Notation: k = 1 , 2 , ..., d ; in 3. and 4, w$_{k}$ = E ( x$_{k}$f$_{z}$ ( z ) | z ); in 4a w = f$_{z}$$_{−}$$_{x}$ ; w$_{k}$ = E ( x$_{k}$w ( z − x ) | ( z − x )) . 14 Model Distributional assumptions Convolution equations Known functions Unknown functions 1. z = x ∗ + u x $^{∗}$⊥ u f$_{x}$ ∗ ∗ f$_{u}$ = f$_{z}$ f$_{z}$, f$_{u}$ f$_{x}$ ∗ 2. z = x ∗ + u z ⊥ u f$_{z}$ ∗ f$_{−}$$_{u}$ = f$_{x}$ ∗ f$_{z}$, f$_{u}$ f$_{x}$ ∗ 3. z = x ∗ + u ; x = x ∗ + u$_{x}$ x $^{∗}$⊥ u ; E ( u$_{x}$ | x $^{∗}$, u ) = 0; E ‖ z ‖ < ∞ ; E ‖ u ‖ < ∞ . f$_{x}$ ∗ ∗ f$_{u}$ = f$_{z}$ ; h$_{k}$ ∗ f$_{u}$ = w$_{k}$, with h$_{k}$ ( x ) ≡ x$_{k}$f$_{x}$ ∗ ( x ); k = 1 , 2 ...d f$_{z}$, w$_{k}$, k = 1 , 2 ...d f$_{x}$ ∗ ; f$_{u}$ 4. z = x ∗ + u ; x = x ∗ + u$_{x}$ ; x $^{∗}$⊥ u ; x $^{∗}$⊥ u$_{x}$ ; E ( u$_{x}$ ) = 0; u ⊥ u$_{x}$ ; E ‖ z ‖ < ∞ ; E ‖ u ‖ < ∞ . f$_{x}$ ∗ ∗ f$_{u}$ = f$_{z}$ ; h$_{k}$ ∗ f$_{u}$ = w$_{k}$ ; f$_{x}$ ∗ ∗ f$_{u}$$_{x}$ = f$_{x}$ ; with h$_{k}$ ( x ) ≡ x$_{k}$f$_{x}$ ∗ ( x ); k = 1 , 2 ...d f$_{z}$ , f$_{x}$ ; w ; w$_{k}$ k = 1 , 2 ...d f$_{x}$ ∗ ; f$_{u}$, f$_{u}$$_{x}$ 4a. Same model as 4., alternative equations: f$_{x}$ ∗ ∗ f$_{u}$ = f$_{z}$ ; f$_{u}$$_{x}$ ∗ f$_{−}$$_{u}$ = w ; h$_{k}$ ∗ f$_{−}$$_{u}$ = w$_{k}$, with h$_{k}$ ( x ) ≡ x$_{k}$f$_{u}$$_{x}$ ( x ); k = 1 , 2 ...d –”– –”–