Digital PDF Book

The Practical Guide

To Truncated Probability Distributions

Author: Claudio San Roman Denegri

First Edition September 2020

Authenticity (SHA256) : 5f13c4980f43624f6c71c76df899deb0c5e5dd228be7667ed2158527dbd2e389

Price : 20 US Dollars (single-user license)

Introductory Videos: Video 1 Video 2     Brief Example: real case

Cover

Cover

Copyright

 

CONTENTS

 

 

 

Page

 

Preface

8

 

 

 

 I

Description of Truncated Distributions

11

 

 

 

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

 

 

 

II

Analysis of Truncated Distributions

19

 

 

 

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

 

 

 

III

Parameter Estimation of Truncated Distributions

29

 

 

 

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

 

 

 

3.2

“Pure” Mathematic Methods

54

3.2.1

The Truncated Generalized Gaussian

54

3.2.2

The Double Truncated Generalized Gaussian

54

 

 

 

 

 

IV

Appendix

55

 

 

 

A1

The Generalized Gaussian Distribution

55

A2

Non-Truncated Distribution Methods to

Estimate Parameters

58

 

A2.1 Moment Method

59

 

A2.2 Maximum Likelihood Estimation Method

60

 

A2.3 The Global Convergence Method

62

 

A2.4 Minimization of The Squared Error

63

 

 

 

V

References

64

 

 

 

 

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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