Local Alternatives and Contiguous Parameter Sequences

Why Study Parameters That Move with the Sample Size?

Many asymptotic results describe an estimator when the data-generating parameter is fixed. But a fixed parameter can conceal how an estimator behaves when the true value is close to a benchmark, such as a null value, and that gap shrinks as the sample grows. Local alternatives make this comparison precise: the parameter moves toward the benchmark at the same rate as the estimator’s sampling uncertainty.

For a root-\(n\) estimator, a difference of order \(1/\sqrt{n}\) is small in the original parameter scale but remains visible after multiplying estimation error by \(\sqrt{n}\). This is why local drift appears as a finite shift in an asymptotic distribution. The same reasoning applies at other convergence rates.

The number \(h\) is the local drift. For a vector parameter, both \(\theta_0\) and \(h\) are vectors of the same dimension. A fixed nonzero \(h\) produces a parameter sequence that converges to \(\theta_0\), since \(h/\sqrt{n}\) tends to zero, while the rescaled difference \(\sqrt{n}(\theta_n-\theta_0)\) remains exactly \(h\).

Contiguity and the Scope of Local Comparisons

Local parameter sequences are often studied when their sampling laws are contiguous to the laws at the reference value. Contiguity rules out events that become virtually impossible under the reference law but retain substantial probability under the alternative.

In regular models, bounded root-\(n\) parameter shifts commonly generate mutually contiguous laws. Intuitively, the alternatives become close to the reference as \(n\) increases, so a sequence of events that is negligible under one law cannot be used to separate the two with probability tending to one. Contiguity is a statement about sequences of probability laws, not an estimator’s limiting distribution.

That distinction matters. Contiguity alone does not imply that an estimator has a normal limit, that its variance is unchanged, or even that its reference-law limit transfers in a particular form. To derive a local limit, we need an expansion or limit theorem that applies under the local laws as well. A common sufficient setup is an asymptotic linear representation and a central limit theorem that remain valid along the parameter sequence.

How Local Drift Shifts an Estimator Limit

Suppose \(\hat{\theta}_n\) estimates the parameter \(\theta_n\) under the local law \(P_{n,\theta_n}\). Assume that, under this sequence of laws, the estimator has a centered limit:

where \(Z\) has mean zero and covariance matrix \(V\). Because \(\theta_n=\theta_0+h/\sqrt{n}\), the following identity holds for every \(n\):

Adding a fixed vector shifts a limiting distribution by that vector. Therefore,

If \(Z\sim N(0,V)\), then the reference-centered limit is \(N(h,V)\). The variance has not changed in this result; the center has moved. In applications, however, a local change in the data-generating law can also change the limiting covariance, so \(V\) must be justified under the local sequence rather than assumed unchanged.

Deriving Drift from an Estimating Equation

The same shift can be seen directly in a stochastic expansion. Suppose \(\hat{\theta}_n\) solves an estimating equation \(n^{-1}\sum_{i=1}^n\psi(W_i,\hat{\theta}_n)=0\), where \(\psi\) is a vector of estimating functions and \(W_i\) denotes the observed data. Assume the equation is correctly centered at the local parameter, the derivative matrix \(G=E[\partial\psi(W_i,\theta_n)/\partial\theta']\) converges to a nonsingular matrix, and a triangular-array central limit theorem applies to the estimating functions under the local laws. Also assume the Taylor remainder is negligible at the root-\(n\) scale.

A first-order Taylor expansion around \(\theta_n\) gives the estimator’s leading term. If the limiting covariance of \(\psi(W_i,\theta_n)\) is \(\Omega\), then

The negative sign in the expansion does not affect the covariance. Under these conditions, the local limit centered at \(\theta_0\) follows by adding \(h\), giving mean \(h\) and the same covariance. The argument depends on the expansion and central limit theorem holding along the local sequence; a fixed-parameter result by itself is not enough.

A Regression Example and Its Interpretation

A familiar regression estimator illustrates why the scale of the local sequence must match the estimator’s rate. Consider the no-intercept model \(Y_i=\beta_n X_i+u_i\), with independent observations, \(E[X_i u_i]=0\), \(E[X_i^2]=2\), and \(E[u_i^2\mid X_i]=3\). Assume the usual moment conditions for the central limit theorem and a nonzero limiting regressor second moment. The ordinary least-squares estimator is \(\hat{\beta}_n=\sum_i X_iY_i/\sum_iX_i^2\).

What Local Limits Do—and Do Not—Establish

A local limit describes estimator behavior for parameter values approaching a reference point at a specified rate. It gives more information than a limit only at one fixed parameter: the shift \(h\) records how the estimator responds to departures from that point that remain visible after rescaling. This perspective is useful when comparing an estimator’s sampling uncertainty with small departures from a benchmark.

The central calculation can be summarized as follows: establish a limit centered at the moving parameter under the moving laws, then add the scaled difference between the moving parameter and the reference value. If the estimator’s rate is \(r_n\) rather than \(\sqrt{n}\), use \(\theta_n=\theta_0+h/r_n\) and center the result on the \(r_n\) scale.

Common Mistake

A common error is to take an asymptotic result proved only under \(P_{n,\theta_0}\), declare it unchanged under every contiguous alternative, and then add \(h\) to the mean. That reasoning skips the key condition: the centered expansion and its stochastic limit must also hold under \(P_{n,\theta_n}\). Contiguity controls how negligible events transfer; it does not, by itself, supply a central limit theorem or fix the estimator’s variance. The correction is to state and verify the local-sequence assumptions first, derive the centered limit under those laws, and only then apply the centering identity.