Weighted Kernel Density Estimation

Why Weight a Kernel Density Estimate?

A kernel density estimate smooths observed losses into an estimated density. When observations represent unequal amounts of exposure, treating every observation equally may not describe the exposure-weighted loss distribution of interest. For example, a loss observation associated with 300 policy-years may reasonably receive more weight than one associated with 50 policy-years, provided the target is the distribution across exposure rather than the distribution across recorded observations.

This tutorial considers exact, nonnegative loss observations \(X_1,\ldots,X_n\), nonnegative weights \(w_1,\ldots,w_n\), and one common bandwidth \(h>0\). The weights may be exposure amounts or other prespecified measures of relative importance. This method does not by itself correct for incomplete observation or selection into a sample; the weight must represent the target population appropriately.

A kernel \(K\) is typically a nonnegative density integrating to one; standard choices are symmetric about zero. For example, the triangular kernel is \(K(u)=(1-|u|)\) when \(|u|\leq 1\), and zero otherwise. The Epanechnikov kernel is \(K(u)=\tfrac{3}{4}(1-u^2)\) when \(|u|\leq 1\), and zero otherwise. The bandwidth \(h\) sets the smoothing scale. The argument \((x-X_i)/h\) is the standardized distance from observation \(i\) to the point \(x\).

Why the Normalization Works

Each observation contributes a kernel centered at \(X_i\), scaled horizontally by \(h\) and vertically by \(1/h\). The normalized weights determine how much probability mass that kernel contributes. The factor \(1/h\) is essential: stretching a kernel by \(h\) would otherwise multiply its area by \(h\).

This derivation also explains why normalizing by the number of observations \(n\) is generally wrong when the weights are unequal: it does not make the kernel contributions sum to one. Multiplying every weight by the same positive constant, however, leaves the estimate unchanged because the constant cancels between numerator and denominator.

Exposure Weights, Interpretation, and Units

The weighted estimate can be understood as smoothing a weighted empirical distribution. Before smoothing, that distribution places mass \(\alpha_i\) at each observed loss \(X_i\). After smoothing, the mass at \(X_i\) is spread over a neighborhood according to the kernel and bandwidth. If the weights are exposure amounts, the estimate describes losses for a randomly selected unit of the represented exposure, under the assumption that the observations and weights appropriately represent the target exposure.

This interpretation differs from simply asking for the distribution across recorded rows. An unweighted estimate assigns mass \(1/n\) to every row. An exposure-weighted estimate assigns mass \(w_i/W\). They coincide when all weights are equal, but not in general. Integer weights can produce the same formula as repeating observations that many times, but exposure weights need not be integer counts and should not automatically be interpreted as replicated claims.

The units of a density are the reciprocal of the units used for the loss variable. If losses are entered in thousands of dollars, the estimate is a density per thousand dollars. To express the corresponding density per dollar, divide the numerical value by 1,000. A density value is not itself a probability; probabilities are obtained by integrating the density over a loss interval.

Exposure-Weighted Versus Unweighted Estimates

The next calculation isolates the effect of unequal exposure. Consider three hypothetical loss observations, \(5,9,10\) thousand dollars, with respective exposure weights \(100,300,100\). Use a triangular kernel, bandwidth \(2\) thousand dollars, and evaluate at \(x=9\) thousand dollars.

Practical Checks and Common Exam Traps

A reliable calculation separates weight normalization from kernel evaluation. Write down the total weight, each normalized weight, the bandwidth, and the standardized distances before adding contributions. This makes it easier to detect whether a zero contribution lies outside a compact-support kernel and whether the bandwidth has been applied exactly once.

• Using \(n\) instead of total weight. The denominator is \(hW\), where \(W=\sum_iw_i\). It becomes \(hn\) only when all weights are equal and can be canceled consistently.
• Forgetting the bandwidth factor. The kernel term is \(K((x-X_i)/h)/h\), not just \(K((x-X_i)/h)\). Omitting \(1/h\) gives the wrong density scale and generally fails the unit-area check.
• Using unnormalized weights as probabilities. The contributions must sum to total mass one. Divide each weight by \(W\), or use the equivalent raw-weight formula with denominator \(hW\).
• Confusing density with probability. A value such as \(0.35\) per thousand dollars is a density height. A probability over an interval requires integrating the estimate over that interval.
• Misreading the target. Exposure weighting changes the distribution being estimated. It is appropriate only when the weights represent the relative exposure relevant to the question; it is not a universal adjustment for unequal or incomplete observations.

For nonnegative losses, a standard symmetric kernel can assign some estimated mass to negative loss values near zero. That is a boundary issue rather than a weight-normalization issue; the weighted formula does not by itself remove boundary bias. Also, a common bandwidth is used throughout these calculations. If a procedure specifies observation-specific bandwidths, each contribution and its normalization must be handled according to that procedure rather than silently substituting a common-bandwidth formula.