NONPROPORTIONAL HAZARD RATE MODELLING (NHRM)
The concept of NHRM
Incorporating the explanatory variables into the PHRM was based on
the reliability function by considering:
g ( z)
(1)
Rt , z   R0 t  ,
t 0
where R0(t) is the reliability function at standard conditions when z = 0 then
g(0) = 1. For incorporating the explanatory variables into the hazard function,
another useful model can be generated from the cumulative distribution
function F(t) namely:
g( z)
(2)
F t , z   F0 t  ,
t 0
where F (t) refers to the standard conditions z= 0.
The probabilty density function takes the form:
g ( z ) 1
(3)
f t , z   g ( z)  F0 t 
,
t 0
where f0(t) is the density function at z =0.
It seems reasonable to specify g(z) according to this model. In the case of the
PHRM model g(z) denotes a function linking vector z to reliabilty and
,consequently, increasing g(z) causes failure time to decrease.
The essence of the NHRM model is that g(z) is a function linking z to the
cumulative function. It means that increasing g(z) corresponds to decreasing
risk and increasing failure time. We suggest the commonly recognized
exponential form of g(z) (2) but the effect on the cumulative function will be
beneficial if z < 0 and harmful if z >0. Considering this fact we replace (2) by:
g(z)=exp(- z)
(4)
The application of this model will be presented on the CWD and Weibull distributions.
For estimating the unknown parameters we use the maximum likelihood function.