# Non-normal distributions with zero skewness and zero excess kurtosis?

Mostly theoretical question. Are there any examples of non-normal distributions that has first four moment equal to those of normal? Could they exist in theory?

• Considering even just a mixture of 2 normals (5 parameters--2 means, 2 variances, and the mixture probability), you can solve for a wide variety of first four moments. Commented Dec 4, 2019 at 22:45
• I gave a discrete example at a later near-duplicate of this question: stats.stackexchange.com/a/378923/225256 Commented May 28, 2021 at 16:21
• @SheridanGrant, a mixture of two normals won't work. Let $X$ be a mixture with $p$ of $N(m,s)$ and $(1-p)$ of $N(n,t)$. Then forcing $X$ and $N(0,1)$ to have the same first, second and third moments leads to $p=n/(n-m)$, $s=\sqrt{(3-m^2+2mn)/3}$, $t=\sqrt{(3-n^2+2mn)/3}$. Under those conditions, the difference in fourth moments between $X$ and $N(0,1)$ is just $2mn(m^2-mn+n^2)/3$; if that is $0$ then $m=0$ or $n=0$, and $X$ is just a single standard normal. Commented May 29, 2021 at 12:13
• @MattF.--huh, cool. I did a quick search for an analysis of "how many moments can you match with a mixture of k Gaussians" and didn't find anything, do you know of any results? Commented Jun 10, 2021 at 6:10

Yes, examples with skewness and excess kurtosis both zero are relatively easy to construct. (Indeed examples (a) to (d) below also have Pearson mean-median skewness 0)

(a) For example, in this answer an example is given by taking a 50-50 mixture of a gamma variate, (which I call $$X$$), and the negative of a second one, which has a density that looks like this:

Clearly the result is symmetric and not normal. The scale parameter is unimportant here, so we can make it 1. Careful choice of the shape parameter of the gamma yields the required kurtosis:

1. The variance of this double-gamma ($$Y$$) is easy to work out in terms of the gamma variate it's based on: $$\text{Var}(Y)=E(X^2)=\text{Var}(X)+E(X)^2=\alpha+\alpha^2$$.

2. The fourth central moment of the variable $$Y$$ is the same as $$E(X^4)$$, which for a gamma($$\alpha$$) is $$\alpha(\alpha+1)(\alpha+2)(\alpha+3)$$

As a result the kurtosis is $$\frac{\alpha(\alpha+1)(\alpha+2)(\alpha+3)}{\alpha^2(\alpha+1)^2}=\frac{(\alpha+2)(\alpha+3)}{\alpha(\alpha+1)}$$. This is $$3$$ when $$(\alpha+2)(\alpha+3)=3\alpha(\alpha+1)$$, which happens when $$\alpha=(\sqrt{13}+1)/2\approx 2.303$$.

(b) We could also create an example as a scale mixture of two uniforms. Let $$U_1\sim U(-1,1)$$, $$U_2\sim U(-a,a)$$, $$B\sim \text{Bernoulli}(\frac12)$$ and let $$M=B U_1+(1-B) U_2$$. Clearly by considering that $$M$$ is symmetric and has finite range, we must have $$E(M)=0$$; the skewness will also be 0 and central moments and raw moments will be the same.

$$\text{Var}(M)=E(M^2)=\frac12\text{Var}(U_1)+\frac12\text{Var}(U_2)=\frac16[1+a^2]$$.

Similarly, $$E(M^4)=\frac{1}{10} (1+a^4)$$ and so the kurtosis is $$\frac{\frac{1}{10} (1+a^4)}{[\frac16 (1+a^2)]^2}=3.6\frac{1+a^4}{(1+a^2)^2}$$

If we choose $$a=\sqrt{5+\sqrt{24}}\approx 3.1463$$, then kurtosis is 3, and the density looks like this:

(c) here's a fun example. Let $$X_i\stackrel{_\text{iid}}{\sim}\text{Pois}(\lambda)$$, for $$i=1,2$$.

Let $$Y$$ be a 50-50 mixture of $$\sqrt{X_1}$$ and $$-\sqrt{X_2}$$:

by symmetry $$E(Y)=0$$ (we also need $$E(|Y|)$$ to be finite but given $$E(X_1)$$ is finite, we have that)

$$Var(Y)=E(Y^2)=E(X_1)=\lambda$$

by symmetry (and the fact that the absolute 3rd moment exists) skew=0

4th moment: $$E(Y^4) = E(X_1^2) = \lambda+\lambda^2$$

kurtosis = $$\frac{\lambda+\lambda^2}{\lambda^2}= 1+1/\lambda$$

so when $$\lambda=\frac12$$, kurtosis is 3. This is the case illustrated above.

(d) all my examples so far have been symmetric, since symmetric answers are easier to create -- but asymmetric solutions are also possible. Here's a discrete example.

(e) Now, here's an asymmetric continuous family. It will perhaps be the most surprising for some readers, so I'll describe it in detail. I'll begin by describing a discrete example and then build a continuous example from it (indeed I could have started with the one in (d), and it would have been simpler to play with, but I didn't, so we also have another discrete example for free).

$$\:\,$$ (i) At $$x=-2,1$$ and $$m=\frac12 (5+\sqrt{33})$$ ($$\approx 5.3723$$) place probabilities of $$p_{-2}= \frac{1}{36}(7+\sqrt{33})$$, $$p_1=\frac{1}{36}(17+\sqrt{33})$$, and $$p_m=\frac{1}{36}(12-2\sqrt{33})$$ (approximately 35.402%, 63.179% and 1.419%), respectively. This asymmetric three-point discrete distribution has zero skewness and zero excess kurtosis (as with all the above examples, it also has mean zero, which simplifies the calculations).

$$\:$$ (ii) Now, let's make a continuous mixture. Centered at each of the ordinates above (-2,1,m), place a Gaussian kernel with common standard deviation $$\sigma$$, and probability-weight given by the probabilities above (i.e. $$w=(p_{-2},p_1,p_m)$$). Phrased another way, take a mixture of three Gaussians with means at $$-2,1$$ and $$m$$ each with standard deviation $$\sigma$$ in the proportions $$(p_{-2},p_1,p_m)$$ respectively. For any choice of $$\sigma$$ the resulting continuous distribution has skewness 0 and excess kurtosis 0.

Here's one example (here the common $$\sigma$$ for the normal components is 1.25):

(The marks below the density show the locations of the centers of the Gaussian components.)

As you see, none of these examples look particularly "normal". It would be a simple matter to make any number of discrete, continuous or mixed variables with the same properties. While most of my examples were constructed as mixtures, there's nothing special about mixtures, other than they're often a convenient way to make distributions with properties the way you want, a bit like building things with Lego.

This answer gives some additional details on kurtosis that should make some of the considerations involved in constructing other examples a little clearer.

You could match more moments in similar fashion, though it requires more effort to do so. However, because the MGF of the normal exists, you can't match all integer moments of a normal with some non-normal distribution, since that would mean their MGFs match, implying the second distribution was normal as well.

• In the uniform example (Example b) the pdf of M as given is not correct. It should be a trapezium. In R simulations with very large numbers the sample kurtosis of M turns out to be 2, not 3 as the example suggests.
– user187833
Commented Apr 19 at 13:15
• @kadirb Although the notation is indeed ambiguous, it is clear in the context that $$\frac{1}{2}U_1+\frac{1}{2}U_2$$ is shorthand for a mixture of distributions (as stated), not a linear combination of random variables as you seem to interpret it.
– whuber
Commented Apr 19 at 21:43
• @kadirdb Apologies for the loose notation. I have amended it -- the natural way would be to average the cdfs or pmfs, but the smallest edit involved leaving it in terms of random variables by introducing a Bernoulli to select between the uniforms. Commented Apr 20 at 7:24