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Context

I am running some stochastic simulations in R to determine appropriate sample sizes for a biological problem of interest to me.

My method iteratively generates "accumulation" curves which eventually tend to an asymptote. These curves are based on computing cumulative means.

Specifics

I wish to compute a 95% confidence interval (CI) for the "true" endpoint ($x$-value) of these curves as they tend to the asymptote. Call this "true" endpoint $\theta$.

I have a 95% CI for the "true" $y$, call it ($L$, $U$), which is formed via taking the 2.5th and 97.5th percentiles of the resulting sample cumulative means.

The iterated function I am using is an estimator of $\theta$. This estimator takes the form:

$\hat{\theta} = \frac{\hat{x}\hat{z}}{\hat{y}}$

$\hat{x}$ is a sequence of the observed number of "types", and is a random variable on [1, 2, ..., $m$], with $m$ not known a priori.

$\hat{z}$ is a sequence of the observed number of unique "types", and is a discrete random variable on [1, 2, ..., $n$], with $n$ not known a priori.

Also, $\hat{x} \geq \hat{z}$, and consequently, $m \geq n$. Sensible guesses are made for the unknown $m$ and $n$.

It is known that $\hat{y}$ approaches $\hat{z}$ asymptotically, so that $\hat{x}$ approaches $\hat{\theta}$ in the limit sense. $\hat{\theta}$ can be thought of as the sample size required to observe all unique types.

Issues

I compute a (likely crude) 95% CI for $\theta$ via ($\frac{\hat{\theta}}{U}$, $\frac{\hat{\theta}}{L}$).

I find a 95% CI for the "true" $y$ of ($L$, $U$) = (10, 13).

My estimate of $\theta$ is $\hat{\theta}$ = 440, with $\hat{z}$ = 18 giving a 95% CI for $\theta$ of ($\frac{440(18)}{13}$, $\frac{440(18)}{10}$) = (610, 792). Something is clearly amiss here... $\hat{\theta}$ should lie within the upper and lower limits of the CI, but it doesn't.

I'm beginning to doubt this approach for finding a CI.

My Question

Any ideas on why my CI approach does not generate a sensible interval and potentially how I can easily obtain one from the given information?

The above situation is similar to the Coupon Collector Problem, as well as the Germain Tank Problem, but I found that neither corresponds to this problem exactly.

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    $\begingroup$ I don't really follow the setup here, but there are potentially several issues to consider. One of the things you need to be very careful about is to avoid treating a succession of cumulative quantities as if they were independent. $\endgroup$ – Glen_b Jan 22 at 23:52
  • $\begingroup$ Thanks. Yes, I suspect that covariance/correlation between observations will be of issue in deriving a meaningful CI around $\theta$. I guess my next question is how best to account for this. I've done some quick googling and searching of CV, but no solution immediately pops up. $\endgroup$ – compbiostats Jan 23 at 1:15
  • $\begingroup$ A clearer understanding of the circumstances would be essential to coming up with a plausible model $\endgroup$ – Glen_b Jan 23 at 2:26
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    $\begingroup$ I have edited my post that corrects the CI for $\theta$. I mistakingly left out the value of $z$, which is used in calculating the interval. I will add further details as to the nature of the variables and how they are generated. $\endgroup$ – compbiostats Jan 23 at 3:41
  • $\begingroup$ Please see the updated post with further details. Please let me know if further clarification is needed. $\endgroup$ – compbiostats Jan 23 at 4:33

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