In this paper, we prove uniform bounds for $\operatorname {GL}(3)\times \operatorname {GL}(2) \ L$-functions in the $\operatorname {GL}(2)$ spectral aspect and the t aspect by a delta method. More precisely, let $\phi $ be a Hecke–Maass cusp form for $\operatorname {SL}(3,\mathbb {Z})$ and f a Hecke–Maass cusp form for $\operatorname {SL}(2,\mathbb {Z})$ with the spectral parameter $t_f$. Then for $t\in \mathbb {R}$ and any $\varepsilon>0$, we have

$$\begin{align*}L(1/2+it,\phi\times f) \ll_{\phi,\varepsilon} (t_f+|t|)^{27/20+\varepsilon}. \end{align*}$$

$L(1/2+it,\phi \times f)$

$|t|-t_f \gg (|t|+t_f)^{3/5+\varepsilon }$

Presented and illustrated is an easy-to-implement and flexible methodology for the analysis of synergistic and antagonistic effects when the effects are defined as nonlinear functions of means. The methodology augments standard mixed-model analyses with the Delta method for standard errors of nonlinear functions of means. Explained is why standard ANOVA methods that have been adopted in the literature are not recommended. To illustrate the methodology, the joint-action effects of fenoxaprop with companion herbicides in two-component mixtures for weed control in rice were evaluated. The companion herbicides were halosulfuron, bispyribac-sodium, bensulfuron, penoxsulam, carfentrazone, quinclorac, and imazethapyr. Weeds evaluated were barnyardgrass and broadleaf signalgrass. Experiments in Louisiana and Mississippi in 2009 revealed a preponderance of antagonistic effects. The analysis showed that mixtures with bispyribac-sodium, penoxsulam, quinclorac, and imazethapyr were generally the most antagonistic and provided the least control.

A generalization of the delta method is proposed. The asymptotic distribution of g(Y n ) is obtained when g is not necessarily a differentiable function but verifies some weaker regularity conditions. An application to diversity profiles is also presented.