Complete Volume - Institute of Business Management

Research
Process Capability Analysis for Non Normal Data
ˆ
⎡ ∂
V ( α ) = ⎢
⎣∂α
V
The three-parameter Weibull distribution probability
density function is given by
f
( x)
β ⎛ x −γ
=
⎞
⎜ ⎟
α ⎝ α ⎠
⎡
⎢
⎛ x −γ
exp −
⎞
⎜ ⎟
⎢⎣
⎝ α ⎠
β −1
β
where x > γ , γ > 0 , α > 0, β > 0 . We may derive the twoparameter
Weibull distribution by assuming the location
parameter “equal to zero”. Many distributions are special cases
of Weibull distribution, for example exponential distribution is a
transformed form of Weibull distribution with shape parameter
. β = 1
We consider the two-parameter Weibull distribution with
β
probability function given as F ( x) = 1 − exp −( x α ) ,
where x>0. A linear regression model was developed using the
cumulative distribution function, as described below.
F x = 1 − exp − x α β
[ ]
( ) ( )
⇒ ln( x) − ln( α ) = [ ln{ − ln ( 1 − F( x)
)}]
β
⇒ ln( x) = ln ( α ) + [ ln{ − ln ( 1 − F( x)
)}] β
The last expression (equation 2) is equivalent to
simple linear regression model, y = β 0 + β1
z + e , where
( ) ( ) [ { ( ( ))}
]
y = ln x , β0 = ln α , β1
= 1 β ,and z = ln −ln
1−F
x
Least square estimators of both parameters were determined and
standard errors were obtained using the Delta method (Stuart
and Ord, 1987) as given below:
2
2
⎤
⎡
2
2
exp( ) (
0
) [ exp(
0
)] (
0
) [ exp(
0)
] ∑ y ⎤
i
β0
⎥ V β = β V β = β ⎢
2 ⎥σ
⎦
⎢⎣
n∑
( y
i
− y ) ⎥⎦
2
2
4
( ˆ
⎡ ∂ ⎛ 1 ⎞⎤
⎡ 1 ⎤ ⎡⎛
1 ⎞⎤
) (
1
) ⎢⎜
⎛
⎟
⎞
β = ⎜ ⎟ V β = − ⎥ V ( β
1
) = ⎜ ⎟ V ( β
1
)
⎢ ⎥
⎣ ∂β
⎝ β1
⎠⎦
⎜
⎢⎣
⎝
β
2
1
⎟
⎠⎥⎦
⎢ ⎥
⎣⎝
β1
⎠⎦
⎡⎛
1 ⎞⎤
= ⎢⎜
⎟⎥
⎣⎝
β1
⎠⎦
4
2
⎡ 1 ⎤
⎢
2
⎥σ
⎢⎣
∑ ( y i − y ) ⎥⎦
2
⎤
⎥
⎥⎦
[ ]
− − − − − − − − − − −
− − −
......2
(3)
− − ( 4)
PAKISTAN BUSINESS REVIEW JULY 2010
239