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Constructing a new probability density function from three-parameter Weibull distribution
- 1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 136-143 © IAEME
136
CONSTRUCTING A NEW FAMILY DISTRIBUTION FROM THREE
PARAMETERS WEIBULL USING ENTROPY TRANSFORMATION
Dhwyia S. Hassun1
1
Professor, Department of Statistics, College of Administration & Economic,
University of Baghdad, Iraq
ABSTRACT
This paper deals with constructing a new family probability density function from Weibull
three parameters (ߠ, ߚ, ܿ), using statistical entropy transformation (which is considered as a measure
of uncertainty), we construct this p.d.f, then prove its integrate from (ߠ ݐ ∞) equal (1). Also the
cumulative distribution function (C.D.F) was derived, and given in simple form which is necessary
for simulation procedure. The constructed . ݀. ݂ is necessary for skewed data, and for data represent
time to failure of device and component after a value of threshold parameter. All the derivation
required are explained. The formula of (ݎ௧
) moments is derived in order to be used for estimation of
parameters by moment method. The maximum likelihood estimators of three parameters were found,
it is implicit functions, which need numerical solution to obtain (ߚመொ. ߠொ, ܿ̂ொ).
Keywords: New Generated . ݀. ݂, Entropy Transformation, Maximum Likelihood & Moment
Estimator, Cumulative Distribution Function.
1. INTRODUCTION
One of the commonly used probability distribution for studying the reliability and
maintainability analysis is the three parameters Weibull. The use of Weibull distribution to describe
real phenomena has a long history. This distribution was originally proposed by the Swedish
physicist WaloddiWeibull. He used it for modeling the distribution of breaking strength of materials.
Since then it has received applications in many areas (Weibull 1951).
This distribution has many applications in engineering, reliability, break age data, and time to
failure data. It is applied in optimality testing of Markov type optimization[15]
, also it is useful to
study the voltage of electric circuits[16]
. The estimating of three parameters Weibull distribution is
complicated, since the equations obtained from the estimation method are too difficult to implement
it, but in this paper we work on modifying the . ݀. ݂ of three parameters Weibull, using entropy like
INTERNATIONAL JOURNAL OF ADVANCED RESEARCH
IN ENGINEERING AND TECHNOLOGY (IJARET)
ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
Volume 5, Issue 6, June (2014), pp. 136-143
© IAEME: http://www.iaeme.com/IJARET.asp
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- 2. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 136-143 © IAEME
137
transformation, this modified . ݀. ݂ is useful in cases of uncertainty found in data obtained from
various applications, so the work here is restricted only about the new generated family with its
derivation and how to estimate its three parameters using method of moments and maximum
likelihood method. But in future we shall continue the work and apply simulation procedure to
compare between various estimators of parameters.
2. OBJECTIVE OF RESEARCH
The objective of this paper is to find a new generated probability distribution function using
entropy like transformation of Weibull three parameters, also to derive its cumulative distribution
function, also to derive a general form of moments from;
ܧሺݔሻ ൌ ܧ ቈ൬
ݕ െ ߠ
ߚ
൰
And use it to find first second and third moments aboutorigin.
Therefore the generated probability density function from Weibull three parameters (c is
shape parameter and ߠ location parameter and ߚ scale parameter), is derived and it is defined by;
ݑሺݐሻ ൌ ൬
ܿ
ߚ
൰ ൬
ݕ െ ߠ
ߚ
൰
ଶିଵ
exp ቊെ ൬
ݕ െ ߠ
ߚ
൰
ቋ ݕ ߠ
0 ݓ⁄
Where;
න ݑሺݐሻ݀ݐ ൌ 1
∞
ఏ
Also we explain how to estimate the parameters (ߠ, ߚ, ܿ), and how to find the cumulative
distribution function of [ݑሺݐሻ] i.e [ݎሺܶ ݐሻ],
ܨሺݐሻ ൌ න ൬
ܿ
ߚ
൰ ൬
ݕ െ ߠ
ߚ
൰
ଶିଵ
exp ቊെ ൬
ݕ െ ߠ
ߚ
൰
ቋ ݀ݕ
௧
ఏ
ܨሺݐሻ ൌ 1 െ exp ቊെ ൬
ݐ െ ߠ
ߚ
൰
ቋ ቈ൬
ݐ െ ߠ
ߚ
൰
1
3. DEFINITION OF CONSTRUCTED . ࢊ. ࢌ FUNCTION
We know that the cumulative distribution function ܿ. ݀. ݂ of .ݎ ݒ Weibull with three
parameters (c is shape parameter and ߠ location parameter and ߚ scale parameter) is defined by;
ܨሺݐሻ ൌ 1 െ exp ቄെ ቀ
௧ିఏ
ఉ
ቁ
ቅ ݐ ߠ (1)
And the reliability function is;
ܴሺݐሻ ൌ exp ቄെ ቀ
௧ିఏ
ఉ
ቁ
ቅ (2)
- 3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 136-143 © IAEME
138
Then the generated function ݃ሺݐሻ defined as;
݃ሺݐሻ ൌ ܨሺݐሻ ܴሺݐሻ ln ܴሺݐሻ (3)
ൌ 1 െ exp ቄെ ቀ
௧ିఏ
ఉ
ቁ
ቅ െ ቀ
௧ିఏ
ఉ
ቁ
exp ቄെ ቀ
௧ିఏ
ఉ
ቁ
ቅ (4)
݃ሺݐሻdefined in equation (3) using entropy like transformation is derived w.r.to ()ݐ to obtain the new
generated . ݀. ݂ ݑሺݐሻ;
݃′ሺݐሻ ൌ ݑሺݐሻ ൌ ቀ
ఉ
ቁ ቀ
௧ିఏ
ఉ
ቁ
ଶିଵ
exp ቄെ ቀ
௧ିఏ
ఉ
ቁ
ቅ ݐ ߠ ߠ, ߚ, ܿ 0 (5)
0 ݓ⁄
Which integrate to one (i.e);
ݑሺݐሻ݀ݐ ൌ 1
∞
ఏ
(6)
Also the cumulative distribution function (C.D.F) is obtained;
ܨ்ሺݐሻ ൌ ݎሺܶ ݐሻ ൌ ݑሺݕሻ݀ݕ
௧
ఏ
(7)
ൌ න ൬
ܿ
ߚ
൰ ൬
ݕ െ ߠ
ߚ
൰
ଶିଵ
exp ቊെ ൬
ݕ െ ߠ
ߚ
൰
ቋ ݀ݕ
∞
ఏ
Using integration by parts we have;
ܨሺݐሻ ൌ 1 െ exp ቄെ ቀ
௧ିఏ
ఉ
ቁ
ቅ ቂቀ
௧ିఏ
ఉ
ቁ
1ቃ (8)
In order to assess the performances of the parameter estimation methods by moments and
M.L and other methods, first we derive a formula for (ݎ௧
) moments of ቀ
௧ିఏ
ఉ
ቁi.eܧ ቀ
௧ିఏ
ఉ
ቁ
which is
defined in equation (10), were;
ܧ ቀ
௧ିఏ
ఉ
ቁ
ൌ ቀ
௧ିఏ
ఉ
ቁ
݂ሺݐሻ݀ݐ
∞
ఏ
(9)
ൌ ൬
ܿ
ߚ
൰ න ൬
ݐ െ ߠ
ߚ
൰
ଶାିଵ
exp ቊെ ൬
ݐ െ ߠ
ߚ
൰
ቋ ݀ݐ
∞
ఏ
Let ݕ ൌ
௧ିఏ
ఉ
݀ݕ ൌ
ௗ௧
ఉ
݀ݐ ൌ ߚ݀ݕ ߠ ൏ ݕ ൏ ∞
ܧሺݕሻ ൌ ൬
ܿ
ߚ
൰ න ݕଶାିଵ
݁ݔሼെݕሽ ߚ ݀ݕ
∞
- 4. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
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139
Let ݖ ൌ ݕ
ݕ ൌ ݖ
భ
݀ݕ ൌ
ଵ
ݖ
భ
ିଵ
ܧሺݕሻ ൌ ܿ න ሺݖ
భ
ሻଶାିଵ
exp ሺെݖሻ ൬
1
ܿ
൰ ቀݖ
భ
ିଵ
ቁ ݀ݖ
∞
ൌ න ሺݖሻଶା
ೝ
ି
భ
ା
భ
ିଵ
݁ି
ݖ ݀ݖ
∞
ൌ න ሺݖሻଶା
ೝ
ିଵ
exp ሺെݖሻ ݀ݖ
∞
ൌ Γ ቀ2
ݎ
ܿ
ቁ
ܧሺݕሻ ൌ Γ ቀ2
ݎ
ܿ
ቁ
ܧ ቀ
௧ିఏ
ఉ
ቁ
ൌ Γ ቀ2
ቁ (10)
When ݎ ൌ 1;
ܧ ቀ
௧ ି ఏ
ఉ
ቁ ൌ Γ ቀ2
ଵ
ቁ (11)
ܧሺݐሻ ൌ ߚ Γ ൬2
1
ܿ
൰ ߠ
And from;
ܧ ൬
ݐ െ ߠ
ߚ
൰
ଶ
ൌ Γ ൬2
2
ܿ
൰
ܧሺݐ െ ߠሻଶ
ൌ ߚଶ
Γ ቀ2
ଶ
ቁ (12)
Also;
ܧሺݐ െ ߠሻଷ
ൌ ߚଷ
Γ ቀ2
ଷ
ቁ (13)
These expectations are useful in obtaining moment estimators for the three parameters (ߠ, ߚ, ܿ):
From ܧሺݐሻ ൌ
∑ ௧
ݐ ൌ ߚመΓ ቀ2
ଵ
̂
ቁ ߠ (14)
And ܧሺݐଶሻ ൌ
∑ ௧
మ
ܧሺݐଶሻ ൌ 2ߠܧሺݐሻ ߚଶ
Γ ൬2
2
ܿ
൰ െ ߠଶ
∑ ݐ
ଶ
݊
ൌ 2ߠ ൜ߚመΓ ൬2
1
ܿ
൰ ߠൠ ߚመଶ
Γ ൬2
2
ܿ
൰ െ ߠ
ଶ
∑ ݐ
ଶ
݊
ൌ 2ߠߚመΓ ൬2
1
ܿ
൰ 2ߠଶ
ߚመଶ
Γ ൬2
2
ܿ
൰ െ ߠଶ
- 5. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
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∑ ௧
మ
ൌ 2ߠݐ ߚመଶ
Γ ቀ2
ଶ
ቁ െ ߠଶ
(15)
∑ ݐ
݊
ൌ ߚመΓ ൬2
1
ܿ̂
൰ ߠ
And from third moment we have;
ܧሺݐଷ
െ 3ݐଶ
ߠ 3ߠݐଶ
െ ߠଷሻ ൌ ߚଷ
Γ ൬2
3
ܿ
൰
ܧሺݐଷሻ ൌ ߚଷ
Γ ൬2
3
ܿ
൰ 3ߠܧሺݐଶሻ െ 3ߠଶ
ܧሺݐሻ ߠଷ
∑ ݐ
ଷ
݊
ൌ ߚଷ
Γ ൬2
3
ܿ
൰ 3ߠ ൜2ߠܧሺݐሻ ߚଶ
Γ ൬2
2
ܿ
൰ െ ߠଶ
ൠ െ 3ߠଶ
൜ߚΓ ൬2
1
ܿ
൰ ߠൠ ߠଷ
ܧሺݐሻ ൌ ߚΓ ൬2
1
ܿ
൰ ߠ
∑ ݐ
ଷ
݊
ൌ ߚଷ
Γ ൬2
3
ܿ
൰ 6ߠଶ
ܧሺݐሻ 3ߠߚଶ
Γ ൬2
2
ܿ
൰ െ 3ߠଷ
െ 3ߠଶ
ߚΓ ൬2
1
ܿ
൰ െ 3ߠଷ
ߠଷ
ܧሺݐሻ ൌ ߚΓ ൬2
1
ܿ
൰ ߠ
6ߠଶ
ܧሺݐሻ ൌ 6ߠଶ
ߚΓ ൬2
1
ܿ
൰ 6ߠଷ
∑ ݐ
ଷ
݊
ൌ ߚଷ
Γ ൬2
3
ܿ
൰ 6ߠଶ
ߚΓ ൬2
1
ܿ
൰ 6ߠଷ
3ߠߚଶ
Γ ൬2
2
ܿ
൰ െ 6ߠଷ
െ 3ߠଶ
ߚΓ ൬2
1
ܿ
൰ ߠଷ
∑ ௧
య
ൌ ߚଷ
Γ ቀ2
ଷ
ቁ 3ߠଶ
ߚΓ ቀ2
ଵ
ቁ 3ߠߚଶ
Γ ቀ2
ଶ
ቁ ߠଷ
(16)
There for to find moment estimators for the three parameters (ߠ, ߚ, ܿ) of new generated . ݀. ݂
the following three equations must solved numerically and simultaneously, these equations (14, 15,
16) are;
∑ ݐ
݊
ൌ ߚመΓ ൬2
1
ܿ̂
൰ ߠ
∑ ݐ
ଶ
݊
ൌ 2ߠݐ ߚመଶ
Γ ൬2
2
ܿ
൰ െ ߠଶ
∑ ݐ
ଷ
݊
ൌ ߚመଷ
Γ ൬2
3
ܿ
൰ 3ߠଶ
ߚመΓ ൬2
1
ܿ
൰ 3ߠߚመଶ
Γ ൬2
2
ܿ
൰ ߠଷ
4. ESTIMATING THREE PARAMETERS BY MAXIMUM LIKELIHOOD METHOD
To find M.L.E for three parameters (ߠ, ߚ, ܿ) of the generated . ݀. ݂ (equation [5]), let
(ݐଵ, ݐଶ, … , ݐ) be value of a random variable (T) with (݊) sample size, from (5) then;
ܮ ൌ ෑ ൬
ܿ
ߚ
൰ ൬
ݐ െ ߠ
ߚ
൰
ଶିଵ
݁ݔ ቊെ ൬
ݐ െ ߠ
ߚ
൰
ቋ
ୀଵ
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ܮ ൌ ൬
ܿ
ߚ
൰
ෑ ൬
ݐ െ ߠ
ߚ
൰
ଶିଵ
݁ݔ ൝െ ൬
ݐ െ ߠ
ߚ
൰
ୀଵ
ൡ
ୀଵ
ݐ ߠ ሺ17ሻ
Taking logarithm of (17) yield;
log ܮ ൌ ݊ log ܿ െ ݊ log ߚ ሺ2ܿ െ 1ሻ log ൬
ݐ െ ߠ
ߚ
൰ െ ൬
ݐ െ ߠ
ߚ
൰
ୀଵ
ୀଵ
ሺ18ሻ
Simplifying equation (18) implies;
log ܮ ൌ ݊ log ܿ െ ݊ log ߚ ሺ2ܿ െ 1ሻ
logሺݐ െ ߠሻ െ ݊ሺ2ܿ െ 1ሻ log ߚ െ ൬
ݐ െ ߠ
ߚ
൰
ୀଵ
ୀଵ
ሺ19ሻ
Then differentiating (19) partially with respect to each parameter we obtain MLE from equating each
partial derivatives to zero.
߲ log ܮ
߲ܿ
ൌ
݊
ܿ
2 logሺݐ െ ߠሻ െ 2݊ log ߚ െ ൬
ݐ െ ߠ
ߚ
൰
log ൬
ݐ െ ߠ
ߚ
൰
ୀଵ
ୀଵ
ሺ20ሻ
߲ log ܮ
߲ߚ
ൌ െ
݊
ߚ
െ
݊ሺ2ܿ െ 1ሻ
ߚ
ܿߚିିଵ
ሺݐ െ ߠሻ
ୀଵ
ሺ21ሻ
߲ log ܮ
߲ߠ
ൌ ሺ2ܿ െ 1ሻ ൬
1
ݐ െ ߠ
൰ ሺെ1ሻ െ ܿߚି
ሺݐ െ ߠሻିଵ
ୀଵ
ୀଵ
ሺെ1ሻ
ฺ െሺ2ܿ െ 1ሻ ሺݐ െ ߠሻିଵ
ୀଵ
ܿߚି
ሺݐ െ ߠሻିଵ
ୀଵ
ൌ 0
ሺݐ െ ߠሻିଵ
ܿߚି
ሺݐ െ ߠሻ
െ ሺ2ܿ െ 1ሻ
ୀଵ
൩ ൌ 0
ୀଵ
֜ ܿߚି
ሺݐ െ ߠሻ
െ ሺ2ܿ െ 1ሻ
ୀଵ
ൌ 0 ሺ22ሻ
Then the equations for maximum likelihood estimators of three parameters which are
obtained from (20, 21, 22) are;
2 logሺݐ െ ߠሻ െ ൬
ݐ െ ߠ
ߚ
൰
log ൬
ݐ െ ߠ
ߚ
൰
ୀଵ
ൌ ݊ሺ2 log ߚ െ
1
ܿ
ሻ
ୀଵ
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ܿ̂ ൫ݐ െ ߠ൯
̂
ൌ ቆ
2݊ܿ̂
ሺߚመሻ̂
ቇ
ୀଵ
൫ݐ െ ߠ൯
̂
ൌ ቆ
2݊
ሺߚመሻ̂
ቇ
ୀଵ
Therefore;
ߚመொ ൌ ቈ
∑ ൫௧ିఏಾಽಶ൯
ො
సభ
ଶ
భ
ො
(23)
Which is implicit function of ߠொand (ܿ̂ொ).
Also we have;
ܿ̂ ቆ
ݐ െ ߠ
ߚመ
ቇ
̂
ൌ 2 ܿ̂ 1
ୀଵ
2
1
ܿ̂
ൌ ቆ
ݐ െ ߠ
ߚመ
ቇ
̂
ୀଵ
1
ܿ̂ொ
ൌ ቆ
ݐ െ ߠ
ߚመ
ቇ
̂
െ 2
ୀଵ
ܿ̂ொ ൌ ቆ
ݐ െ ߠ
ߚመ
ቇ
̂
െ 2
ୀଵ
൩
ିଵ
ሺ24ሻ
Which is also an implicit function can be solved numerically, and from
డ ୪୭
డఏ
ൌ 0
ሺ2ܿ̂ െ 1ሻ ൫ݐ െ ߠ൯
ିଵ
ൌ ܿ̂ሺߚመሻି̂
ୀଵ
൫ݐ െ ߠ൯
̂ିଵ
ୀଵ
ሺ2ܿ̂ െ 1ሻ ൌ ܿ̂ሺߚመሻି̂
൫ݐ െ ߠ൯
̂
ୀଵ
ቆ
ݐ െ ߠ
ߚመ
ቇ
̂
ൌ ൬
2ܿ̂ െ 1
ܿ̂
൰
ୀଵ
ቆ
ݐ െ ߠ
ߚመ
ቇ
̂
ൌ ൬2 െ
1
ܿ̂
൰
ୀଵ
ሺ25ሻ
Which is the same as equation (24), then for fixed value of (ߠ) equation (24 & 25) can be solved
numerically to find (ߚመொ) and (ߠொ) and then (ܿ̂ொ).
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143
CONCLUSION
1. The new generated . ݀. ݂ using entropy transformation is useful for skewness data.
2. The data have location parameter so this distribution is good for representing the distribution of
failure time after (ߠ) value since (ݐ ߠ).
3. The value of (ߠ) may represent the scale for evaluating the reliability of device after time interval
since (ݐ ߠ).
4. Another methods for estimation can be used like Baysian method, Percentile method.
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