This paper gives the i.ni.d. example of "measurements accurate to the nearest foot may be combined with measurements accurate to the nearest inch". Can we simply pool that data to satisfy i.i.d.? (The disadvantage being that pooling throws away information.) More generally, can we combine i.ni.d. random variables into i.i.d. random variables by using the weighted convolution of the non-identical distributions as our new identical distribution?
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$\begingroup$ The title is much more general than the example. Do you want the full generality of the title or something much more like the specifics of the example? $\endgroup$– Glen_bCommented Nov 15, 2014 at 4:07
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$\begingroup$ I'd prefer a general answer. However, if the answer is no in general, I'd be interested to hear under which conditions the answer is yes. $\endgroup$– AntlersCommented Nov 15, 2014 at 4:15
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$\begingroup$ This question needs clarification. For instance the sum of a collection of variables is iid (because there is only one of them!). $\endgroup$– whuber ♦Commented Nov 15, 2014 at 14:25
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1 Answer
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An answer to the general question:
Can independent non-identically distributed random variables be convert to i.i.d. random variables?
Sure, at least for continuous variables. Let $X,Y$ be two independent random variables with distribution functions $G_1$ and $G_2$ respectively. Then $U=G_1(X)$ and $V=G_2(Y)$ are i.i.d. standard uniform.
If $F$ is any continuous invertible cdf then $W=F^{-1}(G_1(X))$ and $Z=F^{-1}(G_2(Y))$ are i.i.d. with distribution $F$.
If that answer is in any way unsatisfying, you probably want to also restrict the general version of your question in some as-yet-unstated way; I'm not quite sure what you'd be looking for beyond this, though.
