Assuming that I have 2 distinct random variables that follow a Weibull distribution, what's the standard error* of the ratio of these two random variables?

Basically I have $X \sim \text{Weibull}, Y \sim \text{Weibull}$ and I want the SE of $Z = \frac{X-Y}{Y}$.

I've followed some answers here on StackExchange and this article to obtain the expected value and the variance of Z, but now I'm pretty much stuck with these 2 values.

Aditional questions:

  1. Should I just compute the standard deviation and assume that it is my standard error? (pretty much what this article says at chapter 4, page 293 between equation 33 and 34 - "...is the standard deviation or the standard error...")

  2. Is it correct to use the expect value of a Weibull distribution instead of some specific quantile since that distribution is very skewed? (my distributions have both a shape between (0,1))

  3. Would a gamma distribution follow the same "algorithm" to obtain the SE?

* by Standard Error (SE) I mean Standard Error of the Mean and Standard Error of some specific Quantile (say 30th)

  • 1
    $\begingroup$ That's not a ratio of distributions, you're asking about the distribution of the ratio (of random variables, allowing for the "-1" at the end being straightforward). Yes, generally standard error simply means "standard deviation of the distribution of" -- so the standard error of $X/Y$ is the standard deviation of the distribution of $X/Y$. $\endgroup$ – Glen_b May 24 '15 at 7:07

The standard error applies to a statistic; so without further information about the nature of the statistic you are interested in, the question does not have a clear answer.

To illustrate: you have defined a random variable $Z = (X-Y)/Y$ where $X, Y$ are (assumed independent) Weibull variables. We can compute the variance of $Z$, from which the standard deviation of the distribution is derived. But there are any number of statistics that can be computed from this distribution; e.g., if we observe a sample $Z_1, Z_2, \ldots, Z_n$, we could compute a sample mean, a sample standard deviation, or an order statistic, etc. These statistics each have their own standard error, which will be a function of the sample size, that characterizes how "close" these statistics will be to the corresponding distributional property (e.g., the standard error of the mean measures the sample mean's "closeness" to the population mean).

Since you do not specify a sample nor a statistic calculated from a sample, it is difficult to guess what you think is the "standard error." If we are talking about a sample of size $n = 1$ and the statistic is the identity, then the standard error is simply the standard deviation of the distribution of $Z$, which in turn is $$\sqrt{\operatorname{Var}[X/Y]}.$$

  • $\begingroup$ Thanks for the answer, it clarifies things better. I forgot to mention: the sample size is about 10k observations for each X and Y and the statistic in question is either a sample mean or 30th quantile. For these statistics, is there a generic way of obtaining the standard error? (standard error of the mean and standard error of the 30th quantile) $\endgroup$ – Thiago Balbo May 23 '15 at 19:02

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