The Quantile Influence Function

Why the Density at a Quantile Matters

A population quantile identifies a location in a distribution: the median, for example, divides the population probability into two halves. But a quantile can be stable or highly sensitive to small changes in the distribution. The key is how much probability is concentrated near that location. If the distribution has little probability density near the quantile, a small change in cumulative probability can require a comparatively large movement in the quantile itself.

The influence function measures the first-order change in a statistical functional under a small perturbation of the distribution. For a quantile, it reveals both the direction and the scale of the response to an observation. Its squared average also gives the semiparametric efficiency bound in the unrestricted model, connecting local sensitivity to the best possible asymptotic precision.

Quantiles and Regularity Conditions

Let \(Y\) have distribution \(P\) and cumulative distribution function \(F(y)=P(Y\leq y)\). For a fixed probability level \(\tau\), with \(0<\tau<1\), define the population \(\tau\)-quantile by

The analysis below assumes that \(F\) is continuous and strictly increasing in a neighborhood of \(q_\tau\), so \(F(q_\tau)=\tau\) and the quantile is locally unique. Assume also that \(F\) has a density \(f\) that is continuous near \(q_\tau\), with \(f(q_\tau)>0\). Positivity is essential: it lets us translate a small change in cumulative probability into a small change in the quantile by dividing by the local density.

For a small contamination fraction \(\varepsilon\), the perturbed distribution is \(P_\varepsilon=(1-\varepsilon)P+\varepsilon\Delta_y\), where \(\Delta_y\) places all probability at \(y\). Write \(F_\varepsilon\) and \(q_{\tau,\varepsilon}\) for its cumulative distribution function and quantile. The influence function is the derivative of \(q_{\tau,\varepsilon}\) at \(\varepsilon=0\), when that derivative exists.

Deriving the Influence Function

For \(y\ne q_\tau\), the contamination point lies either below or above the quantile. In a sufficiently small neighborhood of the original quantile, the contamination indicator is therefore constant: it is one if \(y<q_\tau\) and zero if \(y>q_\tau\). The perturbed quantile satisfies \(F_\varepsilon(q_{\tau,\varepsilon})=\tau\) locally. Differentiating this equation at \(\varepsilon=0\) yields the influence function.

Thus, under the stated conditions, the quantile influence function is

The influence function takes two values, apart from its immaterial value at \(q_\tau\). Below the quantile it is \((\tau-1)/f(q_\tau)\), which is negative. Above the quantile it is \(\tau/f(q_\tau)\), which is positive. The farther the target quantile lies in a region with low density, the larger the absolute influence values.

Derivation Along Smooth Submodels

The same derivative can be expressed in the language of semiparametric efficiency. Consider a differentiable one-dimensional submodel \(P_t\) through \(P=P_0\), with score \(s(Y)\) at \(t=0\). A score has mean zero, \(E[s(Y)]=0\), and finite second moment. Suppose the submodel is regular enough that the CDF and quantile can be differentiated at zero, and write \(q_\tau(t)\) for its quantile.

The derivative of the CDF at a fixed point \(q_\tau\) is \(E[\mathbf{1}\{Y\leq q_\tau\}s(Y)]\). Differentiating the quantile equation \(F_t(q_\tau(t))=\tau\) therefore gives

Since \(E[s(Y)]=0\), the derivative can be rewritten as an inner product with the influence function:

This identity verifies that the influence function represents the derivative in every regular score direction. It is centered because \(P(Y\leq q_\tau)=\tau\), and it is square-integrable when \(f(q_\tau)>0\). In the unrestricted nonparametric model, the tangent space is the space of mean-zero square-integrable functions. The displayed influence function belongs to that space and represents the derivative, so it is the canonical gradient.

The Semiparametric Efficiency Bound

For a regular asymptotically linear estimator of \(q_\tau\), its influence function must represent the derivative in the score directions. In the unrestricted model, the canonical gradient has the smallest variance among such influence functions: any additional component orthogonal to the tangent space can only add variance. The variance of this gradient is therefore the semiparametric efficiency bound for one observation.

To calculate it, note that \(\mathbf{1}\{Y\leq q_\tau\}\) is a Bernoulli random variable with success probability \(\tau\). Its variance is \(\tau(1-\tau)\), and the influence function has mean zero. Hence

Here \(\mathcal{I}^{-1}_{q_\tau}\) denotes the variance bound per observation, not a probability or an interval. For a sample of \(n\) independent, identically distributed observations, the corresponding lower bound on asymptotic variance is \(\tau(1-\tau)/[n f(q_\tau)^2]\). A regular estimator that attains the bound is called semiparametrically efficient.

Using the Influence Function to Understand Precision

For an estimator with the quantile influence function, the asymptotic linear representation is

The leading term is an average of mean-zero influence values. The central limit theorem then gives an asymptotic variance equal to the influence-function variance divided by \(n\), namely \(\tau(1-\tau)/[n f(q_\tau)^2]\). This explains why the bound is not merely an abstract property of a model: it describes the first-order sampling variability that a regular efficient estimator can attain.

Why Density Controls Sensitivity

The quantile equation is \(F(q_\tau)=\tau\). Near \(q_\tau\), a small movement \(\Delta q\) changes the CDF by approximately \(f(q_\tau)\Delta q\). If a perturbation changes cumulative probability by an amount \(\Delta F\), restoring the quantile equation requires approximately \(\Delta q=-\Delta F/f(q_\tau)\). A large density means probability accumulates quickly as the outcome changes, so only a small location movement is needed. A small positive density means the same probability change requires a larger movement.

The efficiency bound makes the same point quantitatively: \(f(q_\tau)\) appears squared in the denominator. Quantiles in sparse regions of a distribution are therefore less precisely estimable, even when the sample size and quantile level are unchanged. This relationship is especially important when comparing estimates across populations or quantile levels: their densities at the target quantiles may differ substantially.

Common Mistake

A frequent error is to reverse the sign of the influence function. One might reason that an observation above the median pulls the median down because it increases the upper tail, or that an observation below pulls it up. That is incorrect. The influence function is positive above the quantile, so adding a small amount of probability there moves the quantile up to first order; it is negative below the quantile, so contamination there moves it down. The sign follows directly from \(\tau-\mathbf{1}\{y\leq q_\tau\}\), not from a vague intuition about changing tail mass.

Another error is to interpret a large density as producing a large variance. The bound is proportional to \(1/f(q_\tau)^2\), so a larger density lowers the bound. Also, the efficiency bound is a lower bound on asymptotic variance for regular estimators in the specified model; it is not a claim that an estimator’s finite-sample variance must equal that value.