In this paper, we study the bias-corrected test developed in Fan(1994). It is based on the integrated squared difference between akernel estimator of the unknown density function of a random vectorand a kernel smoothed estimator of the parametric density functionto be tested under the null hypothesis. We provide an alternativeasymptotic approximation of the finite-sample distribution of thistest by fixing the smoothing parameter. In contrast to the normalapproximation obtained in Fan (1994) in which the smoothingparameter shrinks to zero as the sample size grows to infinity, weobtain a non-normal asymptotic distribution for the bias-correctedtest. A parametric bootstrap procedure is proposed to approximatethe critical values of this test. We show both analytically and bysimulation that the proposed bootstrap procedure works. Consistencyand local power properties of the bias-corrected test with a fixedsmoothing parameter are also discussed.