Inverse Regression

Definition

The problem predicting $x_0$ for a given response $y_0$ is called inverse regression (calibration, discrimination).

Consider a simple linear regression model $Y={\beta }_0+{\beta }_{1}X+ϵ$ The point estimation of $x_0$ for a given $y_0$ is obtained by ${\hat{x}}_0=\frac{y_0-{\hat{\beta }}_0}{{\hat{\beta }}_1}$ Also, the $100(1-\alpha )\%$ confidence interval for $x_0$ is obtained by the second-order inequality $d^2\left[{\hat{\beta }}_1^2-\frac{t^2{s}^2}{{\sum }_{i=1}^n(x_1-\bar{x}{)}^2}\right]-2d{\hat{\beta }}_1(y_0-\bar{y})+\left[(y_0-\bar{y}{)}^2-t^2{s}^2\left(1+\frac{1}{n}\right)\right]\le 0$ where $t=t_{\alpha /2}(n-2)$ and $s^2=\frac{{\sum }_{i=1}^n(Y_i-\hat{Y}{)}^2}{n-2}$. Let $d_1,d_2$ be solutions for the inequality, and we have a confidence interval $\bar{X}+d_1\le X_0\le \bar{X}+d_2$