Moments of the QS Function

Derivation

(1)
\begin{align} Q_n(x) = \int_{0}^{x} a^n f(a) da \end{align}
(2)
\begin{align} = \int_{0}^{x} a^n \, \left( \sum_k \theta_k \, I(a>t_k) (a - t_k)^2 \right) da \end{align}
(3)
\begin{align} = \sum_k \theta_k \left( \int_{0}^{x} a^n \, I(a>t_k) \, (a - t_k)^2 da \right) \end{align}
(4)
\begin{align} = \sum_k \theta_k \left( \int_{0}^{x} \, I(a>t_k) \, (a^{n+2} - 2 t_k a^{n+1} + {t_k}^2 a^n) \, da \right) \end{align}
(5)
\begin{align} = \sum_k \theta_k \left( \int_{t_k}^{min(x,t_k)} (a^{n+2} - 2 t_k a^{n+1} + {t_k}^2 a^n) \, da \right) \end{align}
(6)
\begin{align} = \sum_k \theta_k \left[ \frac{a^{n+3}}{n+3} -2 t_k \frac{a^{n+2}}{n+2} + {t_k}^2 \frac{a^{n+1}}{n+1} \right]_{t_k}^{min(x,t_k)} \end{align}
(7)
\begin{align} = \sum_k \theta_k \: I(x > t_k) \left[ \frac{x^{n+3}-{t_k}^{n+3}}{n+3} -2 t_k \frac{x^{n+2}-{t_k}^{n+2}}{n+2} + {t_k}^2 \frac{x^{n+1}-{t_k}^{n+1}}{n+1} \right] \end{align}

Examples

  • $Q_0(\beta)$ = TFR
  • $Q_0(30)$ = cumulative fertility through exact age 30
  • $\frac{Q_1(30)}{Q_0(30)}$ = mean age of childbearing before age 30
  • $\sqrt{\frac{Q_2(30)}{Q_0(30)} - \left(\frac{Q_1(30)}{Q_0(30)}\right)^2}$ = std deviation of age of childbearing before age 30