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Is it possible to estimate variance of $Y$ if we don't know PMF of $X$? More over does it exist? To make question clear, let's assume some array of time stamps differences $ts_{diff} = ts(n) - ts(n-1)$. we can estimate mean and variance of these differences. But now we want to estimate Y = 1/X - frequency and it's mean and variance. Is it possible?

I looked some close questions like Variance of X/Y and Variance of the reciprocal II

reference to a book would be greatly appreciated!

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    $\begingroup$ If your starting point is the sample variance of the data, why not invert the data (1/data), and calculate the sample variance of that? $\endgroup$ – wolfies Apr 23 '17 at 9:02
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Yes, it is possible. The basic approach uses Taylor expansion of the transformation function. Suppose you have a best-estimate $x$ and an error-bar $\delta x$, and want to know a best-estimate and error-bar for $y = f(x)$. Write a Taylor expansion around $x$:

$$f(x + \delta x) = f(x) + (\delta x) f'(x)$$

This is telling you, as $x$ varies around your best-estimate, about how much the function value will vary around $f(x)$. So interpret the first term as the best-estimate $y$ and the magnitude of the second term as the error-bar $\delta y$. So for $f(x) = 1/x$, $y = 1/x$ and $\delta y$ = $(\delta x) / x^2$. Notice this implies that $(\delta y)/y = (\delta x)/ x$, i.e. for this particular transformation, the fractional accuracy is conserved.

It's important to realize that this is very much an estimate. $(\delta y)^{2}$ isn't exactly the variance of $y$ even if $(\delta x)^{2}$ is exactly the variance of $x$. And the estimate can be very wrong in regions where the one-term Taylor expansion is a poor approximation. Buy you did just ask for an estimate, and in many practically useful cases this estimate is quite good.

Because this is a hand-waving estimate and not a strict bound, I think this technique tends to be better-known to scientists and engineers than to statisticians. It's typically called "error propagation". There is a Wikipedia article, and the first book I could think of on my own shelf that covers it is G. Knoll, "Radiation Detection and Measurement", a handbook for experimental nuclear physics, in Chapter 4 on statistical techniques.

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    $\begingroup$ The question seems as though it might be about estimating a reciprocal average from a sample (i.e. rather than a PDF)? This would be somewhat different (and in some cases you might just invert the $X$ average). $\endgroup$ – GeoMatt22 Apr 23 '17 at 8:07

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