Applications……Page 56
1.13 Conclusions……Page 58
2.1 Introduction……Page 60
2.2 Quantiles and moments……Page 61
2.3 The five-number summary and measures of spread……Page 67
2.4 Measures of skewness……Page 70
2.5 Other measures of shape……Page 72
2.6 Bibliographic notes……Page 74
2.7 Problems……Page 76
3.1 Defining the population……Page 77
The reflection rule……Page 78
The intermediate rule……Page 79
The standardization rule……Page 80
The Q-transformation rule……Page 81
The p-transformation rule……Page 82
The addition rule for quantile density functions……Page 83
3.4 Population moments……Page 84
3.5 Quantile measures of distributional form……Page 87
L-moments……Page 90
Probability-weighted moments……Page 93
3.7 Problems……Page 95
4.1 The process of statistical modelling……Page 98
4.2 Order statistics……Page 99
The order statistics distribution rule……Page 101
The median rankit rule……Page 104
4.3 Transformation……Page 105
4.4 Simulation……Page 109
4.5 Approximation……Page 112
4.6 Correlation……Page 115
4.7 Tailweight……Page 117
The TW(p) function……Page 118
Limiting distributions……Page 120
4.8 Quantile models and generating models……Page 121
4.9 Smoothing……Page 123
4.10 Evaluating linear moments……Page 126
4.11 Problems……Page 128
5.2 The uniform distribution……Page 132
5.3 The reciprocal uniform distribution……Page 133
5.4 The exponential distribution……Page 134
5.5 The power distribution……Page 135
5.6 The Pareto distribution……Page 136
5.8 The extreme type 1 distribution and the Cauchy distribution……Page 137
5.9 The sine distribution……Page 139
5.10 The normal and log-normal distributions……Page 140
5.11 Problems……Page 143
6.2 Position and scale change — generalizing……Page 145
6.3 Using addition — linear and semi-linear models……Page 147
6.4 Using multiplication……Page 154
6.5 Using Q-transformations……Page 155
6.6 Using p-transformations……Page 157
6.7 Distributions of largest and smallest observations……Page 159
Conditional probabilities……Page 161
Truncated distributions……Page 162
6.9 Conceptual model building……Page 164
6.10 Problems……Page 166
7.2 The logistic distributions……Page 168
7.3 The lambda distributions……Page 169
The three-parameter, symmetric, Tukey-lambda distribution……Page 170
The four-parameter lambda……Page 171
The generalized lambda……Page 173
The five-parameter lambda……Page 176
7.4 Extreme value distributions……Page 177
7.5 The Burr family of distributions……Page 180
7.6 Sampling distributions……Page 181
Introduction……Page 182
The geometric distribution……Page 183
The binomial distribution……Page 184
7.8 Problems……Page 185
The context……Page 186
Numerical summaries……Page 187
Skewness……Page 188
Interpretation……Page 189
Starting points……Page 190
Identification plots……Page 191
Identification plots for common distributions……Page 192
Using p-transformations……Page 196
8.4 Identification involving component models……Page 197
8.5 Sequential model building……Page 199
8.6 Problems……Page 203
9.2 Matching methods……Page 205
9.3 Methods based on lack of fit criteria……Page 210
9.4 The method of maximum likelihood……Page 219
9.5 Discounted estimation……Page 222
9.6 Intervals and regions……Page 225
9.7 Initial estimates……Page 229
9.8 Problems……Page 230
10.1 Introduction……Page 234
Density probability plots……Page 235
Residual plots……Page 237
Unit exponential spacing control chart……Page 238
10.3 Application validation……Page 239
10.5 Testing the model……Page 241
Testing using the uniform distribution……Page 242
Tests based on the criteria of fit……Page 243
10.6 Problems……Page 246
Definitions……Page 248
p-Hazards……Page 249
11.3 Hydrology……Page 252
Capability……Page 254
Control charts……Page 256
11.5 Problems……Page 258
12.1 Approaches to regression modelling……Page 261
12.2 Quantile autoregression models……Page 270
12.3 Semi-linear and non-linear regression quantile functions……Page 271
12.4 Problems……Page 276
13.1 Introduction……Page 279
The circular distributions……Page 281
The Weibull circular distribution……Page 284
The generalized Pareto circular distribution……Page 285
The elliptical family of distributions……Page 287
13.3 Additive models……Page 289
13.4 Marginal/conditional models……Page 290
13.5 Estimation……Page 291
13.6 Problems……Page 295
CHAPTER 14: A Postscript……Page 297
Indefinite Integrals……Page 303