Diagonal correction for a quadratic density derivative estimate (${\hat{J}}_{2,b}=\frac{1}{n(n-1)b^5}{\sum }_{i\ne j}A((X_i-X_j)/b)$)

For $A(v)=\int L^{\prime \prime }(t)L^{\prime \prime }(t+v)dt$, this U-statistic has expectation $\Vert L_b\ast f^{\prime \prime }{\Vert }_2^2$. It removes the diagonal term $\Vert L^{\prime \prime }{\Vert }_2^2/(n{b}^5)$ from the integral of a squared second derivative kernel density estimator, but leaves smoothing bias of an estimator.