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t-year probability of decrement by cause j: t_qxj

Source: R/t_qxj.R
t_qxj.Rd

Computes \({}_t q_x^{(j)}\), the probability that a life aged x decrements by a specific cause j within t years, using a multiple decrement table produced by md_table.

Usage

t_qxj(md, x, t, cause, frac = NULL, tidy = FALSE, check = TRUE, tol = 1e-10)

Arguments

md

A multiple decrement table produced by md_table. Must contain columns x, p_total, and the requested cause.

x

Numeric vector. Starting age(s) (integer-valued).

t

Numeric vector. Time horizon(s) in years (t >= 0). Can be non-integer if frac is supplied.

cause

Character. Name of the cause column in md (e.g., "q_death").

frac

Character. Fractional-age assumption for non-integer t: one of "UDD", "CF", "Balducci", or NULL. If NULL (default), t must be integer-valued.

tidy

Logical. If TRUE, returns a tibble with columns x, t, cause, frac, and tqxj.

check

Logical. If TRUE, performs input validation (default TRUE).

tol

Numeric tolerance for integer checks (default 1e-10).

Value

Numeric vector of \({}_t q_x^{(j)}\) (or tibble if tidy=TRUE).

Details

Let \(q_x^{(j)}\) be the annual decrement probability for cause \(j\) and \(q_x^{(\tau)}\) be the total decrement probability. For integer \(t\), $$ {}_t q_x^{(j)} = \sum_{k=0}^{t-1} \left(\prod_{r=0}^{k-1} p_{x+r}^{(\tau)}\right) q_{x+k}^{(j)}. $$

For non-integer \(t = n + s\) with \(n = \lfloor t \rfloor\) and \(s \in [0,1)\), this function supports fractional-age assumptions specified by frac and uses the additional convention that the within-year cause proportions remain constant: \({}_s q_x^{(j)} = w_j \, {}_s q_x^{(\tau)}\) where \(w_j = q_x^{(j)} / q_x^{(\tau)}\) (and 0 when \(q_x^{(\tau)}=0\)).

Supported fractional-age assumptions for the total decrement:

  • "UDD": \({}_s q_x^{(\tau)} = s q_x^{(\tau)}\).

  • "CF": \({}_s p_x^{(\tau)} = (p_x^{(\tau)})^s\).

  • "Balducci": \({}_s q_x^{(\tau)} = \frac{s q_x^{(\tau)}}{1-(1-s)q_x^{(\tau)}}\).

Examples

qx_df <- tibble::tibble(
  x = 30:35,
  q_death = c(0.001, 0.0012, 0.0014, 0.0017, 0.0020, 1.0000),
  q_disability = c(0.002, 0.0021, 0.0022, 0.0023, 0.0024, 0.0000)
)
md <- md_table(qx_df, radix = 1e5, close = TRUE)
t_qxj(md, x = 30, t = 5, cause = "q_death")
#> [1] 0.007243163
t_qxj(md, x = 30, t = 2.5, cause = "q_death", frac = "CF", tidy = TRUE)
#> # A tibble: 1 × 5
#>       x     t cause   frac     tqxj
#>   <int> <dbl> <chr>   <chr>   <dbl>
#> 1    30   2.5 q_death CF    0.00289