# How to combine normal distributions to have a mixture with specified kurtosis

I want to generate random samples from Normal Distributions $N(\mu_i,\sigma_i)$ by fixing kurtosis parameters ($\beta$s), as I need to simulate data by varying $\beta$ for my problem. I am trying to proceed as given in An and Ahmed (2008): Consider a mixture of two normal distribution $p_1N(\mu_1,\sigma_1)+p_2N(\mu_2,\sigma_2)$. If $\mu_1=\mu_2$, then kurtosis parameter of this mixture distribution is $\beta^*=3\frac{(p_1\sigma_1^4+p_2\sigma_2^4)}{(p_1\sigma_1^2+p_2\sigma_2^2)^2}$. In this setting $\beta^*$ has minimum value of 3 when either $p_1$ is 0 or 1 and a maximum value of $\frac{3}{4}(\frac{\sigma_1^2}{\sigma_2^2}+\frac{\sigma_2^2}{\sigma_1^2}+2)$ when $p_1=\frac{\sigma_2^2}{\sigma_1^2+\sigma_2^2}$.

• Correct me if I am wrong, but normal distribution has a fixed kurtosis. The excess kurtosis is always equal to zero. See mathworld.wolfram.com/Kurtosis.html Do you want to combine several normal distributions so their combination has a specific kurtosis? – Karel Macek Jun 25 '15 at 9:20
• yes I want to combine several normal distributions – Nighat Zahra Jun 25 '15 at 9:23
• What is fixed and what can be changed? I suppose that the kurtosis of the mixture is given. Can I change $k$ or is it fixed? Can I change $\mu_i$ as well as $\sigma_i$? – Karel Macek Jun 25 '15 at 9:37
• This is strictly insoluble, I guess. A given kurtosis could arise from many quite different distributions if the means and SDs can both vary, even for a fixed $k$. What is more those mixtures can have varying skewness. I would back up one level and tell us why you think you must approach whatever problem you have in this way. What is the real problem, in short? I suspect the advice will be to use something much simpler, say just $t$ distributions with differing degrees of freedom. – Nick Cox Jun 25 '15 at 10:36
• Could you incorporate this additional information into the question, and also clarify what you mean by Unrestricted and Restricted here? – Anthony Jun 25 '15 at 15:05