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I have a hazard function $\lambda$ for survival time T, is there a way to find the hazard function $\lambda_\alpha$ for T + $\alpha$, where alpha is some integer.
I'm thinking $$S_\alpha(t)=P(T+\alpha > t)=P(T>t-\alpha)=\exp(-\int_0^{t-\alpha}\lambda(s)ds)$$$$=\exp(-\int_0^{t}\lambda(s)ds+\int_{t-\alpha}^{t}\lambda(s)ds)$$
and I'm not really sure how to continue.
The hazard function $\lambda(t)$ is related to the survival function $S(t)$ as \begin{align}\lambda(t) &= \frac{-\frac{\mathrm d(St)}{\mathrm dt}}{S(t)}\\ S(t) &= \exp\left(-\int_0^t\lambda(s)\,\mathrm ds\right)\end{align} and so \begin{align}S_\alpha(T) &= S(T+\alpha)\\ &= \exp\left(-\int_0^{T+\alpha}\lambda(s)\,\mathrm ds\right) \\&= \exp\left(-\int_0^{T}\lambda(s)\,\mathrm ds - \int_T^{T+\alpha}\lambda(s)\,\mathrm ds\right)\\ &= S(T)\cdot \exp\left(-\int_T^{T+\alpha}\lambda(s)\,\mathrm ds\right)\end{align} where you calculated everything except the very last step in writing your question. Unfortunately, no further simplification is possible without more specific information about the hazard function. But, since we do know that $\lambda(t) \geq 0$ for all $t$, we do get the cold comfort of knowing that $S_\alpha(T)$ is a nonincreasing function of $\alpha$.
