Goodness-of-fit tests for long memory moving average marginal density

A goodness-of-fit test for marginal distribution of linear random fields with long memory

“…A slightly different version of Y nϕ was used in Koul et al (2013), where ϕ nt is replaced by ϕ(t/n). For that version, Theorem 2.1 requires the additional condition that the bias µφ n = µn Note Y n0 =Ȳ n ,W n1 =X n , EW nϕ = 0, andφ = 0 implies that Y nϕ is a consistent and unbiased estimator of µ, i.e., Y nϕ → p µ, E Y nϕ = µ.…”

“…observations, the goodness-of-fit testing problem has been well studied, see, e.g., Durbin (1973Durbin ( , 1975, and D' Agostino and Stephens (1986), among others. Koul and Surgailis (2010) and Koul, Mimoto, and Surgailis (2013) discussed the problem of fitting a known distribution function (d.f.) to the marginal d.f.…”

“…of a stationary long memory moving average time series when its mean µ is known and unknown. In particular, Koul et al (2013) provided a class of weighted least squares estimators of µ for which the weak limit of the first order difference between the residual empirical and null distribution functions is a non-degenerate Gaussian distribution, yielding a simple Kolmogorovtype test for fitting a known distribution up to an unknown mean. In the same context, the latter paper also obtained the asymptotic chi-square distribution of test statistics based on integrated square difference between kernel type estimators of the marginal density of long memory moving averages with discrete time t ∈ Z := {0, ±1, · · · } and the expected value of the density estimator based on errors.…”

“…based on (Y t −Ȳ n )/ σ n , t ∈ A n , and letD n := sup x |F n (x) − F 0 (x)|. It follows from DSL, similarly as in the case ν = 1 studied by Koul and Surgailis (2010) and Koul et al (2013), that under H 0 , n ν/2−dD n → p 0, and hence n ν/2−dD n cannot be used asymptotically to test for H 0 . To circumvent this difficulty, in Section 2 we provide a class of weighted least squares estimators Y nϕ of µ for which the normalized weak limit of the spatial empirical process F nϕ based on residuals (Y t − Y nϕ )/ σ n , t ∈ A n , has a non-degenerate Gaussian distribution under H 0 (Theorem 2.1), implying…”

“…This test is an extension of the test proposed in Koul et al (2013) from the time series case to the random fields.…”