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!

| cite | improve this question | | | | |
  • 2
    $\begingroup$ 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

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.

| cite | improve this answer | | | | |
  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.