Nuisance parameters
Section of Estimation, part of Statistical inference
Nuisance parameters
| Type | Section |
| Taught in | Estimation |
| Part | Statistical inference |
Often only one coordinate of \theta is the target. For X\sim\operatorname{N}(\mu,\sigma^2) with both parameters unknown, a question about \mu treats \sigma^2 as a nuisance parameter: it is unknown, it is not the quantity to be reported, and it still appears in \operatorname{Var}(\overline X)=\sigma^2/n. The standard error replaces it by s/\sqrt n. The t pivot is the device that removes it from the distribution of (\overline X-\mu)/(s/\sqrt n). A sufficient statistic for the whole of \theta may be larger than a sufficient statistic for the target with the nuisance held fixed. The normal sample needs both \overline X and s^2 once \sigma^2 is unknown, and only \overline X when \sigma^2 is known.
The information matrix makes the cost of a nuisance parameter visible. The variance bound for one coordinate is the corresponding diagonal entry of \mathcal{I}(\theta)^{-1}, which is at least as large as the reciprocal of the diagonal entry of \mathcal{I}(\theta). Estimating the nuisance parameter cannot tighten the bound for the target, and it widens the bound when the off-diagonal information is not zero.
Match moments, or maximise the likelihood. Then judge the estimator by its sampling distribution: bias, variance, and their combination, the mean squared error. Unbiased is not the same as best.
Used in
- Taught in Estimation
- All 6 questions on Estimation
See also
Estimator · Method of moments · Sample correlation coefficient · Likelihood function · Maximum likelihood · Invariance of maximum likelihood · Unbiased estimator · Mean squared error