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Expected future lifetime from an annual life table

Source: R/e_x.R
e_x.Rd

Computes the curtate or complete expected future lifetime at integer age \(x\), optionally restricted to a temporary horizon of \(t\) years.

Usage

e_x(
  lt,
  x,
  t = NULL,
  type = c("curtate", "complete"),
  frac,
  tidy = FALSE,
  check = TRUE,
  tol = 1e-10
)

Arguments

lt

A life table object as produced by lifetable (must contain columns x and lx).

x

Integer age(s).

t

Optional nonnegative numeric duration(s). If NULL (default), the whole-life expectancy is computed (i.e., horizon extends to \(\omega - x\)). If a numeric value is provided, the \(t\)-year temporary life expectancy is returned.

type

Character: "curtate" (default) or "complete".

frac

Fractional-age assumption for type = "complete": "UDD", "CF", "CML" (alias of CF), or "Balducci". If not specified and lt carries a frac attribute (set by lifetable), that value is used.

tidy

Logical. If TRUE, returns a tibble.

check

Logical. If TRUE, performs basic input checks.

tol

Numeric tolerance for integer checks.

Value

A numeric vector of expected future lifetimes, or a tibble if tidy = TRUE with columns x, t, type, frac, ex.

Details

Curtate life expectancy (Finan, Section 23.7): $$e_x = \sum_{k=1}^{\omega - x} {}_k p_x = \frac{1}{\ell_x} \sum_{k=1}^{\omega - x} \ell_{x+k}.$$

The \(t\)-year temporary curtate expectancy is (Finan, Sec. 23.7): $$e_{x:\overline{t}|} = \sum_{k=1}^{t} {}_k p_x.$$

Complete life expectancy (Finan, Section 23.3): $$\breve{e}_x = \int_0^{\omega - x} {}_t p_x \, dt = \frac{T_x}{\ell_x}.$$

The integral is decomposed year-by-year. Within each year, the within-year survival integral \(\int_0^s {}_u p_y \, du\) is evaluated in closed form under the selected fractional-age assumption (Finan, Section 24):

  • UDD (Sec. 24.1): \(\int_0^s {}_u p_y \, du = s - \frac{1}{2} s^2 q_y\)

  • CF (Sec. 24.2): \(\int_0^s {}_u p_y \, du = (1 - p_y^s) / (-\ln p_y)\)

  • Balducci (Sec. 24.3): \(\int_0^s {}_u p_y \, du = \frac{p_y}{q_y} \ln \left( \frac{p_y + q_y s}{p_y} \right)\)

Integrals use stable exponential/logarithmic increments for small durations. Extinct cohorts contribute zero to later intervals, even if their conditional probabilities are undefined. The CF/Balducci zero-survival interval has integral zero; UDD retains its linear decline. Unknown survivor counts propagate as missing expectations. When annual probabilities must be derived from survivor counts, survival uses the direct count ratio rather than \(1-q_y\). Its logarithm retains the CF integral even when the annual ratio is below floating-point range.

Under UDD, the complete expectancy satisfies the well-known approximation (Finan, Example 20.24): $$\breve{e}_x \approx e_x + \frac{1}{2}.$$