3. UDD in the Single Decrement Models

A third 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 each of the associated single decrement tables.
Under this assumption, for integer age (x) and fraction (t) (( 0 leq t lt 1)), we have
begin{eqnarray*}
_t p_x^{prime(j)} = 1- ~_t q_x^{prime(j)} = 1- t times q_x^{prime(j)} .
end{eqnarray*}
Now, recall that ( ~_t p_x^{prime(j)} = exp left{ – int_0^t mu_{x+t}^{(j)} ~ du
right} ). Taking derivatives of each sides, it is easy to check that
begin{eqnarray*}
_t p_x^{prime(j)} mu_{x+t}^{(j)} = q_x^{prime(j)} .
end{eqnarray*}
Thus,
begin{eqnarray*}
~ q_x^{(1)} &=& int_0^1 ~ _t p_x^{(tau)} mu_{x+t}^{(1)} ~dt ~=~ int_0^1 ~ left(prod_{j=1}^n ~_t p_x^{prime(j)} right) mu_{x+t}^{(1)} ~dt \
&=& q_x^{prime(1)} int_0^1 ~ left(prod_{j=2}^n (1- t times q_x^{prime(j)}) right) ~dt .
end{eqnarray*}
So, for example, with (n=2), we have
begin{eqnarray*}
~ q_x^{(1)} &=& q_x^{prime(1)} left(1- frac{1}{2} q_x^{prime(2)} right) .
end{eqnarray*}
and, by symmetry,(~ q_x^{(2)} = q_x^{prime(2)} left(1- frac{1}{2} q_x^{prime(1)} right) .) With (n=3), we have
begin{eqnarray*}
~ q_x^{(1)} &=& q_x^{prime(1)} left(1- frac{1}{2} (q_x^{prime(2)}+q_x^{prime(3)}) + frac{1}{3} q_x^{prime(2)} times
q_x^{prime(3)} right) ,
end{eqnarray*}
and similarly for (q_x^{(2)}) and (q_x^{(3)}).

[raw] [/raw]