A Test for Exponential Regression and its Robustness

Abstract

We consider the regression model y _i = α + β exp (γ x_i) + e_i (γ < 0) with i. i. d. error variables e _i with E(e_i) = 0, V(e_i) = σ². By simulation experiments we investigate the possibility of using the elements of the sequence V_A(n) (n = 4,…) as approximations of the variance v($\hat \vartheta$) of the least squares estimator $\hat \vartheta$ of θ’ = (α, β, γ) = (θ₁θ₂θ₃). Here V_A (n) is equal to

$${{\rm{V}}_{\rm{A}}}\left( {\rm{n}} \right) = {\left[ {\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {g'} \left( {{{\rm{x}}_{\rm{i}}},\vartheta } \right)} \right]^{ - 1}}$$

$${{\rm{g}}_{\rm{j}}}\left( {{{\rm{x}}_{\rm{i}}},\vartheta } \right) = {\partial \over {\partial {\vartheta _{\rm{j}}}}}\,{\rm{f}}\left( {{{\rm{x}}_{\rm{i}}},\vartheta } \right)$$

$${\rm{f}}\left( {{{\rm{x}}_{\rm{i}}},\vartheta } \right) = \alpha + \beta \,\exp \,\left( {\gamma \,{{\rm{x}}_{\rm{i}}}} \right)$$

We find that approximations for the estimation problem are already good for n = 4 and very good from n = 6 on. Tests and confidence intervals in respect of γ can easily be constructed with sufficient accuracy from n = 4 on. We also investigate the robustness of the proposed test against non-normality.