Assume that the stochastic variables Xi (i=1,…,n) are independent, with cumulative density function (CDF) Fi(Xi) and probability density function (PDF) fi(Xi). The failure rate of each variable is increasing, i.e., hi(Xi)= fi(Xi)/(1- Fi(Xi)) is an increasing function of Xi. Assume the random variable Y=Sum(aiXi)=a1*X1+…ai*Xi+an*Xn, where ai>0 is a constant. Assume the CDF of Y is G(Y), and PDF of Y is g(Y). Define the failure rate of Y is: h(Y)= g(Y)/(1- G(Y)). Whether h(Y) is increasing in Y? How to prove?
To begin with, the IFR property is preserved under change of scale (multiplying X by a positive constant). Thus your problem reduces to whether a finite sum of independent IFR random variables is IFR. The answer is YES and clearly it suffices to prove it for the sum of 2 such random variables. Analytically, the proposition to be proved can be stated as follows:
The IFR property of probability distributions is preserved under convolution.
This was originally established by Barlow and Proschan in the 1980s (in Handbook of Reliability edited by Igor Ushakov and Colleagues). The proof is not easy, it uses the fact that IFR implies being a Polya Frequency of order 2 (PF2). There may be an easier proof by now; in any case, it is worthwhile to pursue one.
Is it correct that in your problem ai are constant but Fi may take different values?
Yes. the random variable Xi is independent, with different CDF and PDF. ai are different constant with different value, for example, maybe a1=1,a2=3,...an=10. Specifically, you can assume ai are the same, for instance, ai=1.
To begin with, the IFR property is preserved under change of scale (multiplying X by a positive constant). Thus your problem reduces to whether a finite sum of independent IFR random variables is IFR. The answer is YES and clearly it suffices to prove it for the sum of 2 such random variables. Analytically, the proposition to be proved can be stated as follows:
The IFR property of probability distributions is preserved under convolution.
This was originally established by Barlow and Proschan in the 1980s (in Handbook of Reliability edited by Igor Ushakov and Colleagues). The proof is not easy, it uses the fact that IFR implies being a Polya Frequency of order 2 (PF2). There may be an easier proof by now; in any case, it is worthwhile to pursue one.
The answer is YES, look at theorem 4.2 in Barlow&Proschan's book.
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