Neyman Allocation in Stratified Sampling

You have budget to audit 500 loans out of a portfolio of 50,000, split between two segments: large corporate loans and small retail loans, and you want the most precise possible estimate of the portfolio's average default probability. You could sample proportionally to segment size, or evenly, but neither is obviously best — because precision depends not just on how many loans are in each segment, but on how variable the default outcomes are within each one. If corporate loan outcomes are wildly different from each other (some default, most don't, in unpredictable ways) while retail loans are fairly uniform, then auditing more corporate loans buys you far more precision per loan sampled. Neyman allocation is the formula that tells you exactly how to split your fixed budget to minimize the uncertainty of your final estimate.

An analogy: polling a noisy room

Imagine polling opinions in two rooms before a vote — one room is unanimous (everyone agrees), the other is split and unpredictable. Polling ten more people in the unanimous room teaches you almost nothing new; you already know the answer. Polling ten more people in the split room teaches you a lot, because there's real uncertainty left to resolve. If you only have 100 total interviews to spend, common sense says spend more of them in the noisy room. Neyman allocation is that common sense turned into an exact formula, weighted by both the room's size and its internal disagreement.

The formula, one piece at a time

Divide the population into strata (subgroups), indexed $h$, where stratum $h$ has $N_h$ members and a standard deviation $\sigma_h$ of the quantity you're measuring within that stratum. You have a total sample size $n$ to allocate. Neyman allocation sets the sample size in stratum $h$ to

$$n_h = n \cdot \frac{N_h \sigma_h}{\sum_k N_k \sigma_k}.$$

In plain English: give each stratum a share of your total sample proportional to its size times its internal variability — a big, noisy stratum gets a large share, a small or uniform stratum gets a small share, and a stratum that's both small and uniform gets almost nothing, because there's little left to learn there and few members to learn it from. This allocation is the one that minimizes the variance of the overall estimate for a fixed total sample size $n$, which is why it's the default choice whenever stratum variances are known or can be estimated from a pilot sample.

Worked example 1: the loan audit

Corporate loans: $N_1 = 5{,}000$, with default outcomes highly variable, $\sigma_1 = 0.45$. Retail loans: $N_2 = 45{,}000$, with more uniform outcomes, $\sigma_2 = 0.15$. Total budget $n = 500$. Compute the weights: $N_1 \sigma_1 = 5{,}000 \times 0.45 = 2{,}250$, and $N_2 \sigma_2 = 45{,}000 \times 0.15 = 6{,}750$, summing to $9{,}000$. Then

$$n_1 = 500 \times \frac{2{,}250}{9{,}000} = 125, \qquad n_2 = 500 \times \frac{6{,}750}{9{,}000} = 375.$$

So despite corporate loans being only 10% of the portfolio's count, Neyman allocation assigns them 25% of the audit sample, because their higher variability makes each corporate observation worth more information than a retail one.

Worked example 2: comparing to proportional allocation

Proportional allocation (sampling in proportion to $N_h$ alone, ignoring variance) would give corporate loans only $500 \times (5{,}000/50{,}000) = 50$ samples and retail loans $450$. Compare the resulting standard errors of the combined estimate using the standard stratified-variance formula; with Neyman's 125/375 split the corporate segment's large $\sigma_1$ is estimated from a sample five times bigger than proportional allocation would give it, sharply cutting its contribution to the total variance, while the retail segment loses relatively few samples because its outcomes were uniform anyway. In realistic loan-audit settings this reallocation typically cuts the standard error of the overall default-rate estimate by 15–30% for the same total sample size $n = 500$ — the same audit budget, spent more intelligently, buys a materially tighter estimate.

What this means in practice

Neyman allocation shows up anywhere a fixed sampling or auditing budget must be split across heterogeneous subgroups: model-risk teams choosing which loan segments to audit, backtests that stratify by volatility regime and want to sample rebalance dates efficiently, or A/B tests stratified by user segment. The catch is that Neyman allocation needs the stratum standard deviations $\sigma_h$ up front, which in practice come from a small pilot sample or last year's data — an imperfect but usually good-enough proxy.