The name is absent

21

The summary of the relationships among functions 1- 5 is shown below:
t

F(V) = j f(s)ds = I-S(V)-                             (2.10)

0
∞

S(V) = I-F(V) = У f(s)ds = e~^^x^^ds = e~^κ^-          (2.11)

t

r(∣∖        f(∕∖          dF(t)          dS(t)

ʌ(ŋ ^^ S{t) ~ 1 - F(t) ~ 1 - F(t) ^^ S(t) '                ⁽²'¹²⁾

ʌ(ŋ ɪ [ X(t)d_s = -l∏(S(t)) = -ln(l - F(t)∖,             (2.13)

Jq

fW = ɪ =-ɪ = W(ⁱ) = ʌ(θ(i ~ r(ʧ);     (2.14)

UjV               Uj V

In Equation 2.10, F(i) is the probability that a person did not survive longer than
time t. That is, it is the complement of S(V) by definition.

Note that any one of f(V), F(V), λ(V) or Λ(i) is enough to specify the survival func-
tion, because knowing any one, you can calculate the other three functions. Figure 2.3
shows the pdf, hazard function, survival function and cdf for a Weibull distribution
with the scale parameter p = 1 and the shape parameter 7 = 0.5.

2.3.3 Kaplan-Meier estimator

The Kaplan-Meier (K-M) estimator is a typical way of estimating a survival function.
It is defined below:

Let ⅛ = 0,

- ∏ (1 - V^j = ∏ V                <²'¹⁵>

i-.t_i<t            ^г i-.t_i<t ^l

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