Bootstrap goodness-of-fit tests with estimated parameters based on empirical transforms

Abstract

Abstract. Several test statistics have been proposed recently which employ a weighted distance that depends on an empirical transform, as well as on estimated parameters. The empirical characteristic function is a typical example, but alternative empirical transforms have also been employed, such as the empirical Laplace transform when dealing with non-negative random variables or the empirical probability generating function corresponding to discrete observations. We propose a general formulation that covers most of the transform-based test statistics which have appeared in the literature. Under this formulation, the asymptotic properties of the test statistics, such as the limiting null distribution and the consistency under general alternatives, are derived. Since large-sample critical values are extremely complicated (if not impossible) to compute, two effective bootstrap versions of the test procedures are derived, which can be used to approximate the critical values, for any given sample size, and to calculate the power under contiguous alternatives. The validity of these bootstrap procedures is shown analytically.

Fast goodness-of-fit tests based on the characteristic function

“…It is also interesting to observe that the assumptions in Meintanis and Swanepoel (2007) and Jiménez-Gamero et al…”

“…−→ denotes the almost sure convergence; if R ⊂ R d , 13 for some d ∈ N, then intR denotes the interior of R and I(R) denotes the indicator function of the set R; if x ∈ R d , then 14 ∥x∥ denotes the Euclidean norm; L p (R d , w) = {ς : R d → R,  |ς (x)| p w(x)dx < ∞}; if ς 1 , ς 2 ∈ L 2 (R d , w) then, 15 ⟨ς 1 , ς 2 ⟩ w =  ς 1 (t)ς 2 (t)w(t)dt and ∥ς 1 ∥ 2 w = ⟨ς 1 , ς 1 ⟩ w ; P θ and E θ denote the probability and expectation when the 16 data have cdf F (x; θ ), respectively; P * , E * and var * denote the conditional probability, mean and variance, given the data, 17 respectively; let {A n } be a sequence of random variables and let ϵ ∈ R, then A n = O P (n −ϵ ) means that n ϵ A n is bounded in 18 probability, A n = o P (n −ϵ ) means that n ϵ A n P −→ 0 and A n = o(n −ϵ ) means that n ϵ A n a.s.…”

“…Specifically, assuming that the data are continuous, they propose to 13 apply the Rosenblatt transformation (Rosenblatt, 1952). 14 Recall that if the random vector X = (X 1 , X 2 , . .…”

Weighted quantile correlation test for the logistic family

“…We compare the new test in the location-scale case with Meintanis tests based on the empirical characteristic function and the empirical momentum generating function from [16]. To the comparison Table 3 Table 2.…”

“…More tests on assessing the fit to the logistic distribution may be found in Aguirre and Nikulin [2] and Meintanis [16]. The main goal of the present paper is to introduce weighted quantile correlation tests for the location and the location-scale logistic families, introduced below.…”