# Confidence interval from sample quantiles

## 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.

• 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. – Glen_b Jan 22 at 23:52
• 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. – compbiostats Jan 23 at 1:15
• A clearer understanding of the circumstances would be essential to coming up with a plausible model – Glen_b Jan 23 at 2:26
• 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. – compbiostats Jan 23 at 3:41
• Please see the updated post with further details. Please let me know if further clarification is needed. – compbiostats Jan 23 at 4:33