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A362785
Size of the support of the Kaplan-Meier product-limit estimator indexed by sample size n.
0
2, 3, 5, 8, 15, 25, 49, 83, 134, 205, 409, 681, 1361, 2307, 3597, 5088, 10175, 16711, 33421, 55211, 76889, 115397, 230793, 383753, 536994, 820907, 1189517, 1597245, 3194489, 5137823, 10275645, 16487301, 22679853, 33790243, 48842489, 60737510, 121475019, 204647341, 303830465, 391169317
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
Sequence in context: A282239 A191792 A151518 * A082095 A177486 A006982
KEYWORD
nonn
AUTHOR
Yuxin Qin, May 03 2023
STATUS
approved