Linear Population Between Census Dates/Reporting Dates
- We can use the linear approximation between reporting dates for the trapezium approximation.
\[ \int^1_0 E^c_x = \frac{P(x,0)+P(x,1)}{2}\]
- Can also use constant population, which is useful if only given a single reporting date, aka there is no points to interpolate between.
\[ \int^1_0 E_x^c = P(x, y), \quad y\in (0,1)\]
Uniform Distribution of Birthdays
- Used in the transformation of the age definition. For example if we were adjusting nearest birthdays to be last birthday:
- Note \(P(x,t)\) is the reported data, nearest birthday
- Note \(P'(x,t)\) is the adjusted data, last birthday
\[ P'(x,t) = \frac{P(x,t)+P(x+1,t)}{2} \]
- Also used when considering the \(f\) measure.
- For age interval, we assume that average age would be \(\text{Start} + \frac{1}{2}\) in the middle of the interval when adjusting mortality rate
\[ \mu_{x+f},\quad f = \frac{1}{2}\text{ for age interval } (x,x+1)\]
- For calender interval, we assume the average age at the beggining of the interval to be \(\text{Start} + \frac{1}{2}\)
\[ q_{x+f},\quad f = \frac{1}{2}\text{ for age interval } (x,x+1)\text{ at census date}\]
Uniform Distribution of Deaths within a single year
- Used to change central exposed to inital exposed to risk
\[ E_x\approx E^c_x + \frac{1}{2}d_x\]