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Two groups of critics (call them A and B) rate a set of products (labeled $i=1,\dots,n$) using different scales.

Assume that product $i$ has an innate quality $Q_i$ and that critic $j$ in group A gives the product the score

$$S^A_{ij} = \alpha^A + \beta^A Q_i + \epsilon^A_{ij}$$

and similarly critic $j$ in group B gives the same product the score

$$S^B_{ij} = \alpha^B + \beta^BQ_i + \epsilon^B_{ij}$$

In the general case the number of critics scoring each product can vary, e.g. say that $n^A_i$ critics from group A and $n^B_i$ critics from group B score product $i$.

If we only observe the scores then there is not enough information to determine all of $\alpha^A, \alpha^B, \beta^A, \beta^B$ but we can set $\alpha^A=0$ and $\beta^A=1$ and try to determine values for $\alpha^B$ and $\beta^B$ which are most consistent with the observed data.

If we make the usual 'nice' assumptions about the errors (e.g. IID normally distributed) then what is the best estimator for $\alpha^B$ and $\beta^B$?

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