Reliability Analysis of Car Subsystem by Weibull …
149
5. For analysing the data, we use two parameter Weibull distribution and find
reliability as a relation
R(t) = e
−t
η
β
(2)
where R(t) = reliability function, t = time, η = scale parameter, β = slope
parameter
F(t) = 1 − R(t)
(3)
where F(t) = unreliability function.
To calculate reliability, we have to estimate the parameters of the Weibull distribution can be found graphically via probability plotting paper, or analytically, either
using least squares or maximum likelihood. For our work, we use rank regression
on Y on principle of least square method. Performing rank regression on Y requires
that straight line mathematically be fitted to be a set of data points such that the
sum of squares of vertical deviations from the points to be minimized. The first
step is to bring our function linear form. For two parameter Weibull distribution, the
cumulative density function is
F(t) = −β ln(η) + β ln(t)
(4)
y = ln[− ln(1 + F(t))]
(5)
a = −β ln(η)
(6)
b = β
(7)
x = ln(t)
(8)
Which results in the linear equation of y = a + bx, the least square parameter
estimation method (also known as regression analysis) was used, and following
equation for regression on Y was used:
ˆ
a =
N
i=1 y i
N
− ˆ
b
N
i=1 y i
N
= y − ˆ
bx
(9)
ˆ
b =
N
i=1 x i y i −
N
i=1 x i
N
i=1 y i
N
N
i=1 x
2
i −
N
i=1 x i
2
N
(10)
149
5. For analysing the data, we use two parameter Weibull distribution and find
reliability as a relation
R(t) = e
−t
η
β
(2)
where R(t) = reliability function, t = time, η = scale parameter, β = slope
parameter
F(t) = 1 − R(t)
(3)
where F(t) = unreliability function.
To calculate reliability, we have to estimate the parameters of the Weibull distribution can be found graphically via probability plotting paper, or analytically, either
using least squares or maximum likelihood. For our work, we use rank regression
on Y on principle of least square method. Performing rank regression on Y requires
that straight line mathematically be fitted to be a set of data points such that the
sum of squares of vertical deviations from the points to be minimized. The first
step is to bring our function linear form. For two parameter Weibull distribution, the
cumulative density function is
F(t) = −β ln(η) + β ln(t)
(4)
y = ln[− ln(1 + F(t))]
(5)
a = −β ln(η)
(6)
b = β
(7)
x = ln(t)
(8)
Which results in the linear equation of y = a + bx, the least square parameter
estimation method (also known as regression analysis) was used, and following
equation for regression on Y was used:
ˆ
a =
N
i=1 y i
N
− ˆ
b
N
i=1 y i
N
= y − ˆ
bx
(9)
ˆ
b =
N
i=1 x i y i −
N
i=1 x i
N
i=1 y i
N
N
i=1 x
2
i −
N
i=1 x i
2
N
(10)