Error Bars on a Fill Rate

Question

Out of 400 passive child orders, 80 were filled, an estimated fill rate of hat p = 0.20. Compute the standard error of this estimate and an approximate 95% confidence interval. What assumption is doing the heavy lifting?

Accepted Answer

A fill indicator is Bernoulli(p) with variance p(1-p), so the sample proportion has textSE(hat p) = sqrtfrachat p(1-hat p)n = sqrtfrac0.2 times 0.8400 = sqrt0.0004 = mathbf0.02. The 95% interval is 0.20 pm 1.96 times 0.02 approx [0.16, 0.24]. Even 400 orders leave a 4-point fill rate uncertain to pm4 points, proportions are data-hungry. The assumption doing the heavy lifting: independence. Fills cluster, the same quiet morning or the same fast market hits whole batches of orders at once, so the 400 orders might carry the information of far fewer independent trials. Positive correlation between outcomes makes the true SE larger than 0.02; the honest fix is to cluster by day or by regime (compute the SE from daily fill rates), or block-bootstrap. Secondary checks worth saying: the normal approximation needs nhat p and n(1-hat p) comfortably large (here 80 and 320, fine); near hat p = 0 or 1 use Wilson or exact intervals instead, the naive interval can poke outside [0,1]. Desk relevance: fill rates, hit rates, and adverse-selection frequencies are the bread-and-butter Bernoulli estimates of execution research, and the first interview upgrade is always the same sentence: quote the erro

Error Bars on a Fill Rate

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