npsurv (version 0.3-3)

uh: U-shaped Hazard Function

Description

uh creates an object of class uh, which stores a U-shaped hazard function.

print.uh prints an object of class uh.

Usage

uh(alpha, tau, nu, eta, mu, upper=Inf, deg=1, collapse=TRUE)
## S3 method for class 'uh':
print(x, ...)

Arguments

alpha

a nonnegative value, for the constant coefficient.

tau

vector of nonnegative real values, for left knots.

nu

vector of nonnegative values, for masses associated with the left knots.

eta

vector of nonnegative real values, for right knots.

mu

vector of nonnegative real values, for masses associated with the right knots.

upper

a positive value, at which point the hazard starts to become infinite.

deg

nonnegative real number for spline degree (i.e., p in the formula below).

collapse

logical, indicating if identical knots should be collapsed.

x

an object of class uh.

...

other auguments for printing.

Value

• uh returns an object of class uh. It is a list having elements alpha, tau, nu, eta, mu, upper and deg, which store their corresponding values as described above.

Details

A U-shape hazard function, as generalized by Wang and Fani (2015), is given by

$$h(t) = \alpha + \sum_{j = 1}^k \nu_j(\tau_j - t)_+^p + \sum_{j = 1}^{m} \mu_j (t-\eta_j)_+^p,$$

where $\alpha,\nu_j,\mu_j \ge 0$, $\tau_1 < \cdots < \tau_k \le \eta_1 < \cdots < \eta_m,$ and $p \ge 0$ is the the spline degree which determines the smoothness of the U-shaped hazard. As p increases, the family of hazard functions becomes increasingly smoother, but at the same time, smaller. When $p = 0$, the hazard function is U-shaped, as studied by Bray et al. (1967). When $p = 1$, the hazard function is convex, as studied by Jankowski and Wellner (2009a,b).

print.uh prints an object of class uh. While alpha, upper and deg are printed as they are, tau and nu are printed as a two-column matrix, and so are eta and mu.

References

Bray, T. A., Crawford, G. B., and Proschan, F. (1967). Maximum Likelihood Estimation of a U-shaped Failure Rate Function. Defense Technical Information Center. Jankowski, H. K. and Wellner, J. A. (2009a). Computation of nonparametric convex hazard estimators via profile methods. Journal of Nonparametric Statistics, 21, 505-518.

Jankowski, H. K. and Wellner, J. A. (2009b). Nonparametric estimation of a convex bathtub-shaped hazard function. Bernoulli, 15, 1010-1035. Wang, Y. and Fani, S. (2015). Nonparametric maximum likelihood computation of a U-shaped hazard function. (In preparation)

See Also

Uhaz, icendata, plot.uh

Examples

(h0 = uh(3, 2, 3, 4, 5, 7, deg=0))              # deg = 0
plot(h0, ylim=c(0,20))
(h1 = uh(4, 2, 3, 5, 6, 7, deg=1))              # deg = 1
plot(h1, add=TRUE, col="green3")
(h2 = uh(1, 1:2, 3:4, 5:6, 7:8, 9, deg=2))      # deg = 2
plot(h2, add=TRUE, col="red3")