Cramer-von mises-type tests with applications to tests of independence for multivariate extreme-value distributions

A Nonparametric Test of Serial Independence for Time Series and Residuals

“…In fact this can be achieved by first computing the cumulants given by (6), and then applying the Cornish Fisher asymptotic expansion, or by inversion of the characteristic function. This inversion is obtained by the numerical integration method proposed by Imhof (1961), or the improved version of this algorithm introduced by Deheuvels and Martynov (1996). The following provides the cumulant of order m of the !…”

A Karhunen–Loeve decomposition of a Gaussian process generated by independent pairs of exponential random variables

“…Therefore, the only case where (1.11) is useful for such applications is when there exist sufficiently simple closed-form expressions for the λ k 's. If such is not the case, one must use different techniques (see, e.g., [18,40]), which, besides being more time-consuming for the computer than a direct approach, do not yield more than a very superficial insight concerning the specific form of the KL decomposition. Unfortunately, for most Gaussian processes of interest with respect to statistics, the values of the λ k 's are unknown, even though their existence remains guaranteed through the knowledge of R(u, v) (see, e.g., [1, p. 76]).…”

“…A general description of the numerical methods which may be used to evaluate the above quantities, is to be found in [38][39][40] and [18]. The approach which has been followed here for the computation of the constants in Table 1 is based on the Smirnov formula [50,51]…”

Goodness-of-fit test for switching diffusion

“…More about Cramér-von Mises theory and the use of Karhunen-Loève expansion can be found in Martynov (1992) and Deheuvels and Martynov (1996).…”