distributionsfittinghistogramskewness
I am trying to find a possible candidate as a fitting function for a distribution that looks like the following
I know that this isn't a straightforward question, but I would like a simple function to start working with, before starting to optimize the procedure.
You could look into the skew-normal distribution (see wikipedia, estimation for skew normal) and you could use it in the same way you used the normal distribution.
But, lacking any knowledge of how the $(x_i, y_i)$ pairs were obtained, there is no principled statistical way of estimating parameters. It doesn't look like you have IID data! So this is probably more a problem of function approximation, more numerical analysis than statistics (unless you tell us some context).
The upper tail of your distribution cuts off much too quickly to be reasonably modelled by a gamma or a lognormal (and it's much too right skew in the rest of the distribution for a Weibull to fit the rapid cut off as well). Look at a normal Q-Q plot of the logs, as well as a histogram of the far upper tail:
I expect none of the default distributions in fitdistr will be suitable here -- and no matter what algorithm you use to estimate parameters, you can't "fix" that they don't fit.
The light extreme tail might not cut off quite quickly enough to look like a truncated lognormal (I think that it wouldn't as gently curve toward flatness like that) -- the tail gets considerably lighter before I think you'd expect to see with a truncated lognormal.
That said, a truncated lognormal isn't a bad model -- it's just not quite describing the distribution. There may be a combination of $\mu$, $\sigma$ and truncation point that does well enough to use for some purposes, perhaps including whatever you're using it for. [Edit: After playing around a bit, actually a truncated lognormal does better than I expected. Not perfect by any means, but not so bad either. You can get a fair approximation of the gentle curve toward flatness there. A truncated might also be adequate. A shifted truncated lognormal should do better if you don't think the truncated lognormal is adequate, but that's a lot of parameters to be fitting to 126 observations.]
Best Answer
There is a family of distributions called the skew normal which includes an additional parameter for skewness. The normal distribution is a special case of the skew normal. Note that this distribution has limited flexibility on how much skewness there can be, with the skewness bounded between $-1$ and $1$ across the range of parameter values. For more info, read here.