9
$\begingroup$

It is well-known that a linear combination of 2 random normal variables is also a random normal variable. Are there any common non-normal distribution families (e.g., Weibull) that also share this property? There seem to be many counterexamples. For instance, a linear combination of uniforms is not typically uniform. In particular, are there any non-normal distribution families where both of the following are true:

  1. A linear combination of two random variables from that family is equivalent to some distribution in that family.
  2. The resulting parameter(s) can be identified as a function of the original parameters and the constants in the linear combination.

I'm especially interested in this linear combination:

$Y = X_1 \cdot w + X_2 \cdot \sqrt{(1-w^2)}$

where $X_1$ and $X_2$ are sampled from some non-normal family, with parameters $\theta_1$ and $\theta_2$, and $Y$ comes from the same non-normal family with parameter $\theta_Y = f(\theta_1, \theta_2, w)$.

I'm describing a distribution family with 1 parameter for simplicity, but I'm open to distribution families with multiple parameters.

Also, I'm looking for example(s) where there is plenty of parameter space on $\theta_1$ and $\theta_2$ to work with for simulation purposes. If you can only find an example that works for some very specific $\theta_1$ and $\theta_2$, that would be less helpful.

$\endgroup$
  • $\begingroup$ Using a standard mathematical construction, any family of distributions can be included within a unique smallest family that is closed under linear combinations. This suggests that your question is not likely to yield interesting or helpful answers. Instead, consider telling us why you are interested in this property and what specific "non-normal family" of distributions you are working with. $\endgroup$ – whuber Apr 10 '15 at 17:37
  • 1
    $\begingroup$ Thanks. I'm really looking for common non-normal families (e.g., Weibull). I'll also try to clarify that the resulting parameter(s) should be functions of the original parameters for a wide variety of original parameters. That is, there should be plenty of parameter space to work with for simulation purposes. $\endgroup$ – Anthony Apr 10 '15 at 18:25
  • 4
    $\begingroup$ Assuming we are talking about arbitrary linear combinations of independent random variables, there are the (Lévy) stable distributions. The entire class of such distributions is fully characterized by their characteristic function taking a certain form. Only a select few have densities with known closed-form expressions. $\endgroup$ – cardinal Apr 10 '15 at 18:48
  • 2
    $\begingroup$ The alpha-stables mentioned by @cardinal are an answer, and if I understand correctly, the only answer if the parameters are required to be location and scale, but are there other answers if the parameters don't need to be location+scale? (Though this is perhaps so far from what OP wanted that this should be a separate question). $\endgroup$ – Juho Kokkala Aug 20 '16 at 5:36
  • 2
    $\begingroup$ @Juho I believe the answer in general is yes. Sums of distributions correspond to (pointwise) sums of cumulant generating functions (defined as the logarithm of the characteristic function), so the closure of a set of distributions under summing is contained naturally within the set of all distributions that are (real) linear combinations of those cgf's. $\endgroup$ – whuber Aug 21 '16 at 14:46
3
$\begingroup$

It is well-known that a linear combination of 2 random normal variables is also a random normal variable. Are there any common non-normal distribution families (e.g., Weibull) that also share this property?

The normal distribution satisfies a nice convolution identity: $X_1\sim N\left[\mu _1,\sigma _1^2\right],X_2\sim N\left[\mu _2,\sigma _2^2\right]\Longrightarrow X_1+X_2\sim N\left[\mu _1+\mu _2,\sigma _1^2+\sigma _2^2\right]$. If you are referring to the central limit theorem, then for example, those gamma distributions with the same shape coefficient would share that property and convolve to be gamma distributions. Please see A cautionary note regarding invocation of the central limit theorem. In general, however, with unequal shape coefficients, gamma distributions would "add" by a convolution that would not be a gamma distribution but rather a gamma function multiplying a hypergeometric function of the first kind as found in Eq. (2) of convolution of two gamma distributions. The other definition of adding, that is forming a mixture distribution of unrelated processes would not necessarily exhibit any central limit, for example, if the means are different.

There are probably other examples, I haven't done an exhaustive search. Closure for convolution does not seem to be far fetched. For linear combination, the product of Pearson VII with a Pearson VII is another Pearson VII.

$\endgroup$
  • 3
    $\begingroup$ You can add inpdependent Gammas random variables with the same scale parameter and get another gamma with that same scale parameter, but you cannot take arbitrary linear combinations. There are a number of well-known distributions for which you can take sums but not arbitrary linear combinations and stay within that family. (There's already a deleted answer here that make the same error) $\endgroup$ – Glen_b Aug 20 '16 at 3:09
  • 1
    $\begingroup$ It is true that the convolution of two gamma distributions, see Eq. 2, yields something other than a gamma distribution, if that is what you mean. $\endgroup$ – Carl Aug 20 '16 at 3:32
  • $\begingroup$ The article clearly states that a linear combination of gammas is not gamma (aside from the same exception I mentioned already) and appears completely consistent with what I said. I'm not sure what you're asking me about, but the article supports my claim that your answer seems to assert something that's not the case. $\endgroup$ – Glen_b Aug 20 '16 at 3:33
  • $\begingroup$ Not asking, saying what the sum is in general. I modified the answer to say "some." If that is not good enough, I will delete my humble attempt at helping. And that I am asking, "Good enough, or not?" $\endgroup$ – Carl Aug 20 '16 at 3:37
  • 2
    $\begingroup$ It's now a bit on the light side for an answer. You might want to move some of the information from your comment up to the answer (the information relating to what was in the paper and the link to it, at the least, though I'd include a proper reference) $\endgroup$ – Glen_b Aug 20 '16 at 4:26
2
$\begingroup$

It is well-known that a linear combination of 2 random normal variables is also a random normal variable. Are there any common non-normal distribution families (e.g., Weibull) that also share this property?

I sounds like you are looking for the class of Levy-stable distributions. This is the class $\mathscr{P}$ of all distributions $\mathcal{P} \in \mathscr{P}$ that satisfy the stability property:

$$X_1, X_2, X_3 \sim \text{IID } \mathcal{P} \quad \quad \implies \quad \quad (\forall a) (\forall b) (\exists c>0) (\exists d): \ aX_1 + bX_2 \overset{\text{Dist}}{\sim} c X_3 + d.$$

In other words, for every distribution in this class, if you take a linear function of two independent random variables with that distribution, then this has the same distribution as an affine function of a single random variable with that distribution. (Note that this stability requirement can be tightened by setting $d=0$, which gives the subclass of strictly stable distributions.)

The Levy-stable distributions can be considered to be a family of distributions in its own right, and in this sense it is the only family of distributions with this stability property, since (by definition) it encompasses all distributions with this property. The normal distribution falls within the class of Levy-stable distributions, as does the Cauchy distribution, the Landau distribution, and the Holtsmark distribution.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.