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Consider the two-component mixture $$ F(z)=\lambda F_1(z)+(1-\lambda)F_2(z) $$ where all the $F$'s are CDFs and $\lambda\in [0,1]$.

A1: Assume that $F(z)=0$ $\forall z\leq 0$.


Claim:

A1 implies that $F(\cdot)$ is not compatible with

  • $F_1(\cdot)$ such that $F_1(z)>0$ for some $z\leq 0$ and $\lambda>0$.

  • (In light of the previous bullet point) $F_1(\cdot)$ and $F_2(\cdot)$ both Logistic. This is because the Logistic CDF is zero only in the limit for any value of the scale and location.


Question: is the claim correct?

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If for some $z\le 0 $ ,$F(z) = 0$ and $F_{1}(z) > 0$ , then $F_{2}(z) < 0$ , but a CDF can't have this property .

Since the CDF of the logistic distribution has positive values on the negative domain the second claim is also true .

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  • $\begingroup$ Thanks. Could you add to your answer a reference to the two bullet points of my claim? In particular, I understand from your answer that my first bullet point is correct. What about the second? $\endgroup$ – TEX Feb 7 at 11:52
  • $\begingroup$ what do you mean by Logistic ... is ther a formula for this CDF $\endgroup$ – phdmba7of12 Feb 7 at 11:54
  • $\begingroup$ @phdmba7of12 Yes, en.wikipedia.org/wiki/Logistic_distribution $\endgroup$ – TEX Feb 7 at 11:55
  • $\begingroup$ if this is for applied purposes, the $F_1$ and $F_2$ would likely be truncated yes $\endgroup$ – phdmba7of12 Feb 7 at 12:00
  • $\begingroup$ I edited the answer . The second point is also true . $\endgroup$ – Popescu Claudiu Feb 7 at 12:06

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