Let F and G denote two distribution functions defined on the same probability space which are absolutely continuous with respect to the Lebesgue measure with probability density functions f and g, respectively. Ahmad and Van Belle (1974) proposed a measure of the closeness between F and G as follows:  . Ahmad (1980) proposed to estimate λ by  , where F_n(x) and G_n(x)) are empirical distribution functions of F(x) and G(x) respectively and   are the well-known kernel estimates of f(x) and g(x) respectively. This paper generalizes the estimator   to a family of modified estimators of λ indexed by a constant γ  , say, where 0≤γ≤1, which includes   special case (when γ = 0). We derive the limiting distribution of normalized   for 0 < y = 1by using the theory of U-statistics and show that the limiting distribution of   for γ = 0, i.e., of   , when normalized, isdegenerate. Consequently,   cannot be used to construct an asymptotically valid goodness-of-fit test. The normalized estimator   for any 0 <γ≤l, however, does have a limiting normal distribution and therefore can be used to construct an asymptotically valid two sample goodness-of-fit test. The modifications of λ proposed by Ahmad (1980) for one sample case suffer from the same problem. So, in this paper, we also generalize Ahmad's estimators of λ for one sample case and apply the resulting estimators in hypotheses testing. All the tests proposed in this paper shown to be consistent.