Expected future lifetime for two independent lives

Computes the expected future lifetime for two independent lives aged x and y, for either joint-life (first death) or last-survivor (second death).

Usage

e_xy(
  lt,
  x,
  y,
  t = NULL,
  type = c("curtate", "complete"),
  frac,
  cohort = c("first", "last"),
  tidy = FALSE,
  check = TRUE,
  tol = 1e-10
)

Arguments

lt: A life table data frame with columns x and lx.
x: Integer actuarial age for life 1.
y: Integer actuarial age for life 2.
t: Optional nonnegative numeric duration(s). If NULL, uses the maximum horizon allowed by the table.
type: Character: "curtate" or "complete".
frac: Fractional-age assumption for type = "complete", as in t_pxy: "UDD", "CF", "CML", or "Balducci". If not specified and lt carries a frac attribute, that value is used. An inherited "CML" attribute is normalized to "CF".
cohort: Two-life cohort: "first" (joint-life) or "last" (last survivor).
tidy: Logical. If TRUE, returns a tibble.
check: Logical. If TRUE, performs basic input checks. Table consistency and scalar logical controls are always checked.
tol: Numeric tolerance for integer checks.

Value

Numeric vector, or tibble if tidy = TRUE.

Details

Curtate expectation (Finan, Section 56.4 / Section 57): $$e_{xy} = \sum_{k=1}^{\infty} {}_kp_{xy}, \quad e_{\overline{xy}} = \sum_{k=1}^{\infty} {}_kp_{\overline{xy}}.$$

Complete expectation (Finan, Section 56.4): $$\mathring{e}_{xy} = \int_0^{\infty} {}_tp_{xy} \, dt.$$

The integral is decomposed year-by-year and evaluated analytically under UDD, constant force, or Balducci, retaining small survival in log form. Observed annual lx ratios determine the fractional survival. Survival is always measured from the initial ages, conditional on both lives being alive at time zero. In particular, last-survivor survival at later times does not imply that both lives remain alive.

Horizons end at the last recorded age: the older life limits joint-life expectation and the younger life limits last-survivor expectation. Survival beyond that age is taken as zero, as in t_px; an open table therefore gives a truncated expectation. Missing initial ages or zero initial survivors give NA, including when t = 0. Tables must have unique consecutive annual integer ages and finite, nonnegative, nonincreasing lx. Observed px/qx must be probabilities consistent with annual lx ratios within 1e-10; unknown probabilities may be NA. The age tolerance tol does not relax this mortality consistency threshold.

**Key identity** (Finan, Example 57.4): $$\mathring{e}_{\overline{xy}} = \mathring{e}_x + \mathring{e}_y - \mathring{e}_{xy}.$$

This can be used to cross-validate results.

Examples

lt <- data.frame(
  x  = 60:66,
  lx = c(100000, 99000, 97500, 95500, 93000, 90000, 86000)
)

# Curtate joint-life expectancy (Finan, Sec. 56.4)
e_xy(lt, x = 60, y = 62, type = "curtate", cohort = "first")
#> [1] 3.601538

# Curtate last-survivor expectancy (Finan, Sec. 57)
e_xy(lt, x = 60, y = 62, type = "curtate", cohort = "last")
#> [1] 5.746923

# Verify identity (Finan, Example 57.4):
# e_{xy-bar} = e_x + e_y - e_xy
e_joint <- e_xy(lt, x = 60, y = 62, type = "curtate", cohort = "first")
e_last  <- e_xy(lt, x = 60, y = 62, type = "curtate", cohort = "last")
e_x_val <- e_x(lt, x = 60, type = "curtate")
e_y_val <- e_x(lt, x = 62, type = "curtate")
c(last_surv = e_last, sum_minus_joint = e_x_val + e_y_val - e_joint)
#>       last_surv sum_minus_joint 
#>        5.746923        5.746923 

# Complete joint-life expectancy under UDD
e_xy(lt, x = 60, y = 62, type = "complete", frac = "UDD", cohort = "first")
#> [1] 3.691013

# Temporary: 3-year curtate joint-life
e_xy(lt, x = 60, y = 62, t = 3, type = "curtate", cohort = "first")
#> [1] 2.781231

# Tidy output
e_xy(lt, x = 60, y = 62, type = "curtate", cohort = "first", tidy = TRUE)
#> # A tibble: 1 × 7
#>       x     y     t type    frac  cohort    ex
#>   <int> <int> <dbl> <chr>   <chr> <chr>  <dbl>
#> 1    60    62    NA curtate UDD   first   3.60