# Test hypothesis that values given as mean +- confidence interval comes from a given distribution

I've performed $m$ fits of datasets $Y=\theta X+b$ coming from different experiments. As a result, I have $m$ estimates $\theta_1,\theta_2,\dots \theta_m$ where $\theta_{k}=(\theta_{k,1},\dots\theta_{k,n})$ is a vector of slopes. For each slope $\theta_{k,i}$, I have a confidence interval $[\underline{\theta}_{k,i},\bar{\theta}_{k,i}]$.

I want to test a hypothesis, which these slopes (i.e. $\theta_{\cdot,j}$ with data $\theta_{1,j},\dots\theta_{m,j}$) come from Gamma distribution. The easiest way is to use Kolmogorov-Smirnov or Chi-square tests for mean values. Nevertheless, I don't want to lose information, which contains in confidence intervals $[\underline{\theta}_{k,i},\bar{\theta}_{k,i}]$.

Do you have any idea how to deal with such a problem?

• What is the origin of the multiple fits? Did you use different subsets of the training sets? – Karel Macek Jan 18 '17 at 14:27
• It is different participants in a study. I study performance in task Y versus performance in task X during several days. – zlon Jan 18 '17 at 14:30
• I see, you can run the Kolmogorov-Smirnov with estimated parameters (en.wikipedia.org/wiki/… or stats.stackexchange.com/questions/20648/… ) I would say that it would be the most appropriate way. – Karel Macek Jan 18 '17 at 14:33
• I know it. But my problem is not to estimate parameters of possible distribution. My problem is to test not only means, but means +-CI. For example i have slope1+-CI1, slope2+-CI2,slopeN+-CIN. So, i know how to test that slope(1-N) come from Gamma distribution. The question is how to test that slope(1-N)+-CI(1-N) come from a Gamma distribution. In other words if my CI are really big (+/- several means), than good test for means has no sense. – zlon Jan 18 '17 at 14:48
• What other distribution can the slopes have if not Gamma? – Karel Macek Jan 18 '17 at 15:05

In fact, you have $m$ Student distributions, one for each $\theta_{k,i}$ out of your data and you want to know if the mixture of these distributions is equivalent to a Gamma distribution.
Essentially, they are of different shapes. However, you try the following for each slope $i$
• Generate $N$ of samples from each of them. You will have $m\cdot N$ results.