Let F denote a distribution functiondefined on the probability space (Ω,,P), which is absolutelycontinuous with respect to the Lebesgue measure inRd with probabilitydensity function f. Letf0(·,β) be aparametric density function that depends on anunknown p × 1 vector β. In thispaper, we consider tests of the goodness-of-fit off0(·,β) forf(·) for some β based on (i) theintegrated squared difference between a kernelestimate of f(·) and thequasimaximum likelihood estimate off0(·,β) denoted byIn and (ii) theintegrated squared difference between a kernelestimate of f(·) and thecorresponding kernel smoothed estimate off0(·, β) denoted byJn. It is shown inthis paper that the amount of smoothing applied tothe data in constructing the kernel estimate off(·) determines the form of thetest statistic based onIn. For each testdeveloped, we also examine its asymptotic propertiesincluding consistency and the local power property.In particular, we show that tests developed in thispaper, except the first one, are more powerful thanthe Kolmogorov-Smirnov test under the sequence oflocal alternatives introduced in Rosenblatt [12],although they are less powerful than theKolmogorov-Smirnov test under the sequence of Pitmanalternatives. A small simulation study is carriedout to examine the finite sample performance of oneof these tests.