Suppose that $X_1, X_2, \ldots, X_n$ are i.i.d. random variables, with failure rate $r(t)$. Suppose $r(t)$ is increasing. Is it true that the sum $X_1 + X_2 + \ldots + X_n$ also has an increasing failure rate?
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It can't be true in general, because the central limit theorem says that the sum approaches a Normal distribution, which is not IFR.
Thomas Lumley
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I found a reference that states the result: [On Increasing-Failure-Rate Random Variables](https://www.jstor.org/stable/pdf/30040857.pdf): "...Barlow and Proschan (1975) showed that the sum of a fixed number of independent IFR random variables is IFR, Shanthikumar (1988) showed that the sum of a geometric number of independent, identically distributed decreasing-failure-rate (DFR) random variables is DFR,..." I am closing this question. – Harish Guda Apr 20 '21 at 03:08
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I found a reference that states the result: On Increasing-Failure-Rate Random Variables (J. Applied Probability): "...Barlow and Proschan (1975) showed that the sum of a fixed number of independent IFR random variables is IFR, Shanthikumar (1988) showed that the sum of a geometric number of independent, identically distributed decreasing-failure-rate (DFR) random variables is DFR,..." I am closing this question.