  This paper proposes a max-test for testing (possibly infinitely) many zero
parameter restrictions in an extremum estimation framework. The test statistic
is formed by estimating key parameters one at a time based on many empirical
loss functions that map from a low dimension parameter space, and choosing the
largest in absolute value from these individually estimated parameters. The
parsimoniously parametrized loss identify whether the original parameter of
interest is or is not zero. Estimating fixed low dimension sub-parameters
ensures greater estimator accuracy, does not require a sparsity assumption, and
using only the largest in a sequence of weighted estimators reduces test
statistic complexity and therefore estimation error, ensuring sharper size and
greater power in practice. Weights allow for standardization in order to
control for estimator dispersion. In a nonlinear parametric regression
framework we provide a parametric wild bootstrap for p-value computation
without directly requiring the max-statistic's limit distribution. A simulation
experiment shows the max-test dominates a conventional bootstrapped test.
