Under UDD in the multiple decrement model, we can calculate the dependent and independent rates using the below formula:

$$q_x^k=\frac{(aq)^k_x}{1-\frac{1}{2}(aq)_x^{-k}}$$

where \(q_x^k\) represents the independent rate and \((aq)^k_x\) represents the dependent rate and \((aq)_x^{-k}\) represents the total probability of decrement (from all causes, excluding decrement k). Our textbook provides another method (under UDD), this is given below:

$$p_x^{*(j)}=(p_x^{00})^{p_x^{0j}/p_x^{0\bullet}}$$

Where the notation above has the same meaning as defined in the textbook.

Are these two formulae equivalent?