The seminonparametric (SNP) density estimator, proposed by \textit{A. R. Gallant} and \textit{D. W. Nychka} [ibid. 55, 363-390 (1987; Zbl 0631.62110)], has been used for structural, reduced form, and efficient method of moments estimation in economics, finance, and the health sciences. In assessing this body of applied work it is important to know if the choice of SNP for its computational convenience was well-advised. Rate results are useful in making this assessment. For an estimator \(\widehat{f}\) of a density \(f_0\), we derive the rate at which the \(L_1\)-norm \(\int^\infty_{-\infty} |f_0 - \widehat{f}|dx\) converges almost surely to zero, which is to be distinguished from results that deduce a rate from the leading term of an asymptotic expansion of \({\mathcal E} \int^\infty_{-\infty} |f_0 - \widehat{f}|dx\). Because our objective is to gain a qualitative understanding of the behavior of SNP relative to other density estimators, we restrict attention to the canonical case of a univariate density estimated from a random sample.