The toy model is the TM2 model as introduced by Weigel and Bowler
(2009) with a slight modification to allow for forecasts with negative
correlation skill. In this toy model, the observations \(x\) and forecasts
\(f_i\) are defined as follows:
$$x = \mu_x + \epsilon_x$$
$$f_i = \alpha / |\alpha| \mu_x + \epsilon_{\beta} + \epsilon_i$$
where
$$\mu_x ~ N(0, \alpha^2)$$
$$\epsilon_x ~ N(0, 1 - \alpha^2)$$
$$\epsilon_{\beta} ~ N(0, \beta^2)$$
$$\epsilon_i ~ N(0, 1 - \alpha^2 - \beta^2)$$
$$\alpha^2 \le 1$$
$$0 \le \beta \le 1 - \alpha^2$$