Specification test for the heteroscedastic extremes model

Develop a rigorous specification test for the heteroscedastic extremes model in which covariates rescale the conditional tail through a scedasis function while conditional tail indices remain constant across the covariate space.

Background

The proposed framework assumes that covariates alter the scale of the response tail without changing its shape, implying that conditional and marginal distributions share a common extreme-value index. The estimation theory controls approximation error under a second-order condition, but the paper does not provide a formal test of whether this structural assumption is appropriate.

The discussion points to testing whether conditional tail indices are constant across the covariate space, in analogy with existing work on heteroscedastic extremes, but leaves the construction of a rigorous testing procedure unresolved.