Local Alternatives and Noncentral Likelihood Limits

Why Local Alternatives Matter

A test’s limiting distribution under a null hypothesis describes its false-positive behavior, but not its power against alternatives that become closer to the null as the sample grows. To study that boundary between null behavior and detectable departures, let the parameter move toward the null at the rate $n^{-1/2}$. At this scale, the departure and the estimator’s sampling error are of comparable size, so the limiting test statistic is generally noncentral chi-square rather than central chi-square.

The noncentrality parameter records how far the local alternative moves away from the null in units determined by Fisher information. It therefore depends both on the direction of the departure and on how precisely the model estimates that direction.

Local Alternatives and the Null Geometry

Let $X_1,\ldots,X_n$ be independent and identically distributed observations from a regular parametric model with parameter $\theta\in\mathbb{R}^p$. Write $\ell_n(\theta)$ for the sample log-likelihood, and suppose the null hypothesis is specified locally by $q$ smooth restrictions $g(\theta)=0$, where $g:\mathbb{R}^p\to\mathbb{R}^q$. At the null point $\theta_0$, assume $g(\theta_0)=0$ and the $q\times p$ derivative matrix $A=Dg(\theta_0)$ has rank $q$. The null’s tangent space at $\theta_0$ is $T=\{v:Av=0\}$.

Definition: A sequence of local alternatives about $\theta_0$ has the form $\theta_n=\theta_0+h/\sqrt{n}$, for a fixed vector $h\in\mathbb{R}^p$. The vector $h$ is the local displacement. Its component normal to the null tangent space determines the first-order departure from the null.

Assume the model is differentiable in quadratic mean near $\theta_0$, the per-observation Fisher information $I=I(\theta_0)$ is finite and positive definite, and the usual likelihood regularity conditions hold: the score central limit theorem and information identity apply, the local log-likelihood admits a uniform quadratic expansion for bounded local parameter vectors, and the relevant unrestricted and restricted maximum likelihood estimators localize in this expansion. These conditions exclude such complications as nonidentifiability and a singular information matrix. Under these assumptions, the local log-likelihood has the form

\ell_n(\theta_0+t/\sqrt{n})-\ell_n(\theta_0) = t^T\Delta_n-\frac{1}{2}t^TIt+o_p(1),

uniformly for $t$ in bounded sets, where $\Delta_n=n^{-1/2}\sum_{i=1}^n s_{\theta_0}(X_i)$ is the normalized score. Under the local alternatives $\theta_n=\theta_0+h/\sqrt{n}$, regularity gives $\Delta_n\xrightarrow{d}N(Ih,I)$. This shift in the score distribution is why the null limits become noncentral.

Likelihood-Ratio Limit Under a Local Alternative

Let $\hat{\theta}$ be the unrestricted maximum likelihood estimator and $\tilde{\theta}$ the maximum likelihood estimator subject to $g(\theta)=0$. The likelihood-ratio statistic is $2\{\ell_n(\hat{\theta})-\ell_n(\tilde{\theta})\}$. The following result identifies its local limit and its noncentrality parameter.

Theorem: Under the regularity conditions above, under $\theta_n=\theta_0+h/\sqrt{n}$, $$ 2\{\ell_n(\hat{\theta})-\ell_n(\tilde{\theta})\} \xrightarrow{d}\chi^2_q(\lambda), \qquad \lambda=(Ah)^T(AI^{-1}A^T)^{-1}(Ah). $$ Here $\chi^2_q(\lambda)$ denotes a noncentral chi-square distribution with $q$ degrees of freedom and noncentrality parameter $\lambda$.

Proof. In the local coordinate $t=\sqrt{n}(\theta-\theta_0)$, the null is approximated by $T=\{t:At=0\}$, since $g(\theta_0+t/\sqrt{n})=At/\sqrt{n}+o(n^{-1/2})$ for bounded $t$. The quadratic log-likelihood to maximize is $Q(t)=t^T\Delta_n-\tfrac12t^TIt$. Set $z=I^{-1/2}\Delta_n$ and $w=I^{1/2}t$. Then $Q(t)=w^Tz-\tfrac12\|w\|^2$, and the transformed null tangent space is $S=I^{1/2}T$. Completing the square gives $Q(t)=\tfrac12\|z\|^2-\tfrac12\|w-z\|^2$. Thus the unrestricted maximum is $\tfrac12\|z\|^2$, while the restricted maximum is $\tfrac12\|P_Sz\|^2$, where $P_S$ is orthogonal projection onto $S$. Twice their difference is $\|P_{S^\perp}z\|^2+o_p(1)$.

Under the local alternatives, $z\xrightarrow{d}N(I^{1/2}h,I_p)$. The space $S^\perp$ is the column space of $I^{-1/2}A^T$. Therefore the squared length of the projected normal vector has a noncentral chi-square limit with $q$ degrees of freedom. Its noncentrality is the squared length of the projected mean: $$ \|P_{S^\perp}I^{1/2}h\|^2 =(Ah)^T(AI^{-1}A^T)^{-1}(Ah). $$ This proves the result. $\square$

The formula makes the role of information explicit. The matrix $AI^{-1}A^T$ is the covariance scale for estimating the restrictions; departures are measured relative to that scale. If $Ah=0$, the local displacement lies in the null tangent space, so $\lambda=0$: it creates no first-order departure detectable by these restrictions.

Wald and Score Limits

The same local noncentrality appears for the Wald and score statistics under the same regularity conditions. For the Wald statistic, use $\sqrt{n}\,g(\hat{\theta})$ and its estimated covariance. The local asymptotic distribution of $\sqrt{n}(\hat{\theta}-\theta_0)$ is $N(h,I^{-1})$, so a first-order expansion gives $\sqrt{n}\,g(\hat{\theta})=A\sqrt{n}(\hat{\theta}-\theta_0)+o_p(1)$. This vector converges to a normal distribution with mean $Ah$ and covariance $AI^{-1}A^T$; its covariance-standardized squared length converges to $\chi^2_q(\lambda)$.

For the score statistic, the null fit retains the component of the score compatible with movement along the null tangent space. The test uses the remaining, information-adjusted component. In the same whitened coordinates used in the proof, that component is $P_{S^\perp}z$. Its squared length has the same noncentral chi-square limit. Thus, under local alternatives, all three statistics have the same first-order noncentrality parameter. This is an asymptotic statement; their finite-sample values need not agree.

Worked Example: A Normal Mean With Known Variance

Suppose $X_i$ are independent $N(\mu,4)$ observations. Test $H_0:\mu=0$ against a two-sided alternative, and consider $\mu_n=2/\sqrt{n}$. Here $p=q=1$, $A=1$, the per-observation information is $I=1/4$, and the local displacement is $h=2$. Therefore $$ \lambda=(Ah)^2(AI^{-1}A^T)^{-1} =\frac{2^2}{4}=1. $$

The likelihood-ratio statistic can also be calculated directly. The unrestricted estimate is $\bar X$, and the null fixes the mean at zero. Since the variance is 4, $$ 2\{\ell_n(\bar X)-\ell_n(0)\}=\frac{n\bar X^2}{4}. $$ Under $\mu_n=2/\sqrt{n}$, $\sqrt{n}\bar X/2$ is exactly $N(1,1)$, so the statistic is the square of a $N(1,1)$ variable. Its distribution is exactly $\chi^2_1(1)$ for each $n$, consistent with the limiting result.

Worked Example: A Nuisance Mean in a Bivariate Normal Model

Let $X_i$ be independent $N_2(\theta,\Sigma)$ observations with known $$ \Sigma=\begin{pmatrix}4&1\\1&1\end{pmatrix}. $$ Test $H_0:\theta_1=0$, leaving $\theta_2$ unrestricted, against local alternatives $\theta_n=(a,b)^T/\sqrt{n}$. Here $A=(1,0)$, and $I=\Sigma^{-1}$. The formula uses $AI^{-1}A^T=A\Sigma A^T=4$, so $$ \lambda=\frac{(Ah)^2}{4}=\frac{a^2}{4}. $$ The nuisance displacement $b$ does not appear.

To check directly, profiling the likelihood over $\theta_2$ leaves the likelihood for the first coordinate, whose observations have marginal distribution $N(\theta_1,4)$. The likelihood-ratio statistic for $\theta_1=0$ is therefore $n\bar X_1^2/4$. Under $\theta_{1,n}=a/\sqrt{n}$, $\sqrt{n}\bar X_1/2$ has distribution $N(a/2,1)$, and its square has noncentrality $(a/2)^2=a^2/4$. This verifies that the information formula accounts for the nuisance parameter correctly.

Worked Example: A Local Alternative for a Binomial Probability

Let $X_1,\ldots,X_n$ be independent Bernoulli observations with success probability $p$. Test $H_0:p=0.4$, and consider $p_n=0.4+1/\sqrt{n}$, for $n$ large enough that $p_n\leq1$. The per-observation Fisher information at $p_0=0.4$ is $I=1/\{p_0(1-p_0)\}=1/0.24=25/6$. With $A=1$ and $h=1$, $$ \lambda=h^2I=\frac{25}{6}\approx4.167. $$

For a direct check, under $p_n$, the sample proportion has mean $p_n$ and variance $p_n(1-p_n)/n$. The standardized local displacement relative to its limiting standard error is $$ \frac{\sqrt{n}(p_n-p_0)}{\sqrt{p_0(1-p_0)}} =\frac{1}{\sqrt{0.24}}, $$ whose square is $1/0.24=25/6$. The score and Wald statistics consequently have a one-degree-of-freedom noncentral chi-square limit with this noncentrality.

What the Noncentrality Tells Us

For a fixed local direction $h$, larger Fisher information in the tested direction produces a larger $\lambda$. A larger noncentrality generally means greater power at a fixed significance level. Conversely, a displacement largely along directions that the null allows to vary contributes little or nothing to the test. The information-adjusted geometry, rather than the raw Euclidean length of $h$, determines what the test can detect.

This result also explains why a fixed alternative and a local alternative answer different questions. A fixed parameter departure does not shrink with sample size and can yield power tending to one in a consistent test. Under an $n^{-1/2}$ departure, power instead has a nondegenerate limiting value governed by the noncentral chi-square distribution. This local power comparison is useful when assessing how sensitive a test is to small departures from a scientific null.

Common Mistake

A common error is to use the central chi-square limit under a local alternative simply because the test has $q$ degrees of freedom under the null. The degrees of freedom remain $q$, but the distribution is generally noncentral: the local shift changes the mean of the limiting, information-standardized statistic. The central limit is recovered only when $Ah=0$, so $\lambda=0$. Another error is to use $\|h\|^2$ as the noncentrality; the correct quantity accounts for both the null restrictions and the information matrix.

Key takeaway: For smooth $q$-dimensional restrictions, a local alternative $\theta_0+h/\sqrt{n}$ gives a noncentral chi-square limit with $$ \lambda=(Ah)^T(AI^{-1}A^T)^{-1}(Ah). $$ The null’s tangent space removes allowed directions, and Fisher information sets the scale for the tested departure.

Check Your Understanding

Use the definitions and results above to answer the following questions.

Why are alternatives at distance $n^{-1/2}$ especially useful for studying a test’s limiting power?
For the restriction $g(\theta)=0$, what role does the derivative matrix $A=Dg(\theta_0)$ play in the noncentrality formula?
If $Ah=0$, what is the noncentrality parameter, and how should the local displacement be interpreted?
In the bivariate normal example, why does the nuisance displacement $b$ not affect the noncentrality for testing $\theta_1=0$?
What mistake is made by using $\|h\|^2$ as the noncentrality parameter?