A spline density function estimate is considered. Limit theorems are obtained for a quadratic norm of the normalized deviation of the estimate from its expected value. Results obtained can be used in a test of goodness of fit. Some of the commonly used boundary conditions for the spline function are considered. It appears that in some situations certain boundary conditions are undesirable.
"A Global Measure of a Spline Density Estimate." Ann. Statist. 6 (5) 1138 - 1148, September, 1978. https://doi.org/10.1214/aos/1176344316