A362785 Size of the support of the Kaplan-Meier product-limit estimator indexed by sample size n.

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

Table of n, a(n) for n=1..40.

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

Adjacent sequences: A362782 A362783 A362784 * A362786 A362787 A362788

Keyword

nonn

Author

Yuxin Qin, May 03 2023

Status

approved