simple representation of R p ∞ in terms of a well-known distribution-free Gaussian process for which critical values are readily available. To overcome this problem, we propose to compute critical values with the assistance of a multiplier bootstrap. The proposed procedure has good theoretical and empirical properties, is computationally easy to implement, and does not require computing new parameter estimates at each bootstrap replication. More precisely, in order to estimate the critical values, we propose to approximate the asymptotic behavior of ˆ R p $_{n}$( u ) by that of where { V$_{i}$ } n i =1 is a sequence of i.i.d. random variables with zero mean, unit variance and bounded support, independent of the original sample { ( D$_{i}$, X ′ $_{i}$) $^{′}$} n i $_{=1}$. A popular example involves i.i.d. Bernoulli variates { V$_{i}$ } with P ( V = 1 − κ ) = $_{κ/}$√$_{5 and}$ P ( V = κ ) = 1 − $_{κ/}$√$_{5, where}$ κ = ( $^{√$_{5 + 1}$}$) / 2 , as suggested by Mammen (1993). ˆ p ∗ ˆ p With ˆ R p ∗ n ( u ) at hands, the bootstrapped version of our test statistics Γ ( ˆ R p n ) is simply given by Γ ( ˆ R p, ∗ n ) . For instance, the bootstrapped versions of CvM$_{n}$ and KS$_{n}$ in (2.8) and (2.9), respectively, are given by ˆ R p ∗ n ( u ) ≡ 1 √$_{n}$ n ∑ i =1 ε$_{i}$ ( ˆ θ$_{n}$ ) P $_{n}$1 { q ( X$_{i}$, ˆ θ$_{n}$ ) ≤ u } V$_{i}$, (3.5) CvM ∗ n = 1 n n ∑ i =1 [ ˆ R p ∗ n ( q ( X$_{i}$, ˆ θ$_{n}$ ))] $^{2}$, KS ∗ n = sup u ∈ Π ∣ ˆ R p ∗ n ( u ) ∣ . c Γ ∗ n,α ≡ inf { c$_{α}$ ∈ [0 , ∞ ) : lim n $_{→∞}$P ∗ n { Γ ( ˆ R p, ∗ n ) > c$_{α}$ } = α } , ˆ R p ∗ n ⇒ ∗ R p ∞ a.s. , The asymptotic critical values are then estimated by where P ∗ n means bootstrap probability, i.e. conditional on the original sample { ( D$_{i}$, X ′ $_{i}$) $^{′}$} n i $_{=1}$. In practice, c Γ ∗ n,α is approximated as accurately as desired by ( Γ ( ˆ R p, ∗ n )) B (1 − α $_{)}$, the B (1 − α ) − th order statistic from B replicates {( Γ ( ˆ R p, ∗ n )) l } B l =1 of Γ ( ˆ R p, ∗ n ) . The next theorem establishes the asymptotic validity of the multiplier bootstrap procedure proposed above. Theorem 4 Assume Assumptions 3.1-3.3. Then, where R p ∞ is the Gaussian process defined in Theorem 1, and ⇒ ∗ denotes the weak convergence 17