Two statistical inference problems in nonlinear mixed models (NLMM) are
considered under only moment conditions on random effects and random errors. First,
higher-order moment estimates of random effects and random errors in NLMM are
proposed and they turn out to be strongly consistent. Second, a difference-type test
T m Ds is developed to test whether some sub-vector of random effects exists or not,
which is easy to implement without requiring the Monte Carlo method. Its theoretical
properties including the power properties are obtained. Moreover, in the special case
of testing the existence of random effects, two kinds of tests are also constructed: the
global difference-type test T m DG , which is a special case of T m Ds , and the modified
score-type test ST n0 , which is motivated by ST nr u in Russo et al. (TEST 21:519–545,
2012). The simulation study indicates that T m Ds is the most powerful. A real data
analysis is also conducted to investigate the applicability of the procedures.

• Inference of nonlinear mixed models for clustered data under moment conditions