# Generating skew-normal distribution in Matlab

My apologies if this is a trivial question, but I am having trouble with this for a while now.

I need to use a skew-normal distribution in research in MATLAB and the only way I found after googling was to use Pearsrnd, as given in here.

Now, I did the math and wrote function skewnormal function in MATLAB as follows:

    %% The helper function calculating parameters for skew-normal using pearsrnd
function [m,s, sk, kurt] = skewnormal(a, e, w)
c = sqrt(2/pi); % it is used a lot in what follows
d = a/sqrt(1+a*a); % temp variable
m = e + d*w*c; % mean
s = w*sqrt(1 - d^2*c^2); % variance
sk = (4 - pi)/2*(d*c*w/s)^3; % skewness
kurt = 3 + 2*(pi-3)*(d*c*w/s)^4; % kurtosis accounted for the matlab convention.
end


Then, when I use the above in my code and return the type of the Pearsrnd, it returns $$1$$ - which is apparently Four parameters Beta distribution in Pearsrnd.

I did look for answers here that could immediately answer but I did not find any.

Can anyone fix my attempt at generating Skew-Normal distribution, since I am clearly doing something wrong?

• If the question is just how to use pearsrnd just drop an e-mail to MathWorks support. These are pretty much bread and butter for them. Apr 16 '20 at 12:11
• @dezdichado does the answer I've provided give you what you were looking for or is there additional context not in the OP? Reason I ask is b/c my answer hasn't been marked correct and we're coming up on the end of bounty period, see e.g. stats.stackexchange.com/help/bounty for policies on bounties. If I've missed something please let me know. Apr 21 '20 at 17:09

I think you can use the following approach: First you need the command to generate a "standard normal", e.g. N(0,1)

https://www.mathworks.com/help/stats/normrnd.html

then given 2 random normals, you can use the following formula:

$$Y = \mu + \sigma \left(\delta|T_0| + T_1\sqrt{1-\delta^2}\right),$$where

$$\delta = \frac{\lambda}{\sqrt{1+\lambda^2}}$$ from here, where $$T_0,T_1$$ are both standard normal random variates. Also, the values $$\mu, \sigma, \lambda$$ are parameters/arguments to the function so I assume those 3 parameters are given.

In code this looks something like:

t0 = normrnd(0,1)
t1 = normrnd(0,1)
d  = _lambda / sqrt(1 + _lambda*_lambda)
y  = mu + sigma * (d * abs(t0) + t1 * sqrt(1 - d * d))


Of course, if you want to you can wrap that into a function in matlab. I don't really know matlab so if the syntax isn't the greatest, please adjust accordingly. Hopefully, the logical calculations are simple enough that you can modify appropriately for your uses.

EDIT: The $$\delta$$ or equivalently, the $$\lambda$$ is the parameter that determines the extent and nature (positive or negative) of the skew.