摘要：We discuss inference for additive models with random scaling
factors. The additive effects are of the form (1+¡ã)f(z) where f is a nonlinear
function of the continuous covariate z modeled by P(enalized)-splines and
1 + ¡ã is a random scaling factor. Additionally, monotonicity constraints on
the nonlinear functions are possible.
Our work is motivated by the situation of a retailer analyzing the impact of
price changes on a brand¡¯s sales in its orange juice product category. Relating
sales to a brand¡¯s own price as well as to the prices of competing brands in
the category, we estimate own- and cross-item price response functions flexibly
to represent nonlinearities and irregular pricing effects in sales response.
Monotonicity constraints are imposed so that a brand¡¯s own price is inversely
related and the prices of competing brands are directly related to the number
of items sold, as suggested by economic theory. Unobserved store-specific
heterogeneity is accounted for by allowing the price response curves to vary
between different stores.
关键词：P-Splines, Monotonicity Constraints, Multiplicative Random Effects,
Price Response, Own- and Cross-item Price Effects.
