Siler.Rddsiler gives the probability density function, psiler the cumulative density function, qsiler the quantile function, rsiler random draws, hsiler the hazard, chsiler the cumulative hazard, ssiler the survival, and nllsiler the negative log-likelihood.
hsiler(x, a) chsiler(x, a, x0 = 0) ssiler(x, a, x0 = 0) dsiler(x, a, x0 = 0) psiler(x, a, x0 = 0) qsiler(qvect, a, x0 = 0) rsiler(N, a, x0 = 0) nllsiler(a, x, x0 = 0)
| x | the ending point |
|---|---|
| a | parameterization vector |
| x0 | the initial point |
| qvect | vector of quantiles |
| N | number of observations |
A numeric vector of length equal to the input
The Siler hazard is
$$\lambda(x) = a_1 \exp(-a_2 \, x) + a_3 + a_4 \exp(a_5 \, x)$$
where \(a_i\) (index notation) is the parameter vector. The cumulative hazard is found by integrating the hazard from some initial point \(x_0\) to \(x\),
$$\Lambda(x_0,x) = -\frac{a_1}{a_2} (e^{-a_2 x} - e^{-a_2 x_0}) + a_3 (x-x_0) + \frac{a_4}{a_5} (e^{a_5 x} - e^{a_5 x_0})$$.
If \(x_0\) (optional) is not input it is set equal to 0. The survival and cumulative density function are, respectively,
$$S(x_0,x) = \exp(-\Lambda(x_0,x)) = \frac{S(0,x)}{S(0,x_0)} = \frac{S(x)}{S(x_0)}$$
and
$$F(x_0,x) = 1 - S(x_0,x)$$.
The probability density function is
$$f(x_0,x) = \frac{d}{dx} F(x_0,x) = \lambda(x) S(x_0,x)$$.