---
tag: T9640YF4TX
kind: definition
name: Wald test
msc: [62F03, 62F05]
defines: DPW8KNHE6P
uses: [0FREPFKNS7, G1Z3WSTE8Q, 087P03NEND, 0RFMMBHFTV]
sources:
  - url: https://en.wikipedia.org/wiki/Wald_test
---

## Definition

Let $X$ be data from a model with parameter $\theta$ in an open set $\Theta \subseteq \R^k$, and let $\theta_0 \in \Theta$.
Let $\hat\theta$ be the [[maximum likelihood estimate|0FREPFKNS7]] of $\theta$, and let $I(\hat\theta)$ be the [[Fisher information|G1Z3WSTE8Q]] of $X$ evaluated at $\hat\theta$, assumed invertible.
The *Wald statistic* for $H_0 \colon \theta = \theta_0$ is
$$W = (\hat\theta - \theta_0)^{T} I(\hat\theta) (\hat\theta - \theta_0).$$
The *Wald test* at level $\alpha$ rejects $H_0$ when $W$ exceeds the $1 - \alpha$ quantile of the [[chi-squared distribution|0RFMMBHFTV]] with $k$ degrees of freedom.

## Remarks

Under $H_0$ and the usual regularity conditions for i.i.d. samples, $W$ converges in distribution to $\chi^2_k$, so the test has asymptotic level $\alpha$.
For $k = 1$ the test rejects when $\abs{\hat\theta - \theta_0}/\widehat{\mathrm{se}} > z_{1 - \alpha/2}$, with $\widehat{\mathrm{se}} = I(\hat\theta)^{-1/2}$ and $z_{1 - \alpha/2}$ a standard normal quantile.
The observed information may replace $I(\hat\theta)$ without changing the limit law.
Example: for $X$ binomial with parameters $n$ and $p$, one has $\hat p = X/n$ and $W = (\hat p - p_0)^2 n/(\hat p(1 - \hat p))$.
Unlike the [[likelihood-ratio test|YM1BNA80RC]], the Wald test is not invariant under a reparametrization of $\theta$.