EXERCISE
Find the log-likelihood function for the regression model of log-Normal Distribution considering the right-censored observations
ATTEMPT:
So, we know that the probability density function of log-normal distribution is:$$f(t)=\dfrac{1}{\sqrt{2πσ^2}\cdot t}\cdot exp(-\frac{(lnt-μ)^2}{2σ^2})$$
At first, we have to show that $S(t)$ (reliability function) is in this form: $$S(t;x)=S_0(tg(x))$$ where $S_0$ is a basic function and we usually have $g(x)=exp(-β'x)$. In other words we have to show that we have an accelerated life model
My first question is how can I show this, considering that I have to use the log-Normal distribution
Afterwards, we take the log-likelihood function with right-censored observations $$L(μ,σ)=\prod\limits_{i=1}^n f(t_i)^{δ_i}S(t_i)^{1-δ_i} \Rightarrow l(μ,σ)=lnL(μ,σ)=\sum\limits_{i=1}^n( δ_ilnf(t_i)+(1-δ_i)lnS(t_i) )$$
I also know that I have to use that in accelerated life models, we have that: $$lnT_x=μ_0+β'x+σε$$
So, if $T_x$ follows Normal distribution, $lnT_x$ will follow log-Normal distribution.
So I have to use that we speak about regression model but actually I do not have any idea how to use this. Can anyone give me a thorough solution/explanation of this problem because I do not have previous experience in this type of problems for any distribution.
Thanks, in advance!
