Another way to build a multi-decrement table from the associated single decrement tables (and vice-versa) is to assume uniform distribution of decrements (UDD) in the multi-decrement table (MDT).
\nUnder this assumption, for integer age \\(x\\) and fraction \\(t\\) \\(( 0 \\leq t \\lt 1)\\), we have
\n\\begin{eqnarray*}
\n_t q_x^{(j)} = t \\times q_x^{(j)} .
\n\\end{eqnarray*}
\nIf the (UDD) assumption holds for all decrements, then
\n\\begin{eqnarray*}
\n~ _t q_x^{(\\tau)} = 1 – t \\times q_x^{(\\tau)} .
\n\\end{eqnarray*}Recall Kolmogorov’s forward equation \\(
\n\\frac{\\partial}{\\partial t} ~ _t q_x^{(j)} = \\frac{\\partial}{\\partial t} ~ _t p_x^{0j}
\n= ~ _t p_x^{00} \\mu_{x+t}^{0j}= ~ _t p_x^{(\\tau)} \\mu_{x+t}^{(j)}\\). Thus, we may express the transition force as
\n\\begin{eqnarray*}
\n\\mu_{x+t}^{(j)} = \\frac{q_x^{(j)}}{1 – t \\times q_x^{(\\tau)}} .
\n\\end{eqnarray*}
\nWith \\( q_x^{\\prime(j)} = 1- \\exp \\left\\{ – \\int_0^1 \\mu_{x+t}^{(j)} ~ dt \\right\\} \\), we have
\n\\begin{eqnarray*}
\n– \\ln(1-q_x^{\\prime(j)}) &=& \\int_0^1 \\mu_{x+t}^{(j)} ~ dt \\\\
\n&=& \\int_0^1 \\frac{q_x^{(j)}}{1 – t \\times q_x^{(\\tau)}} ~ dt ~=~ \\frac{q_x^{(j)}}{q_x^{(\\tau)}} (-\\ln(1- q_x^{(\\tau)})) .
\n\\end{eqnarray*}
\nThus, as with the constant transition force, we have
\n\\begin{eqnarray*}
\n1- q_x^{\\prime(j)} &=& \\left( p_x^{(\\tau)}\\right)^{q_x^{(j)}\/q_x^{(\\tau)}} .
\n\\end{eqnarray*}<\/p>\n