I know that using central limit theorem we approximate sum of random variables into Gaussian distribution. Is the any other approximation method available for finding the probability distribution function of sum of independent random variables ? Note that the total number of variables are very large and the random variables has exponential distribution with different rates. Please let me know.
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$\begingroup$ Direct convolution? $\endgroup$– Glen_bOct 7, 2015 at 16:14
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$\begingroup$ But in my case total number of variables are very large, like 1000. $\endgroup$– upol94Oct 7, 2015 at 16:15
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1$\begingroup$ What is the reason the Gaussian approximation is inadequate? $\endgroup$– guyOct 7, 2015 at 17:06
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1$\begingroup$ @guy That's a great question. Note, however, that the different rates of the distributions require careful understanding and application of the CLT. If those rates vary a lot, then the CLT is likely to fail. Your suggestion, then, can be turned into actionable advice only through an analysis of how much the variances of those distributions vary, in order to determine whether a Gaussian approximation will meet whatever the OP's accuracy needs may be. $\endgroup$– whuber ♦Oct 7, 2015 at 17:14
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2$\begingroup$ @whuber Right, I was probing for more information, not making any suggestion. If the concern is that there are a few whose rates are small enough to dominate the sum, then that is one thing. On the other hand, if the issue is that the data isn't iid because the rates vary (but are otherwise well behaved), then OP would presumably be interested in learning about the Lindeberg-Feller CLT. $\endgroup$– guyOct 7, 2015 at 17:18
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2 Answers
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There is the Edgeworth series, which can be thought of as an adjustment to the central limit theorem that uses the cumulants of your variables.
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Somewhat related to the answer by Hong Oi, there is saddlepoint approximations, which can be used in most cases where the central limit theorem can be used, and then gives much more accurate approximations. Saddlepoint methods are somewhat better behaved than Edgeworth series; they can never give a negative density, as is possible with Edgeworth. For an example, see my answer to General sum of Gamma distributions
answered Oct 8, 2015 at 12:27