Let $X_1,X_2,X_3,X_4$ be iid and $X_1\sim \text{Gamma}(\alpha,\beta)$. Let us fix $$T_1(X_1,X_2,X_3,X_4)=\frac1n \sum_{i=1}^4X_i=t_1,$$ $$T_2(X_1,X_2,X_3,X_4)=\frac{\left( \prod_{i=1}^4X_i\right)^{1/4}}{t_1} =t_2.$$ Goal is to find the conditional distribution of $X_1|t_1,t_2$. I've used two algorithms to estimate the distribution by drawing samples and plotting the empirical CDFs. The first algorithm is the Gibbs algorithm, from the following paper https://www.jstor.org/stable/20441432?seq=1#metadata_info_tab_contents
The second algorithm is from here https://folk.ntnu.no/bo/forskning/LindTarFestschrift.Revision.pdf
I've fixed $t_1=125.1$ and $t_2=0.007993605$ and the plots are as follows https://docdro.id/m9v2rfb
Since I don't have a true benchmark to compare my results against, I'm having trouble figuring out which one of the algorithms is working incorrectly. Is there a way to estimate the CDF at a single point? For example $F_{X_1|t_1,t_2}(50)$ would work great to determine which curve is correct. Or is there a third algorithm I could use to compare my first two with? Since the curves are different enough, perhaps there's an intuitive explanation for why one of them should be incorrect.
A third and most straight forward method of estimating the conditional gamma distribution is to use the naive sampler. We note that the statistic values $t_1,t_2$ were calculated from a sample $x=(500, 0.1, 0.2, 0.1)$ and the corresponding MLEs are $\hat{\alpha}=0.157956685$ and $\hat{\beta}=0.001262635$.
Let us generate iid samples $X_1,X_2,X_3,X_4$ such that $X_1\sim \text{Gamma}(\hat{\alpha},\hat{\beta})$. We can say that the sample comes from the conditional gamma distribution, if the following conditions are met $$\left| \frac1n\sum_{i=1}^4X_i -t_1\right|<\epsilon_1\wedge \left|\frac{\left(\prod_{i=1}^4X_i\right)^{1/4}}{t_1}-t_2\right|<\epsilon_2.$$ I found that by testing $10^9$ sample sets, $\epsilon_1=1$ and $\epsilon_2=0.0001$ produced $9824$ accepted sample points. That is enough to estimate the ECDF and the results suggest that Algorithm 2 was correct https://docdro.id/uoLOgSP