Any distribution that has an inverse can be simulated from Uni(0,1)

Proof:

Highest density region

https://stats.stackexchange.com/a/148452/61705

Different likelihoods

Maximum Likelihood

Find β and θ that maximizes L(β, θ|data).

Partial Likelihood

If we can write the likelihood function as:

L(β, θ|data) = L1(β|data) L2(θ|data)

Then we simply maximize L1(β|data).

Profile Likelihood

If we can express θ as a function of β then we replace θ with the corresponding function.

Say, θ = g(β). Then, we maximize:

L(β, g(β)|data)

Marginal Likelihood

We integrate out θ from the likelihood equation by exploiting the fact that we can identify the probability distribution of θ conditional on β.

Calculate 10 year probability use Cox model

Also pay attention to time unit here.

Transform or link?

https://ecommons.cornell.edu/bitstream/handle/1813/31620/BU-1049-MA.pdf?sequence=1

Simple linear regression

ranking and empirical distributions

In the absence of repeated values (ties), the cdf can be obtained computationally by sorting the observed data in ascending order, i.e., . Then , where  represents the ascending rank of . Likewise, the p-value can be obtaining by sorting the data in descending order, and using a similar formula, , where  represents the descending rank of .

https://brainder.org/2012/11/28/competition-ranking-and-empirical-distributions/