The background is residual useful life prediction. The following is my problem description.
Degradation signal path: $r(t)=\phi+\theta t$ , where $\phi$ is assumed to be the same for all units，and $\theta$ is varying from unit to unit.
Denoting that $S(t)$ is the survival function.
Model one:
$$S_1(t)=\int \exp\left\{-\int_0^t h_0(\mu) \exp(\phi+\theta \mu) \, d\mu\right\} f(\theta) d\theta\,,$$
where $h_0(\mu)$ is the baseline hazard rate function, and $f(\theta)$ is the probability density function of $\theta$.
Model two:
$$S_2(t)=1-\int P(r(t)\ge Y\mid Y=y)p(Y=y) \, dy=1-\int F(t\mid y)f_Y(y) \, dy\,,$$
where $Y$ is the random degradation signal threshold, $f_Y(y)$ is the probability density function of $Y$, and $F(t\mid y)$ is the cumulative distribution function of failure time $t$.
Now I want to prove that model one is equivalent to model two, or to say, $S_1(t)=S_2(t).$
The idea is as follow.
Proposition: For any given model one, that is to say, $h_0(t)$ and $f(θ)$ are known in model one, we can always find an equivalent model two by manipulating its parameters such as the distribution of $Y$ $f_Y(y)$. On this occasion, we can assume that model two is a special case of model one.
To be specific, after getting $S_1(t)$ , by utilizing the equation $S_1(t)=S_2(t)$, we can obtain the distribution of $Y$ in model two on the condition that $F(t\mid y)$ is known. Here any restrictive parametric distribution form for $Y$ is not pre-assumed. You even can regard $f_Y(y)$ as a black-box function. And $f_Y(y)$ can be obtained as a form of implicit funciton.
Questions:
$S_1(t)$ can be only computed by numerical methods. I have ever tried the numerical integration method to obtain $f_Y(y)$ but failed. Due to the complexity of two models, I don’t know how to simplify the integration and thereby get the distribution of $Y$.
$S_1(t)$ can be only computed by numerical methods. Actually, I even needn’t figure out the specific expression of $f_Y(y).$ What I need is the method to prove the existence of $f_Y(y)$. Are there other ideas except for computing $f_Y(y)$? On the other hand, if I want to negate the proposition above, or to say, to demonstrate that $f_Y(y)$ is non-existent for the equivalence relation, what should I do?
I truly hope someone can provide me some suggestions or ideas. Thank you, all viewers!