Estimation & Confidence Intervals

An estimate is a number computed from observed data; the estimand is the unknown quantity we want. Keeping them separate prevents “we observed 65%” from silently becoming “the true probability is exactly 65%.”

Point estimate and uncertainty #

With $k$ successes in $n$ binary trials, the sample proportion is

$$\hat q=\frac{k}{n}.$$

Under independent Bernoulli sampling, its estimated standard error is

$$SE(\hat q)\approx\sqrt{\frac{\hat q(1-\hat q)}{n}}.$$

A rough large-sample 95% interval is $\hat q\pm1.96SE$. For 65 wins out of 100, $\hat q=0.65$, $SE\approx0.0477$, and the rough interval is about $[0.557,0.743]$. Near 0 or 1 or with small $n$, use a binomial method such as Wilson rather than this symmetric shortcut.

Life example #

If 65 of 100 sampled customers prefer a new design, 65% is evidence about the sampled population under the sampling design. It is not automatically evidence about every country, future year, or customer segment. Sampling bias is not repaired by a narrow formula.

Trading estimate #

A model that won 65 of 100 comparable trades has an observed accuracy of 65%, but the uncertainty is wide relative to many trading edges. Buying at 0.60 based on $q=0.65$ assumes a five-cent edge; sampling error alone can easily be larger.

Application to no-hedge #

Estimate $q$ on the cases the no-hedge strategy actually fills, not on all displayed opportunities. Execution selection can produce a different population. Confidence intervals should respect time dependence and shared events; otherwise they give false precision.