The Practical Guide
To Truncated Probability Distributions
Author: Claudio San Roman Denegri
First Edition September 2020
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Introductory Videos: Video 1 Video 2     Brief Example: real case
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CONTENTS |
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Page |
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Preface |
8 |
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I |
Description of Truncated Distributions |
11 |
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1.1 |
Definition |
12 |
1.2 |
Properties |
12 |
1.3 |
Use of Decibels to Graph Probability Curves |
13 |
1.4 |
Example |
13 |
1.5 |
Industry Context |
16 |
1.6 |
Notes about Censored Data |
18 |
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II |
Analysis of Truncated Distributions |
19 |
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2.1 |
The Mean |
21 |
2.2 |
The Median |
22 |
2.3 |
The Mode |
23 |
2.4 |
The Variance |
24 |
2.5 |
The Asymmetry |
25 |
2.6 |
The Kurtosis |
27 |
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III |
Parameter Estimation of Truncated Distributions |
29 |
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3.1 |
Engineering Methods |
30 |
3.1.1 |
The Artifice Mirror with Mode |
32 |
3.1.2 |
The Artifice Optimal Mirror |
35 |
3.1.3 |
Test of the Artifice Optimal Mirror |
40 |
3.1.4 |
Techniques for Double Truncated Distributions |
47 |
3.1.4.1 |
Estimating the Tail of the Generalized Gaussian Distribution using Polynomials |
48 |
3.1.4.2 |
Estimating β of the Generalized Gaussian Distribution using Shape of Head |
49 |
3.1.4.3 |
Double Truncated Distribution Example |
50 |
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3.2 |
“Pure” Mathematic Methods |
54 |
3.2.1 |
The Truncated Generalized Gaussian |
54 |
3.2.2 |
The Double Truncated Generalized Gaussian |
54 |
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IV |
Appendix |
55 |
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A1 |
The Generalized Gaussian Distribution |
55 |
A2 |
Non-Truncated Distribution Methods to Estimate Parameters |
58 |
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A2.1 Moment Method |
59 |
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A2.2 Maximum Likelihood Estimation Method |
60 |
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A2.3 The Global Convergence Method |
62 |
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A2.4 Minimization of The Squared Error |
63 |
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V |
References |
64 |
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PREFACE
This book contains practical knowledge about truncated distributions. It applies mainly to symmetrical distributions, with special emphasis on the Truncated Generalized Gaussian Distribution. This book and the “pure” mathematical books related to truncated distributions complement each other.
Truncated distributions were complex subjects before. Now, with the knowledge provided by this book, these discontinuous functions are easy to analyze and solve. This book should be read by students and professionals that deal with statistics like economists, biologists, engineers, mathematicians, doctors, sociologists, astronomers, agronomists, psychologists, educators, etc.
To understand this book, it is required to complete university statistic courses.
The first chapter is about the description of truncated distributions, the reader will learn what the truncated distributions are. Analysis of truncated distribution are made in the second chapter, it contains information to understand how parameters change with different levels of truncation. The third chapter is dedicated entirely on how to estimate parameters, and the test of these tools. Also, there is a section about double truncated distributions. The appendix is located at the fourth section, it provides important tools to understand the first three chapters. And finally, the references are included in the fifth section.
A part of this book was written during the difficult days of coronavirus, I am glad that I could finish the first edition and share it with you. I wish the entire community a good health.
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Review 1:
I read this book in a couple of weeks and I found it very interesting. This book has a lot of graphs and practical math, instead of large and complex equations. I learned that the methods presented in this book enable many conventional methods of non-truncated distributions to be applied to truncated distributions, I believe this is a major contribution to mathematics. I strongly recommend this book.
Andres. November 2020
Review 2:
This book showed me the idea of using decibels (or logarithmic scale) to work with probability curves, it was for me a new idea, I could better understand the tails of distributions and compare with other curves. I also liked the use of Monte Carlo method to analyze truncated distributions and to test methods. This book is a key source for students and professionals.
Gabriel. December 2020
Review 3:
Now I understand. When you have data limited by time, capacity, direction, sensor-range or others, you get truncated curves. In my opinion, this book gives excellent tools for this topic.
Aristides. February 2021
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