Known univariate unimodal analytical convolution with gaussian I have data that are distributed with an unknown distribution. The data are from one continuous variable and unimodal. The shape looks like a gaussian, but it is asymmetric and with more long tails.
I can get a good fit of the data with several different alternative functions, but the point is that I need a pdf that I am able to convolve analytically with a normal distribution $N[0,\sigma]$ (I need to smear it with an arbitrary smearing $\sigma$).
The only pdf I have in mind is a normal distribution, and as a consequence also a mixture of normal distributions, since I am able to convolve these distrubution with $N[0, \sigma]$ analytically.
Are there other pdfs that fulfill my needs (flexible enough to fit my data, and able to do analytically convolution with $N[0,\sigma]$)?
 A: You say you have data from a distribution which looks like gaussian, unimodal, but skew. That sounds like a skew-normal distribution could be a good fit.  You then want to convolve this model with a gaussian distribution.  That can be done, analytically, with a skew-normal distribution.  
That can be done multiple ways, but the information to do it seems to be not very well known, it is for instance not included in the wikipedia article https://en.wikipedia.org/wiki/Skew_normal_distribution.  The two methods are 
1  using the stochastic representation
2  using the moment-generating function
We will give the details below.  Information about the skew-normal distribution can be found in the book  https://www.amazon.com/Skew-Normal-Institute-Mathematical-Statistics-Monographs/dp/1107029279/ref=sr_1_1?s=books&ie=UTF8&qid=1481472499&sr=1-1&keywords=skew-normal
Let Z have a standard skew-normal distribution, written $\text{SN}(0,1,\alpha)$. Then $Y= \xi+\omega Z$ is $SK(\xi,\omega^2,\alpha)$.  Its moment-generating function can be found to be (Azzalini-book above, page 26-28)
$$
M_Y(t) = 2 e^{\xi t + \frac12 \omega^2 t^2} \Phi(\delta \omega t)
$$
where $\delta=\delta(\alpha) = \frac{\alpha}{\sqrt{1+\alpha^2}}$, $\delta \in (-1, 1)$.  Then let $X$ have a normal distribution (independent of $Y$), $X \sim \text{N}(\mu, \sigma^2)$ with moment-generating function
$$
M_X(t) = e^{\mu t + \frac12 \sigma^2 t^2}
$$
Then it is easy to find the distribution of the convolution $Y+X$, it is found by taking the product of above two moment-generating functions. I leave the details for you, it is easily seen to be skew-normal. 
The other approach is via stochastic representation.  I will come back to this case!
