Theorem 4.3: biases and variances of the boundary-free MRL estimators

(4.25)-(4.28). The two estimators' biases are identical except that the first carries b_2 and the second b_3; everything else is shared (Remark 4.3). The variance is O(1/n) with the bandwidth entering only at O(h/n), so it is far less bandwidth-sensitive than the bias.

Usage

morie_fauzi_theorem_4_3(
  t,
  S_X,
  S_bar_X,
  m_X,
  b1,
  b2 = NULL,
  b3 = NULL,
  n = 100,
  h = 0.1,
  g_prime = 1,
  f_X = 1,
  mu2 = 1,
  VW = NULL
)

Arguments

t

evaluation points.

S_X

survival at t, strictly positive.

S_bar_X

the cumulative survival at t.

m_X

mean residual life at t.

b1

coefficient b_1 at t.

b2, b3

coefficients; supply either or both.

n

sample size.

h

bandwidth.

g_prime, f_X

transformation derivative and density at t.

mu2

int y^2 K(y) dy.

VW

int V(y) W(y) dy; the Gaussian value if NULL.

Value

list: t, bias_1, bias_2, variance, b4, b5, differ_only_in, variance_leading_order, n, h, method.

References

Fauzi and Maesono (2023), Theorem 4.3.

Examples

morie_fauzi_theorem_4_3(1, 0.4, 0.5, 1.25, 0.3, b2 = 0.7)$bias_1
#> [1] 0.0134375