If you're interested in a plot I wouldn't use random values for the
ex, but some approximation for expected order statistics. In terms of the appearance of a more-or-less straight plot the shift parameter and the rate parameter (or scale parameter, depending on your parameterization) shouldn't matter. I wouldn't bother with anything but expected scores for a standard exponential.
Actually the exact expected order statistics aren't hard to do so you could use those -- but it also doesn't make a huge amount of difference. I'm going to suggest using a quick approximation for all but the maximum order statistic, which we will do exactly.
Note that if you don't shift the scores by 2 the intercept won't be 0 and if the scale of your observations is not the same as your expected scores then the slope won't be 1. This in no way affects the usefulness of the plot -- standard exponential scores are fine. If you want to show a theoretical rate parameter of 0.6, you can plot a line with slope 1/0.6 and intercept 2.
Here's a random sample of simulated shifted exponential random variables with rate 0.6 and shift 2, plotted against approximate expected exponential scores:
y <- 2+rexp(100,0.6)
n <- length(y)
Ei <- qexp(ppoints(n)) # a good approximation apart from the last point
Ei[n] <- log(n)+.5772+0.5/n # better approximation for the last point
plot( y ~ Ei[order(order(y))], ylab="Sample quantiles",
xlab="Theoretical quantiles", main="Exponential Q-Q Plot")
[Another way to get a good approximation for all the points is to use
qexp(ppoints(n,0.4385)) -- this is pretty reasonable for n>20 or so and
gets better for larger n. It's less accurate for the last point at smaller n but accuracy for the last point generally matters less then]
The slope in the plot (the scale) could be approximated as
implying a rate parameter of:
However, if you want to draw a line on the plot, a more robust approach, perhaps one based on quantiles like that in
qqnorm probably makes more sense.
Pretty close to 0.6 (or you could calculate the ML estimate which is
1/mean(y-2), which is a little smaller in this case).
So the intercept and slope in this plot are informative, showing us the shift and scale parameter estimates for the shifted exponential.
If you want exact expected order statistics, see Did's answer here
Order statistics of i.i.d. exponentially distributed sample
-- but for typical sample sizes it makes such a small difference you can't see it, and my approximation is a bit shorter. If you prefer to use that, though, it's quite simple to do:
n <- length(y)
H <- cumsum(1/(1:length(y)))
Hn <- tail(H,1)
Ei <- Hn-rev(c(0,head(H,-1)))