In the following plot you see my empirical data (black) plotted against a hypothesised distribution (blue).
However, a KS-test shows that there is no indication that my sample follows this distribution. Same for the gamma test described by Villasenor and Gonzalez-Estrada (2015).
My question is related to how to explain the mismatch in an easy to understand way (for a report I'm writing). The way I see it, the "mismatch" happens mostly at the peak of the curve. My guess is that since the majority of values lie around this peak, the "penalty" calculated by the KS-test is weighted and is therefore much worse in that area than in other parts of the distribution. If that's true it explains why the minor difference in the peak has such a huge impact even though the rest seems to fit well. Is this reasoning correct? Or is the test not weighted and is there simply too much difference with the distribution overall.
To ask my question in a different way, does a mismatch around the peak density affect the outcome of a KS-test more than a mismatch in areas of lower density?
I am very new to statistic so please forgive my choice of words and limited knowledge. Any help/pointers would be appreciated.
EDIT: following the advice of Nick Cox below, I have created some Q-Q plots. Interestingly the gamma distribution was a really bad fit so I tried some alternative distributions.
Out of these distributions the Q-Q plot for the lnorm distributions seemed the most promising.
However, there is still no statistically significant fit (not even close) and I'm still not sure how to interpret this.