OFFSET
1,1
COMMENTS
No closed form formula is known for a(n), but the values can be determined by an induction algorithm. Let X_(n-1) be the set of KMPLE support values associated with n-1 items on test. Let |X_(n-1)| be the cardinality of X_(n-1). The set of KMPLE support values associated with n items in test is the union of X_(n-1) and the set consisting of the elements of X_(n-1) multiplied by (1- 1/n) = (n-1)/n.
For example, when n = 1, X_1 = {1,0}, so a(1) = |X_1| = 2.
When n = 2, X_2 = {1,0}*(1 - 1/2) union X_1 = {1, 1/2, 0}, so a(2) = 3.
Similarly, when n = 3, X_3 = X_2 * (1 - 1/3) union X_2 = {1, 2/3, 1/2, 1/3, 0}, so a(3) = 5, and so on.
The support values for a particular value of n can be found by using the R function km.support(n), which is included in the 'conf' package (see the link below).
LINKS
Yuxin Qin, H. D. Sasinowska, and L. M. Leemis, The Probability Mass Function of the Kaplan-Meier Product-Limit Estimator, The American Statistician, 2022, 77 (1), 102-110.
Christopher Weld, conf: Visualization and Analysis of Statistical Measures of Confidence, The Comprehensive R Archive Network, 2023.
MAPLE
support := {1};
for n from 2 to 40 do
support := support union {(n - 1) / n * op(support)}:
od:
PROG
(PARI) f(n) = if (n==1, [0, 1], my(v=List(f(n-1)), w=v); for (i=1, #w, listput(v, w[i]*(n-1)/n, )); Set(v));
a(n) = #f(n); \\ Michel Marcus, Aug 01 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Yuxin Qin, May 03 2023
STATUS
approved