GENERATION OF CRACK RANDOM NUMBERS USING
THREE-PARAMETER CRACK DISTRIBUTION
A Thesis
Submitted to the Faculty of Graduate Studies and Research
In Partial Fulfillment of the Requirements
for the Degree of
Master of Science
In
Statistics
University of Regina
By
George Kwesi Teye
Regina, Saskatchewan
December 2013
c Copyright 2013: George Kwesi Teye
UNIVERSITY OF REGINA
FACULTY OF GRADUATE STUDIES AND RESEARCH
SUPERVISORY AND EXAMINING COMMITTEE
George Kwesi Teye, candidate for the degree of Master of Science in Statistics, has
presented a thesis titled, Generation of Crack Random Numbers Using ThreeParameter Crack Distribution, in an oral examination held on December 17, 2013. The
following committee members have found the thesis acceptable in form and content, and
that the candidate demonstrated satisfactory knowledge of the subject material.
External Examiner:
Dr. Yiyu Yao, Department of Computer Science
Co-Supervisor:
Dr. Taehan Bae, Department of Mathematics & Statistics
Co-Supervisor:
Dr. Andrei Volodin, Department of Mathematics & Statistics
Committee Member:
Dr. DiangLiang Deng,
Department of Mathematics & Statistics
Chair of Defense:
Dr. Farshid Torabi, Faculty of Engineering & Applied Science
Abstract
The two-parameter crack lifetime distribution had led to the development of the
three-parameter crack lifetime distribution. The developments has been a very useful and effective tool in statistical analysis in conjunction with engineering concept
relating to the fatigue crack which usually happen in the materials used for most
engineering products like lorry, air craft and other heavy machineries. The crack is
said to occur when excessive load or force is exerted beyond the materials ability or
tendency. This three-parameter crack lifetime distribution was developed from the
already-known two-parameter distributions, which was also used for modeling fatigue
failure. This two-parameter lifetime distribution includes: Length-biased Inverse
Gaussian distributions, the Birnbaum-Saunders distribution and also the Gaussian
distribution.
There is a strong relationship between the two-parameter distributions and the
three-parameter lifetime distribution. However, there has not been much literature
on the generation of the crack random numbers that follow the three-parameter crack
i
distribution.
The research briefly explains a few issues in generating random numbers. The
research consider some values of the proposed parameter in this research for the
simulation procedure and make the comparison by the generated histograms using
the Software package R. The values of these parameters are λ = 2, 5, 10, 20, θ = 1,
5, 10 and p = 0.2, 0.4, 0.6. There are two major ways of generating random numbers
classified in this research as direct and in-direct approach. The non-direct approaches
consist of the acceptance-rejection method, and the direct approach comprises of the
convolution procedure, the inversion, the composite, and many other procedures.
Some of the methods will be briefly reviewed in the second chapter of this thesis. The
situation where the direct approach can not be efficiently used in the generation of the
random variable then the Acceptance-Rejection approach is most likely to be used.
The result of the generation procedure in this research is presented in histogram and
compared.
ii
Acknowledgements
I would like to extend my outermost gratitude and appreciation to Dr. Andrei
Volodin and Dr. Taehan-Bae for their guidance, generous help, expert advice, encouragement and endless patience in making this thesis a success.
I would like to express my enormous thanks to my mother Mary Lugu the best
woman in my world, My Father Mr. Eric Laryea, for his motivation and advice ever
since he has been in my life. My Late Biological Father, Ebenezer Teye Amediavor
and my very good friends, Donna Hackman, Faustina Selanyo Hukportie, for their
sincere affection and moral support.
I am appreciative to my colleagues at the Ministry of Central Services especially,
The Director, Todd Godfrey, for giving me the opportunity to learn how Saskatchewan
government operates. Their continuous support and encouragement has been very
important to me.
iii
Notations
In the following, we will use these notations:
BS(λ,θ) - Birnbaum-Saunders distribution with parameters λ,θ;
IG(λ,θ) - Inverse Gaussian distribution with parameters λ,θ;
LB(λ,θ) - Length Biased inverse Gaussian distribution with parameters λ,θ;
P - Probability function;
F - Cumulative distribution function
f - Probability density function
CR(P ,λ,θ) - Crack distribution with parameters p,λ,θ;
U (0, 1) - Uniform distribution random number on (0,1)
τ - Random variable with one-sided BS-distribution
X̄ - The sample mean of the original samples
X - The random variable;
S - Standard deviation;
e.g. - Example;
iv
Contents
Abstract
i
Acknowledgements
iii
Notations
iv
Table of Contents
v
Table of Contents
1
1 INTRODUCTION
2
1.1
The Statement of Problems and Importance of Study . . . . . . . . .
2
1.2
Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.3
Research Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.4
Research Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.4.1
Research Advantages . . . . . . . . . . . . . . . . . . . . . . .
16
1.4.2
Benefits of the Research . . . . . . . . . . . . . . . . . . . . .
16
v
1.5
Criteria to Compare . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 REVIEW OF LITERATURE
16
17
2.1
Review of the Related Literature . . . . . . . . . . . . . . . . . . . .
17
2.2
Birnbaum-Saunders and Inverse Gaussian Distributions . . . . . . . .
19
2.3
Inverse Gaussian Distributions . . . . . . . . . . . . . . . . . . . . . .
21
2.4
Length-Biased Inverse Gaussian Distributions . . . . . . . . . . . . .
22
2.5
Random Birnbaum-Saunders Numbers Generation Procedure . . . . .
23
2.6
Random Inverse Gaussian numbers generation procedure . . . . . . .
24
2.7
Crack Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.8
Generating Random Variable . . . . . . . . . . . . . . . . . . . . . .
25
3 RESEARCH METHODOLOGY
27
3.1
Research Methodology . . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.2
The Probability Density Function . . . . . . . . . . . . . . . . . . . .
28
3.3
The Cumulative Distribution Function . . . . . . . . . . . . . . . . .
29
3.4
The Moment Generating Function . . . . . . . . . . . . . . . . . . . .
29
3.5
The Maximum Likelihood Estimate of Paramters . . . . . . . . . . .
30
3.6
The Desirable Parameter Estimated Properties . . . . . . . . . . . . .
30
3.7
The First Three Cumulants . . . . . . . . . . . . . . . . . . . . . . .
30
3.8
Parameter (p) Estimation . . . . . . . . . . . . . . . . . . . . . . . .
31
vi
3.9
Generating Procedure
. . . . . . . . . . . . . . . . . . . . . . . . . .
4 RESULT OF SIMULATION
31
35
4.1
In the case of λ = 2, θ = 1 and p = 0.2. . . . . . . . . . . . . . . . . .
35
4.2
In the case of λ = 2, θ = 1 and p = 0.4. . . . . . . . . . . . . . . . . .
36
4.3
In the case of λ = 2, θ = 1 and p = 0.6. . . . . . . . . . . . . . . . . .
38
4.4
In the case of λ = 2, θ = 5 and p = 0.2. . . . . . . . . . . . . . . . . .
38
4.5
In the case of λ = 2, θ = 5 and p = 0.4. . . . . . . . . . . . . . . . . .
40
4.6
In the case of λ = 2, θ = 5 and p = 0.6. . . . . . . . . . . . . . . . . .
40
4.7
In the case of λ = 2, θ = 10 and p = 0.2. . . . . . . . . . . . . . . . .
42
4.8
In the case of λ = 2, θ = 10 and p = 0.4. . . . . . . . . . . . . . . . .
42
4.9
In the case of λ = 2, θ = 10 and p = 0.6. . . . . . . . . . . . . . . . .
44
4.10 In the case of λ = 5, θ = 1 and p = 0.2. . . . . . . . . . . . . . . . . .
44
4.11 In case the of λ = 5, θ = 1 and p = 0.4. . . . . . . . . . . . . . . . . .
46
4.12 In case the of λ = 5, θ = 1 and p = 0.6. . . . . . . . . . . . . . . . . .
46
4.13 In case the of λ = 5, θ = 5 and p = 0.2. . . . . . . . . . . . . . . . . .
48
4.14 In case of the λ = 5, θ = 5 and p = 0.4. . . . . . . . . . . . . . . . . .
49
4.15 In case of the λ = 5, θ = 5 and p = 0.6. . . . . . . . . . . . . . . . . .
50
4.16 In case of theλ = 5, θ = 10 and p = 0.2.
. . . . . . . . . . . . . . . .
51
4.17 In the case of λ = 5, θ = 10 and p = 0.4. . . . . . . . . . . . . . . . .
51
4.18 In the case of λ = 5, θ = 10 and p = 0.6. . . . . . . . . . . . . . . . .
53
vii
4.19 In the case of λ = 10, θ = 1 and p = 0.2. . . . . . . . . . . . . . . . .
53
4.20 In the case of λ = 10, θ = 1 and p = 0.4. . . . . . . . . . . . . . . . .
55
4.21 In the case of λ = 10, θ = 1 and p = 0.6. . . . . . . . . . . . . . . . .
55
4.22 In the case of λ = 10, θ = 5 and p = 0.2. . . . . . . . . . . . . . . . .
57
4.23 In the case of λ = 10, θ = 5 and p = 0.4. . . . . . . . . . . . . . . . .
58
4.24 In the case of λ = 10, θ = 5 and p = 0.6. . . . . . . . . . . . . . . . .
59
4.25 In the case of λ = 10, θ = 10 and p = 0.2.
. . . . . . . . . . . . . . .
60
4.26 In the case of λ = 10, θ = 10 and p = 0.4.
. . . . . . . . . . . . . . .
60
4.27 In the case of λ = 10, θ = 10 and p = 0.6.
. . . . . . . . . . . . . . .
62
4.28 In the case of λ = 20, θ = 1 and p = 0.2. . . . . . . . . . . . . . . . .
62
4.29 In the case of λ = 20, θ = 1 and p = 0.4. . . . . . . . . . . . . . . . .
64
4.30 In the case of λ = 20, θ = 1 and p = 0.6. . . . . . . . . . . . . . . . .
64
4.31 In the case of λ = 20, θ = 5 and p = 0.2. . . . . . . . . . . . . . . . .
66
4.32 In the case of λ = 20, θ = 5 and p = 0.4. . . . . . . . . . . . . . . . .
67
4.33 In the case of λ = 20, θ = 5 and p = 0.6. . . . . . . . . . . . . . . . .
68
4.34 In the case of λ = 20, θ = 10 and p = 0.2.
. . . . . . . . . . . . . . .
69
4.35 In the case of λ = 20, θ = 10 and p = 0.4.
. . . . . . . . . . . . . . .
69
4.36 In the case of λ = 20, θ = 10 and p = 0.6.
. . . . . . . . . . . . . . .
71
4.37 PP-PLOT FOR λ = 2, θ = 5 and p = 0.2 . . . . . . . . . . . . . . . .
71
4.38 PP-PLOT FOR λ = 2, θ = 5 and p = 0.4 . . . . . . . . . . . . . . . .
72
viii
5 CONCLUSIONS AND DISCUSSIONS
5.1
74
Research conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
5.1.1
If parameter λ is variable whereas p and θ are fixed. . . . . . .
75
5.1.2
If parameter θ is variable whereas λ and p are fixed. . . . . . .
76
5.1.3
If parameter p is varies whereas λ and θ are fixed. . . . . . . .
76
5.2
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
5.3
Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
Bibliography
79
ix
Contents
1
Chapter 1
INTRODUCTION
1.1
The Statement of Problems and Importance of Study
In this Research, we will begin by looking at the Theory of Reliability and it applications in Finance, Engineering Environment, Statistics, Economics and Physics.
Lifetime distribution is one of the important distributions to note in the theory of
reliability. Most practical problem and applications are widely explained by lifetime distributions. The information carried by the lifetime distribution is used by
practitioners to make predictions and also forecast the expected lifetime estimate for
machines so they can be replaced before the time of failure. This will prevent some
major losses such as financial, life loss and other economic losses. Since the machinery
and the systems are very expensive, it is important for the practitioners to be able to
have an idea when they expect lifetime failure and make sure that the machine parts
can be replaced or services in order to prevent lost in productivity when machine part
2
break down unexpectedly.
If the manufacturer or industry changes them earlier before its expected failure
time, then it saves them an unexpected cost that will be incurred if machines turn
to get damaged before their estimated time of failures. This enables the industry to
make a provision and also the necessary arrangement towards this failure times. It
enables practitioners by giving them an idea on what the estimated failure lifetime
of the machine. This helps them prevent the decrease in production that can affect
total revenue and industries finance.
It is very important for the practitioners to know the life times of their systems to
make the necessary arrangement towards the risk ahead if there should be a system
failure and all consequences it may come along with. This gives us the interest to emphasis on some useful lifetime distributions. In this research, we will study the family
of the new lifetime distribution. Additionally, the various parameters corresponding
to the machine element or what is known to occur as a crack in materials used for
making the machinery. The examples below gives a brief idea about fatigue cracks in
materials used for the machines and systems.
Example of Lorry Steering Arm Failure
The ability for all materials to crack as a result of failure beyond it static strength
is a common phenomenon in materials used for machines such as lorry and this
example can be seen in the figure below, which is Figure 1.1. This crack is common
3
in all materials for machinery and this is why it is important to study and know
about the static strength of these metals so as to put cautions beyond which the
materials will fail. This resulting cracks leads to various breakdowns in machines
such as lorry and the lorry steering arm failure is a good example to look at in
terms of material failure. These programs will enable the calculation of the fatigue
life of the lorry steering arm under repeated loading or stress beyond the tendency
it could withstand. This program does the calculation on the use of lower bound,
values derived empirically for fatigue endurance static strength. If the user and the
manufacturer have specific information about the fatigue endurance and strength of
the lorry steering arm to be analyzed, then they can use it as a preference to their
program review.
Figure 1.1: Fatigue failure of the bolt
Source: “http://materials.open.ac.uk/mem/mem_mf1.htm"
Example of Brake Caliper Bolt Failure Considering the situation of the failure in
the brake caliper shown in Figure 1.2, in this image to the right is the front brake
4
assembly of a bicycle which broke off and lead to a severe injury to the cyclist, this
was due to the inability of the cyclist to apply the brake due to failure in of the brake
caliper since improper maintenance enhanced the bolt of the brake to loosen causing
it to crack. The second picture enables us to see the magnification of the bolt that
had a crack and broke off making the assembly loosen and leading to the braking of
the bicycle impossible when the cyclist was in motion being a loading factor. This is
an example of life being at risk and injury being occurred due to a crack in material.
Figure 1.2 Fatigue failure of Brake Caliper Bolt
Figure 1.2: Fatigue failure of Brake Caliper Bolt
Source ‘‘http://materials.open.ac.uk/mem/mem_ccf2.htm"
Example of Damage to Car Engine Fatigue Failure
Considering the situation of a failure of the car engine of a saloon car demonstrated
in the Figure 1.3 below, which failed due to pure bending. Assuming the manufacturer
want to make precise prediction how reliable the crankshafts will be after one year of
service at a specific stress level.
5
Figure 1.3: Fatigue failure of the Crank Shaft
Source ‘‘http://materials.open.ac.uk/mem/mem_mf5.htm"
Example of Material Testing
There has been a laboratory set for the goals of providing static, fracture and also
fatigue testing over a time period now. In Figure 1.4 we see a laboratory setup for the
materials that are being tested for their fatigue strength. The idea of the material
testing laboratory is to combine state-of-the-art test equipment, to provide highly
trained professional staff to work in those various laboratories. The combination of
this diverse experience in testing and knowledge of materials sciences help maintains
the facility at the edge of being the leading fatigue life enhancement research.
The varieties of material fatigue include the thermo-dynamic or mechanical fatigue testing, strain gaging, structural mechanics, stress corrosion cracking and many
others.
Materials are used in automotive, manufacturing, production, aerospace, and
6
chemical and oil industries. Materials are used in machinery to improve production and ease work and provide services in a wide range of diverse fields. This led
to the motivation of highly trained fatigue testing experts in all areas to improve on
material quality. This enhanced these experts to come up with the idea of making
materials that are safer, stronger for most successful in making machine parts and
the product. This improvement can be seen in test programs like kidney dialysis machines for human medical needs, military airframe testing spectrum and many more.
The program have been made for materials to be accurate as well as cost effective.
Figure 1.4 Fatigue testing
Figure 1.4: Fatigue testing of materials
Source: ‘‘http://coldwork.com/services/material-testing"
Example of Lorry Spring Failure
Springs like any other material tend to have a combination of bending, torsion and
fatigue cracks that correspond to propagate at right angles to the principal stresses.
7
Hence, we can see the fracture surface of this spring complex even at higher magnification as can be seen in the image shown in Figure 1.5. The morphology is also
influenced by any texture of the material wire used in the manufacturing of this lorry
spring
Figure 1.5 Lorry spring failure
Figure 1.5: Fatigue failure of Lorry spring
Source: ‘‘http://materials.open.ac.uk/mem/mem_mf2.htm"
Example of cracks in a high cost machine element Air Plane
This example can be viewed in Figure 1.6 which demonstrates a high cost machine
with a fatigue crack causing an accident with huge lost of lives. For the past couple
of years cracking in aircraft structures was a major newsworthy topic. And by far
great improvement has been made in the material for airplanes
Example of Fatigue Failure in Pipeline Test
In addition, development made by testing fatigue failure, the fatigue properties
of the material is measured for different pipeline steels at full characteristics and
8
Figure 1.6: The calamity of this machine caused finance and lives lost
Source: ‘‘http://www.metallurgist.com/html/MetalFatigue.htm"
thickness to provide data for making prediction in the performance of the pipeline
as we can see in Figure 1.7 The Specimens are machines obtained directly from the
pipeline based on its longitudinal axis. The fatigue behaviors of pipeline has been
researched and investigated for the strength and weakness as well as it characteristics.
Figure 1.7 To the Left is Fatigue test of the pipeline section and to the right is a sample
of an old pipe
The Crack Lifetime distribution is a very useful statistical tool for engineering
applications. As special cases it contains such well-known parameters in distributions such as Length Biased Inverse Gaussian, Birnbaum-Saunders Gaussian, Inverse
Gaussian. Birnbaum-Saunders.
There are several distributions used to describe the lifetime in the framework of
failure models, as can be seen in many text books since developments of lifetime
9
Figure 1.7: To the Left is Fatigue test of pipeline section and to the right is a
sample of an old pipe
Source: ‘‘http://www.nist.gov/mml/materials_reliability/structural_
materials/pipeline-safety.cfm”
distribution. Some of which are Binomial, Geometric, Exponential, Poisson, Gamma,
Weibull, Normal, Lognormal, Birnbaum-Saunders, Inverse Gaussian distribution, and
etc. In this research, the properties of one new family of three-parameter lifetime
distribution are proposed. This distribution gives a detailed explanation that will
help the development in the crack fatigue failure in physics and engineering. It
is significant to mention that these new three-parameter family of distribution, is
known as Crack Lifetime distribution (CR), contains already known two-parameter
known lifetime distributions, i.e. Inverse Gaussian distribution, Birnbaum-Saunders
distribution and Length Biased Inverse Gaussian distribution.
Beginning with a brief literature on what research has been made in relation to this
three parameter lifetime distribution. Birnbaum and Saunders (1969a), introduced
the “two-parameter distribution” as a failure lifetime distribution caused under the
10
cyclic loading for fatigue failure to occur. This distribution has been a good development and very useful for models such as the reliability theory in a situation of
fatigue failure when we consider the development in cracks that are due to excess
force beyond the material strength.
In the years 1985 and 1986 Desmond came up with a more useful and general
derivation in terms of a biological model, justified and strengthened it for the use
of this distribution. The consideration of the renewal theory was what led to the
derivation of the amount of required cyclical load to influence a fatigue crack under
extensive force to exceed a critical value. Birnbaum and Saunders(1969b) made a
comprehensive theoretical and practical review of the family of distribution fitting to
the solution of the problem is due to the development of the crack.
The Inverse Gaussian usefulness in statistics was attended by various authors and
researchers 18 years ago. There have been several contemporary statistical procedures
used in data analysis, derived and has been extensively involved by the use of the
inverse Gaussian distribution and its significance statistically. Even though the name
turned to be “inverse Gaussian” it turns to be misleading because it is only inverse
in the sense of which they both appear to be Brownian distribution.
The distance traveled by a particle at a specific time or a fixed time period in
Brownian motion is described by the Gaussian distribution whiles the distribution of
the time required for a Brownian motion to reach a specific positive point.
11
The analogies between the statistical properties corresponding to Inverse Gaussian
and the Gaussian distribution with summary statistics and properties reviewed in
Chikara and Folks (1989) and was done by Johnson, Kotz and Balakrishnan (1995).
In theoretical physics Tweedie (1956) extended some results to find the inverse
link that exist between the generated cumulant required for the path to be covered.
The study was made and was published in the detail of the study with the new name
Inverse Gaussian time distribution linked to the Brownian motion.
The study of reliability as we looked at earlier on in our and life testing problem
is very relevant as a result of the Gaussian distribution. This was known to be the
first distribution to be in the passage time in terms of Brownian motion. However,in
reliability theorem, the Inverse Gaussian distribution has been very useful and has
been widely applied in other related works. The family of highly skewed normal distribution can also be presented as a characteristic of the Inverse Gaussian distribution
It is easier to also analyze data statistically based on how skewed a distribution is.
After the review of this two-parameter distribution so far Birnbaum-Saunders is
very useful in the application of crack under an extensive cyclic force.
However inverse Gaussian distribution, been the initial passage distribution for
the Brownian motion. As a characterization of the inverse Gaussian distribution by
Khatri (1962) patrolled the usual characteristics of the normal distribution by it mean
and variance independently, further reflecting this analogy.
12
Amed, Budsaba and Volodin (2008) considered the Birnbaum-Saunders distribution as a one-sided fatigue life distribution. One (two) sided term represents one
(two) side of a plate that applies force until crack has happened. They presented the
point estimation for a one and two-sided Birnbaum-Saunders distribution. Moreover,
the new parameterization of Birnbaum-Saunders was proposed which fits the physical
phenomenon of the crack development.
There was less research conducted in connection with the properties of Crack Lifetime distribution. Panta, Volodin, and Budsaba (2010) presented the method of random number generating and Kumnadee, Volodin and Lisawadi (2010) derived the first
three moments. The Crack Lifetime distribution was not studied in depth.Simulations
are made by Andrei’s work and development mainly for this thesis.
Our research establishes some deeper results especially functions on Crack Lifetime
distribution with rigorous proofs. The relations between Length Biased Inverse Gaussian,Inverse Gaussian distribution distribution, Birnbaum-Saunders distribution, and
Crack Lifetime distribution are shown.
No property of point estimation has been investigated. We study popular point
estimation of this distribution. Since the probability density function of Crack Lifetime distribution is cumbersome and involves three parameters, the estimation is not
easy. For example, formula for the various estimators can not be derived from closed
forms. We first present some theoretical results that provide systems of equations,
13
after that we numerically solve by computer simulations.
Hence, taking note of the importance of this research to study the new family of
lifetime distribution, i.e. the Crack Lifetime distribution that has a relationship to the
three distributions. We also consider proofing rigorously functions and properties on
the Crack Lifetime distribution. The other is to numerically estimate the parameters.
1.2
Research Objectives
The main objectives of this research follows the establishment and investigation
of the computational properties of Crack Lifetime distributions and their application
as far as further studies that could be made in development in the application. We
also look at the generation of crack random distribution by representing the generator
game that can be seen in the next chapter. R software package is used and the output
is displayed after the simulation studies. The results were presented with histogram
chart to make our analysis and comparison in terms of the parameters.
As a fact we look at the various characteristic of the newly generated Crack
Lifetime distributions and how it formation from already known two-parameter Crack
Lifetime distribution. Finally, we develop an equation using the already known twoparameter distribution to be used in the algorithm for the three-parameter Crack
Lifetime distribution.
14
1.3
Research Hypotheses
Random numbers are generated that follow the Crack Lifetime distribution for
all values of the proposed parameters and the method of generation also follows the
already known two-parameter distribution for specific values of parameters. The
shape of the crack distribution differs quite a bit with different values of parameters.
1.4
Research Scope
In the scope of this research looks at the following:
1) This in the chapter two the characteristics of the three-parameter crack random
distribution, such as the mean, variance, and maximum likelihood estimators the
moment generation function and some other characteristics are presented.
2) In chapter four statistical modeling of the random numbers that follow the
three-parameter crack distribution is simulated and results are presented. The following values are assumed for the various parameters which yields the results in the
chapter four, the values are as follow; λ = 2, 5, 10, 20, θ= 1, 5, 10 and p= 0.2, 0.4,
0.6.
3) We run a simulation of the random numbers independently for each fixed values
in the proposed parameter and we run simulations independently and make report
on their corresponding histograms.
15
4) We repeat the simulations 1000 times using R software to construct the histogram.
1.4.1
Research Advantages
1. This research will enable us generate crack random distribution that follows
the three parameter crack distribution
2. The generated crack random numbers that follow the Crack Lifetime distribution as we note will be very useful and has several applications in statistical analysis
in various studies such as engineering concepts of fatigue crack in materials such as
metal under some form of loaded force.
1.4.2
Benefits of the Research
The benefit of this thesis is to show how the Crack Lifetime distribution and its
applications can be used in various practical work and it development in engineering.
1.5
Criteria to Compare
The comparison will be determined by comparing with histograms of the crack
distribution for various values and probability density function
16
Chapter 2
REVIEW OF LITERATURE
2.1
Review of the Related Literature
As we saw in the introduction, this chapter will enable us to review most relevant
development, publications and research on Crack distribution.
In 1969, Birnbaum and Saunders developed and published two-parameter BirnbaumSaunders distribution which was the base for the development which motivated most
research of fatigue failure lifetime distribution which is mostly caused by excessive
loading. The distribution and its development has improved the theory of lifetime
modeling in terms of reliability which had great impact in the engineering and other
fields of interest. It is a development which caught on so much attention due to it
relevance in major aspect of lifetime distribution. Most manufactures and also industries used this lifetime model to determine the life span of their products and used
those as life expectancy model for their materials.
17
In the cause of the development, The comprehensive review, was presented that
is in both practical and theory aspect,it comprised of the fitting of distributions
that belong to the same family and found a solid problem-solving ability to the new
development in crack.
In 1985 and 1986, Desmond derived a more general form based on a biological
model to observe, justify and the strength for the reasons why this distribution is
necessary. This constituted as a result of the theory of reliability being in existence
which was due to the force needed to be exerted in other for a fatigue crack to
occur, and due to the fact that an extinction beyond the crack exceeding the value
of tendency.
The next emergence of the first passage time distribution can be traced to Tweedie
(1941) and Wald (1944). Tweedie, attempting to extend Schrodinger’s results, was
led to notice the inverse main relation between the cumulant function, and the gap
which it will cover at as specific time period. Tweedie (1945) also researched in
the main lineage between the poison distribution, binomial and many other related
distribution. He proposed to call them inverse statistical variants. Later, in 1956,
the name “inverse Gaussian” was used for the initial passage relating to distribution
of the Brownian motion with respect to time. He published a detailed study of the
distribution in 1957 that established many of its important statistical properties.
Folks and Chhikara (1978) made suggestion on the importance of Tweedie’s work, it
18
might be more appropriate to call this distribution as Tweedie’s distribution.
Wald gave a special case of the distribution in 1947. Wald derived it as an approximation of the frequent probability that is as a result of the distribution of the
related sample size. This describes the distribution commonly known to be Wald’s
distribution, particularly in literatures in Russia.
In reference to what has been said, Birnbaum-Saunders distribution appears when
we model the physics of the crack development under periodic or cyclic loading.
Contrary to the Birnbaum-Saunders distribution, the inverse Gaussian appears under
chaotic or a cyclic loading.
2.2
Birnbaum-Saunders and Inverse Gaussian Distributions
Now we consider Birnbaum-Saunders and inverse Gaussian lifetime distributions
from statistical point of view.
Birnbaum and Saunders (1969a) developed in their article entitled, “A new family
of life distributions” which is a two known parameter life distribution due to it relevant
argument that originated from the theory of renewal, through the ideas based on the
number of cycles needed for an excessive load to cause a fatigue crack to develop.
As a fatigue life distribution, the Birnbaum-Saunders model considers a specimen
known to be the material that is orderly exposed to a regularly repeated loading
factor.
19
The cumulative distribution function is for x > 0:
r
FBS (x, θ, λ) = P (τ < x) = 1 − Φ(λ
θ
−
x
r
x
)
θ
(2.1)
r
r
x
θ
= Φ(
−λ
), x > 0, θ > 0, λ > 0
θ
x
(2.2)
This is the unimodal distribution called the Birnbaum-Saunders distribution, and
we denote it BS(λ, θ) .
The probability density function for this distribution is
1
θ 3
x 1
1
fBS (x, θ, λ) = √
[λ( ) 2 + ( ) 2 ]exp− (λ
θ
2
2 2πθ x
r
θ
−
x
r
x 2
)
θ
(2.3)
The shape of the BS-distribution density function is very similar to the probability density function of the the following Gamma distribution mentioned earlier on.
The probability density function is known to be a combination of the inverse Gaussian probability distribution function and Length Biased inverse Gaussian probability
distribution function.
Lisawadi,Volodin, Ahmed and Budsaba (2008) proposed the contemporary parameter of the known Birnbaum-Saunders distribution and strategy that will be required
in estimating it various paramters. Their proposed parameters are important since
they fit the physical phenomena of fatigue cracks. The parameters l > 0 and q > 0
correspond to how thick the material of the machine and the pressure to which the
20
same material of the machine was treated. This study will continue to demonstrate
Crack Lifetime distribution in the form of their parameters.
The relation between the parameters θ, λ and proposed parameters l, q is the
physical interpretation:
l=
1
and q = θ2 λ
θ2
1
θ = √ and λ = lq
l
2.3
Inverse Gaussian Distributions
The Inverse Gaussian family is a versatile one for modeling nonnegative rightskewed data such as the data obtained from reliability and life test studies. This
family shares striking similarities with the Gaussian family. The Inverse family of
distributions, denoted by IG(θ, λ) with the cumulative distribution function known
to be:
r
r
r
r
x
θ
x
θ
−λ
) + e2λ Φ(−
−λ
), x > 0, θ > 0, λ > 0
FIG (x, θ, λ) = Φ(λ
θ
x
θ
x
(2.4)
Corresponding density function
λ
θ 3
1 θ
fIG (x, θ, λ) = √ ( ) 2 exp[− (λ −
2 x
θ 2π x
21
r
x
)], x > 0, θ > 0, λ > 0
θ
(2.5)
We use the notation IG(λ, θ) for the distribution as can be seen in the notation
page of the thesis. The Inverse Gaussian belongs to the class of exponential family. The mean and the corresponding variance of this distribution are θ and σθ3
respectively.
The derivation of the well known Inverse Gaussian distribution was viewed in
the background of the known fatigue failure as this has been of great interest for
researchers over the past years and have made great development in the quality of
material strength over regular period of time. Mostly, as fatigue develops according
to the Wiener process based on a constant drift and diffusion process, and if the
material fails as soon as its accumulated fatigue exceeds the limitation.
2.4
Length-Biased Inverse Gaussian Distributions
The Length Biased Inverse Gaussian distribution denoted by LB with the probability density function to be :
r
r
x 1
1
θ
x 2
1
( ) 2 exp[− (λ
−
) ], x > 0, θ > 0, λ > 0
fLB (x, λ, θ) = √
2
x
θ
2πθ θ
(2.6)
The Birnbaum-Saunders (BS) and inverse Gaussian (IG) distributions are commonly used in practical applications of the reliability theory for modeling a lifetime
for products with failure due to a development of fatigue cracks.
22
2.5
Random Birnbaum-Saunders Numbers Generation Procedure
Here we discuss how to generate random numbers for the one-sided BS-distribution
without using Acceptance-Rejection method.
Let fix λ and θ known to be the shape and scale parameter respectively. If u is
known to be a uniformly distributed random number with mean 0 and variance 1,
then a random number with one-sided BS-distribution can be obtained as, y = θ · x
where x is a positive solution of the quadratic equation.
x−
λ
= Φ−1 (u)
x
If the random variable τ belongs to the one-sided BS-distribution and U is uniformly distributed with mean 0 and variance 1, then we can establish the following
stochastic relation between τ and U
r
r
τ
θ
Φ(
−λ
)=U
θ
τ
Let
r
Y =
τ
, and α = Φ−1 (U )
θ
Obviously Y > 0 and hence the positive root of this equation is considered
α
Y = +
2
r
23
α2
+λ
4
Hence, we recommend the following procedure. Fix the parameters and first we generate a random number α with standard normal distribution N(0,1). After we obtain
a standard normal N(0,1) random number α, we obtain a BS-distributed random
number by the formula
α
τ = θ( +
2
2.6
r
α2
+ λ)2
4
Random Inverse Gaussian numbers generation procedure
The following is an IG(λ, θ) random number generator procedure.
1) Generate a random number a uniform [0,1] and independently a standard normal
number α .
√
θ
2) Calculate u = λθ + [α2 − α4 + 4λα2 ]
2
λ2 θ 2
λθ
then take take IG =u, otherwise IG =
3) If a <
(λθ + u)
u
2.7
Crack Distribution
The Crack distribution has the density function
fCR (x, p, λ, θ) = pfIG (x, λ, θ) + (1 − p)fLB (x, λ, θ), x > 0, θ > 0, λ > 0, 0 ≤ p ≤ 1
(2.7)
For Crack Lifetime distribution CR(p, θ, λ), It is easy to see that:
24
1
CR(0, λ, θ) = LB(λ, θ), CR(1, λ, θ) = IG(λ, θ) and CR( , λ, θ) = BS(λ, θ)
2
2.8
Generating Random Variable
There are several algorithms used in the principle of generating a random variable,
some of which we will explain briefly, and the algorithms are, the inversion method,
convolution method, the Approach of Composite generating method, Approach of
Acceptance-Rejection, the Approach of using Vector for the generation procedure of
random variable. This thesis stress on how to generate the Crack distributed random
variable.
The Approach of Acceptance-Rejection is said to be one of the very effective way
To Generating crack random numbers by Simulation. This method is said to be
efficient in the sense that it uses the mass function of the required distribution in
the generation process after which it uses the proportion based on the principle of
probability to accept the stimulated value. Just like as we see in Hypothesis testing
to see if an argument is statistically significant or not procedure and comprises of
limits for Accepting and Rejecting the null hypothesis based on the interval. Under
this approach we have p to be probability of accepting the stimulated value and then
q to be the probability of rejecting the stimulated value.
The Approach of Inversion that by its self-definition is as a result of transforming
the algorithm inversely to be able to generate the random number. This method is
25
used for in generating the random variable for most binomial distribution
26
Chapter 3
RESEARCH METHODOLOGY
3.1
Research Methodology
In this chapter we will consider the Crack Random Distribution and computation
simulation since not much research has been done in this area.
We will begin with the Three-Parameter Crack Lifetime distribution and its properties. In this aspect we take a look at the characteristics and functions associated
with the Crack Lifetime distribution.
The development made by Ahmed, Budsaba and Lisawadi and Voldin (2008) on
the Birnabaum-Saunders distribution led to the proposal of these new parameters.
They provided a characteristics link between the classical parameters and their proposed parameters. Whiles in this research more emphasis will be based on their
proposed parameters of the Crack Lifetime distribution. This proposed parameters
27
are known to be λ,θ. The λ parameter is the nominal treatment pressure on the machine element whiles θ is said to correspond to how thick the element of the machine
is. The parameter λ and θ are both greater than zero. In addition to the two parameter discussed earlier is the parameter p which is termed as the weight parameter and
its interval is from 0 to 1 inclusive 0 ≤ p ≤ 1. These resulted in the crack random distribution with the parameters λ, θ, p simply known to be the three-parameter Crack
Lifetime distribution. All functions and parameter estimation will be in this known
form of the proposed parameters.
The Crack Lifetime distribution like any other distribution has its cumulative distribution function, probability density function moment generation function, other
estimators such as the well-known maximum likelihood estimation,and also the reciprocal property of the crack random variable.
3.2
The Probability Density Function
The probability density function of the Crack Random distribution is denoted
as fCR (x, p, λ, θ) with x being the random variable whiles p, λ, θ are the proposed
parameters as we looked at earlier on. This came about as a mixture of the probability
density function of both the Length Biased Inverse Gaussian distribution and the
Inverse Gaussian distribution. This expression below demonstrate the relationship
between the density function of the Crack Random distribution and that of the two
28
Gaussian distribution function
fCR (x, p, λ, θ) = pfIG (x, λ, θ) + (1 − p)fLB (x, λ, θ), x > 0, θ > 0, λ > 0, 0 ≤ p ≤ 1
3.3
The Cumulative Distribution Function
Cumulative distribution function is defined to be the integral over a specific range
of values of the probability density function of a distribution. We apply that principle
in the Crack Lifetime distribution by integrating its density functionfCR (x, p, λ, θ) .
We can also demonstrate that by differentiating the distribution function of crack
random distribution yields the density function of the Crack Lifetime distribution.
And the cumulative distribution function is denoted by FCR (x, p, λ, θ) .
3.4
The Moment Generating Function
The characteristic function being one of the functions associated with the crack
random numbers ϕx (t) = E[etx ] as well as the moment generation function. Moment
generation function is similar to the characteristic function but different in notation
it is expressed as φx (t) = E[etx ].
29
3.5
The Maximum Likelihood Estimate of Paramters
To find the first maximum likelihood estimators for the three-parameter crack
distribution. We will start by finding the log of the function and after which we
will find the partial derivative of the various parameters p, λ and θ respectively and
equate it to zero.
3.6
The Desirable Parameter Estimated Properties
Like any other parameters in various distributions when estimated have special
properties and these apply to some and others. In reference to the Crack Lifetime
distribution, the proposed parameters for this distribution seem to have the following
properties namely; Estimators are unbiased, have their variance being the minimum
and also so is the mean square estimated error. Having an idea of what this properties
are to enable you to double check if the estimator actually follows the Crack Lifetime
distribution.
3.7
The First Three Cumulants
One of the important characteristics of the three-parameter crack random distribution is it cumulant. To find the three cumulants of the distribution the MacLaurins
series is being expanded to the third power. The proof of these characteristics will
30
not in this Research, but there are interesting papers that have proofs of some this
characteristics in research papers that were supervised by Andrei Volodin.
3.8
Parameter (p) Estimation
The parameter p in the two-parameter Crack Lifetime distribution is estimated
poorly and therefore we will be looking forward to take into consideration the p in
the distribution to be known. This effect will be seen in the output as we can see in
the generator using the software package R.
3.9
Generating Procedure
Computationally the criteria we require is statistical modeling of the random
numbers following three-parameter Crack distribution considering the following values
of parameters λ = 2, 5, 10, 20, θ = 1, 5, 10 and p = 0.2, 0.4, 0.6. For fixed values of the
three parameters we run simulations of corresponding random numbers independently,
the simulations are repeated 1,000 times for constructing and reporting the histogram
of the Crack distribution by using the program R. The following are the steps we use
in the generating procedure.
Step1: For each of the generating procedure of the data Let λ > 0, θ > 0, 0 < p ≤ 1.
Step2: We generate the random numbers for the one-sided Birnbaum Saunders distribution
31
We begin by generating a random number denoted by α from standard normal distribution N(0,1). After which, we obtain a Birnabaum Saunders-distributed random
number by means of the formula below
α
x = θ( +
2
r
α2
+ λ)2
4
Step3: We begin our generation procedure by means of the Birnbaum-Saunders
distribution,
We then generate random variable which follows the Crack distribution. Hence,
we have to calculate the well known Birnbaum-Saunders distribution by means of the
density function for this distribution. And this is given by ;
g = fBS (x, θ, λ) =
θ 3
√1 [λ( ) 2
2 2πθ
x
r
r
x 1
1
θ
x 2
+ ( ) 2 ]exp− (λ
−
)
θ
2
x
θ
Step4: Compute the crack distribution from the the mixture of the two probability
distribution function. And this is represented by :
fCR (x, p, λ, θ) = pfIG (x, λ, θ) + (1 − p)fLB (x, λ, θ)
Step5: Generate the uniform distribution u ∼ U (0, 1)
Step6: If u <
f
where c = 2 · max[p, (1 − p)] when c is computed to be from the
c·g
formula below
fCR (x, p, λ, θ) = pfIG (x, λ, θ) + (1 − p)fLB (x, λ, θ)
32
1
1
fBS (x, θ, λ) = fIG (x, λ, θ) + fLB (x, λ, θ)
2
2
fCR (x, p, λ, θ)
pfIG (x, λ, θ) + (1 − p)fLB (x, λ, θ)
=
1
1
fBS (x, θ, λ)
fIG (x, λ, θ) + fLB (x, λ, θ)
2
2
1
1
2 · max[p, (1 − p)]( fIG (x, λ, θ) + fLB (x, λ, θ))
2
2
≤
1
1
fIG (x, λ, θ) + fLB (x, λ, θ)
2
2
≤ 2·max[p, (1 − p)] = c
Step7: Use all the random number x from the step above to generate histogram
charts and summary in figure 3.1 shows the generating procedure in flow chart on the
next page
33
Figure 3.1 Procedure of generating random numbers that follow Crack
distribution
Start
Fix l  0, t  0,0  p  1
i=0
Generate a ~ N (0,1)
a

a2
x  t 
l
2

4


g
2
3
1
2


2
2
1 t
x 
l  t    x   exp 
 

  l

t  
2 2 t   x 
t 
 2 x



1
2
x 

 x 2
 1  t

f  p  x  (1  p)  exp   l


t  
t
 2 x

1
Generate u ~ U (0,1)
If
u
f
cg
Yes
i=i+1
c(i)  x
Yes
i  1000
No
Plot histogram using c(i ) , 1  i  1000
34
No
Chapter 4
RESULT OF SIMULATION
In this chapter, we consider the outcome of the results in our research, by simulation.The scope of this research took into consideration 36 cases with the proposed
parameters as we read from the previous chapter. This parameters assume the values
λ = 2, 5, 10, 20, θ = 1, 5, 10 and p = 0.2, 0.4, 0.6. We also use the statistical software
package R to run simulations 1000 times repeatedly and the present the histogram
chart we generated from the crack random numbers that follow the three-parameter
crack distribution.
We can conclude the results of our research with the following findings:
4.1
In the case of λ = 2, θ = 1 and p = 0.2.
Figure 4.1 The resulted histogram plot which was generated using the crack
random numbers with the density function of the three-parameter Crack distribution
35
is given below.(λ = 2, θ = 1 and p = 0.2)
4.2
In the case of λ = 2, θ = 1 and p = 0.4.
Figure 4.2 The histogram which is from generating the random numbers and
density function of Crack distribution. (λ = 2, θ = 1 and p = 0.4.)
36
37
4.3
In the case of λ = 2, θ = 1 and p = 0.6.
Figure 4.3 The histogram which is from generating the random numbers and
density function of the Crack distribution ( λ = 2, θ = 1 and p = 0.6.)
4.4
In the case of λ = 2, θ = 5 and p = 0.2.
Figure 4.4 The histogram which is from generating the random numbers and
density function of the Crack distribution. ( λ = 2, θ = 5 and p = 0.2.)
38
39
4.5
In the case of λ = 2, θ = 5 and p = 0.4.
Figure 4.5 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 2, θ = 5 and p = 0.4.)
4.6
In the case of λ = 2, θ = 5 and p = 0.6.
Figure 4.6 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 2, θ = 5 and p = 0.6.)
40
41
4.7
In the case of λ = 2, θ = 10 and p = 0.2.
Figure 4.7 The histogram which is from generating the random numbers and
density function of the Crack distribution. ( λ = 2, θ = 10 and p = 0.2.)
4.8
In the case of λ = 2, θ = 10 and p = 0.4.
Figure 4.8 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 2, θ = 10 and p = 0.4.)
42
43
4.9
In the case of λ = 2, θ = 10 and p = 0.6.
Figure 4.9 The histogram which is from generating the random numbers and
density function of the Crack distribution. ( λ = 2, θ = 10 and p = 0.6.)
4.10
In the case of λ = 5, θ = 1 and p = 0.2.
Figure 4.10 The histogram which is from generating the random numbers and
density function of the Crack distribution. ( λ = 5, θ = 1 and p = 0.2.)
44
45
4.11
In case the of λ = 5, θ = 1 and p = 0.4.
Figure 4.11 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 5, θ = 1 and p = 0.4.)
4.12
In case the of λ = 5, θ = 1 and p = 0.6.
Figure 4.12 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 5, θ = 1 and p = 0.6.)
46
47
4.13
In case the of λ = 5, θ = 5 and p = 0.2.
Figure 4.13 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 5, θ = 5 and p = 0.2.)
48
4.14
In case of the λ = 5, θ = 5 and p = 0.4.
Figure 4.14 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 5, θ = 5 and p = 0.4.)
49
4.15
In case of the λ = 5, θ = 5 and p = 0.6.
Figure 4.15 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 5, θ = 5 and p = 0.6.)
50
4.16
In case of theλ = 5, θ = 10 and p = 0.2.
Figure 4.16 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 5, θ = 10 and p = 0.2.)
4.17
In the case of λ = 5, θ = 10 and p = 0.4.
Figure 4.17 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 5, θ = 10 and p = 0.4.)
51
52
4.18
In the case of λ = 5, θ = 10 and p = 0.6.
Figure 4.18 The histogram which is from generating the random numbers and
density function of the Crack distribution. ( λ = 5, θ = 10 and p = 0.6.)
4.19
In the case of λ = 10, θ = 1 and p = 0.2.
Figure 4.19 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 10, θ = 1 and p = 0.2.)
53
54
4.20
In the case of λ = 10, θ = 1 and p = 0.4.
Figure 4.20 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 10, θ = 1 and p = 0.4.)
4.21
In the case of λ = 10, θ = 1 and p = 0.6.
Figure 4.21 The histogram which is from generating the random numbers and
density function of the Crack distribution. ( λ = 10, θ = 1 and p = 0.6.)
55
56
4.22
In the case of λ = 10, θ = 5 and p = 0.2.
Figure 4.22 The histogram which is from generating the random numbers and
density function of the Crack distribution. ( λ = 10, θ = 5 and p = 0.2.)
57
4.23
In the case of λ = 10, θ = 5 and p = 0.4.
Figure 4.23 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 10, θ = 5 and p = 0.4.)
58
4.24
In the case of λ = 10, θ = 5 and p = 0.6.
Figure 4.24 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 10, θ = 5 and p = 0.6.)
59
4.25
In the case of λ = 10, θ = 10 and p = 0.2.
Figure 4.25 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 10, θ = 10 and p = 0.2.)
4.26
In the case of λ = 10, θ = 10 and p = 0.4.
Figure 4.26 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 10, θ = 10 and p = 0.4.)
60
61
4.27
In the case of λ = 10, θ = 10 and p = 0.6.
Figure 4.27 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 10, θ = 10 and p = 0.6.)
4.28
In the case of λ = 20, θ = 1 and p = 0.2.
Figure 4.28 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 20, θ = 1 and p = 0.2.)
62
63
4.29
In the case of λ = 20, θ = 1 and p = 0.4.
Figure 4.29 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 20, θ = 1 and p = 0.4.)
4.30
In the case of λ = 20, θ = 1 and p = 0.6.
Figure 4.30 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 20, θ = 1 and p = 0.6.)
64
65
4.31
In the case of λ = 20, θ = 5 and p = 0.2.
Figure 4.31 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 20, θ = 5 and p = 0.2.)
66
4.32
In the case of λ = 20, θ = 5 and p = 0.4.
Figure 4.32 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 20, θ = 5 and p = 0.4.)
67
4.33
In the case of λ = 20, θ = 5 and p = 0.6.
Figure 4.33 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 20, θ = 5 and p = 0.6.)
68
4.34
In the case of λ = 20, θ = 10 and p = 0.2.
Figure 4.34 The histogram which is from generating the random numbers and
density function of the Crack distribution. (λ = 20, θ = 10 and p = 0.2.)
4.35
In the case of λ = 20, θ = 10 and p = 0.4.
Figure 4.35 The histogram which is from generating the random numbers and
density function of the Crack distribution. ( λ = 20, θ = 10 and p = 0.4. )
69
70
4.36
In the case of λ = 20, θ = 10 and p = 0.6.
Figure 4.36 The histogram which is from generating the random numbers and
density function of the Crack distribution. ( λ = 20, θ = 10 and p = 0.6.)
4.37
PP-PLOT FOR λ = 2, θ = 5 and p = 0.2
Figure 4.4 The pp-plot which is used to model the emperical distribution of the
crack random numbers against the distribution function of the three-parameter crack
71
distribution. ( λ = 2, θ = 5 and p = 0.2.)
4.38
PP-PLOT FOR λ = 2, θ = 5 and p = 0.4
Figure 4.5 The pp-plot which is used to model the emperical distribution of the
crack random numbers against the distribution function of the three-parameter crack
distribution. ( λ = 2, θ = 5 and p = 0.4.)
72
73
Chapter 5
CONCLUSIONS AND DISCUSSIONS
The main objective of this thesis is generating crack random numbers, this random
numbers follow the three-parameter crack distribution as we discuss in the early
chapters. For some specific values of parameters crack random numbers are generated
and then modeled with a histogram of the corresponding density function. We also
compare the shape of the generated distributions for various values of parameters.
Thus, we conclude that result from the observations follow the Research Scope as
mentioned in chapter one, which considers the histogram shapes which are based on
random numbers generation that follow three-parameter Crack distribution and the
density function graph of the Crack distribution. The considerations are based on
the parameters of the Crack distribution and as follows:
1. If parameter p varies whereas λ and θ are fixed.
2. If parameter p varies whereas λ and θ are fixed.
74
3. If parameter p varies whereas λ and θ are fixed.
5.1
5.1.1
Research conclusion
If parameter λ is variable whereas p and θ are fixed.
As a result from Chapter 5, when we consider the case of parameter λ variable
whereas θ and p are fixed, we found that in each case, the histogram shape based on
the random numbers that follow three-parameter crack distribution generation, The
shape of the probability density function graph of crack distribution were similar.
Moreover, we found that the histogram shapes and the shape of the probability density function chart for the crack distribution changed.They went from being skewed
to the right to a bell shape when the value of λ increases. It can be seen in the
pp-plot that the crack random numbers generated follow the three parameter crack
distribution. The pp-plot models the empirical distribution function of the random
number generated to that of the proposed three parameter crack random numbers by
Andrei Volodin in his paper. The 45 degrees angle formed by this two distribution
function in the qq plot help make the conclusion that the crack random numbers
generated follow the three parameter crack distribution.
75
5.1.2
If parameter θ is variable whereas λ and p are fixed.
From the result when we take into consideration the situation where the parameter
θ is variable whereas λ and p are fixed, we found that in each case, the histogram
shape of the random numbers that follow three-parameter Crack distribution and
its corresponding density function graph were similar. If the value of parameter θ
increases, the graph shapes will not have a significant change which was obvious from
the simulation in chapter 4.
5.1.3
If parameter p is varies whereas λ and θ are fixed.
Considering the case of parameter p variable whereas θ and λ are fixed, the shape
of the histogram chart based on the random numbers that follow three-parameter
crack distribution generation and the shape of the density function graph were all
similar.So we can conclude that when the value of p increase, the graph shape will
not change.
5.2
Discussion
When considering all the cases, it can be seen in all the charts that all the shapes
of the histogram produced using the parameters that follow the three-parameter crack
random distribution and the shape of the density function graph of crack distribution
were all similar. This shows that this approach can be used to generate the crack
76
random numbers that follow the three parameter Crack distribution for all values of
parameters. For the values assumed by the parameters used in this thesis,We generate
random numbers from already known two-parameter distributions: Length Biased Inverse Gaussian Birnbaum-Saunders, Inverse Gaussian, and found that the histogram
shape and the shape of the density function graph of crack random distribution will
change it shape from being skewed to the right to a bell shape when the parameter λ
increases. Also, the crack random distribution provides a class of distributions with
different assumed shapes. It may as well be used as a general model to fit data. The
crack distribution could be applied not only for modeling a fatigue failure in crack development concepts in engineering, but also modeling future prices on a stock market
as wellas Actuarial modeling in Insurance whiles only Inverse Gaussian was formerly
used for this purpose.
5.3
Future Research
The simulation studies presented in this research suggest some directions forthe
future research as follows:
1. In this research, discussion was solely only the study of random numbers of
Crack distribution generation. For future study we may suggest random numbers of
Crack distribution generation by other methods with different algorithms.
77
2.This thesis, took into consideration the shapes produced by the histogram in relation to the crack random numbers generated that tend to follow the three-parameter
Crack distribution. The procedure used in this thesis is the probability density function graph of corresponding Crack distribution. Therefore, for future related studies
consideration can be made to how the similarity in are the values of mean and it
corresponding variance based on the sample obtained by the generation approach, as
well as the true values of the related mean, variance and if possible the how skewed
is the three-parameter Crack distribution
78
Bibliography
[1] Ahmed S. E., Budsaba K., Lisawadi S., and Volodin, A. I. (2008, July). Parametric estimation for the Birnbaum-Saunders lifetime distribution based on a
new parameterization. Journal of Thailand Statistician, 6, 213-240.
[2] Bartoszynski, R., and Niewiadomska-Bugaj, M. (2008). Probability theory and
statistical inference (2nd ed.). New York: Wiley.
[3] Birnbaum, Z. W. and Saunders, S. C. (1969). A new family of the life distribution.
J. Appl. Probab. 6, pp. 319-327.
[4] Birnbaum, Z. W. and Saunders, S. C. (1969). Estimation for a family of life
distribution with application to fatigue. J. Appl. Probab. 6, pp. 328-337.
[5] Bogdanoff, D. and Kozin, F. (1985). Probabilistic models of cumulative damage.
John Wiley and Sons, New York-Chichester.
79
[6] Chang, D. S., and Tang, L. C. (1993, September). Reliability bounds and critical
time for the Birnbaum-Saunders distribution. IEEE Transactions on Reliability,
42(3), 464-469.
[7] Chhikara, R. S. and Folks, J. L. (1989). The inverse Gaussian distribution: theory, methodology, and applications. Statistics: Textbooks and Monographs, v.
95. New York.
[8] Desmond, A. (1985). Stochastic models of failure in random environment. Canad.
J. Statist. 13, pp. 171-183.
[9] Desmond, A. (1986). On the relationship between two fatigue-life models. IEEE
Trans. in Reliability 35, pp. 167-169.
[10] Johnson, N.L., Kotz, S. and Balakrishnan, N. (1995). Continuous Univariate
Distributions. Vol. 2 (2nd edition), John Wiley and Sons, New York.
[11] Kennedy, W.J., Gentle, Jr. and Gentle, J.E. (1980). Statistical Computing. Marcel Dekkor, Inc., New York.
[12] Khattree, R. (1989, December). Characterization of inverse-Gaussian and gamma
distributions through their length-biased distributions. IEEE Transactions on
Reliability, 38(5), 610-611.
80
[13] Kumnadee, S., Volodin, A. and Lisawadi, S. (2010, May). Crack distribution parameters estimation by the method of moments. Proceedings of the 11th Statistics and Applied Statistics Conference. Chiang Mai, Thailand
[14] Law, A.M. and Kelton W.D. (1982). Simulation Modelling and Analysis.
MeGraw-Hill Book Company, New York.
[15] Morgan, B.J.T. (1984). Elements of Simulation. Chapman and Hall, New York.
[16] Panta, C., Volodin, A., and Budsaba, K. (2010, May). The generation of random variables having crack distribution by the acceptance-rejection method. Proceedings of the 11th Statistics and Applied Statistics Conference. Chiang Mai,
Thailand..
[17] Richard, A., Kronmal, V., Arthur V. and Peterson, Jr. (1981). A Variant of the
Acceptance-Rejection Method for Computer Generation of Random Variables.
J. Amer. Statist. Vol. 76, No. 374, pp. 446-451.
[18] Shuster, J. (1968). On the inverse Gaussian distribution function. J. Amer.
Statist. Assoc. 63, pp. 1514-1516.
[19] Tweedie, M. C. K. (1956). Some statistical properties of inverse Gaussian distributions. The Annals of Mathematical Statistics, Vol. 28, No. 2 (Jun., 1957), pp.
362-377.
81
[20] Wald, A. (1947). Sequential analysis. John Wiley and Sons, Inc., New York.
82