- #1

- 14

- 0

[itex]R = e^\eta r[/itex]

where [itex]r[/itex] is predicted and [itex]R[/itex] is observed. [itex]\eta[/itex] is mean zero.

I can find [itex] \eta [/itex] as follows: [itex]\ln( r) - \ln( R) = \eta [/itex]. I'm summing the squares of the [itex]\eta[/itex]'s.

However, there are some markets where I only observe the sum of sales of multiple stores. The simplest example would be for two stores:

[itex]R_{1}=e^{\eta_{1}}r_{1}[/itex]

[itex]R_{2}=e^{\eta_{2}}r_{2}[/itex]

where I only observe [itex] R_1 + R_2 [/itex]. I would like to calculate [itex] \eta_1^2 + \eta_2^2 [/itex] (or at least its expected value). Is there any way I can do this? Is there anything at all that I can say about [itex] \eta_1^2 + \eta_2^2 [/itex] or [itex] \eta_1 + \eta_2 [/itex] ? I'm OK with making assumptions about the distribution of [itex]\eta[/itex] if that's necessary.

Thanks!