On coupon collector’s and Dixie cup problems under fixed and random sample size sampling schemes

On coupon collector’s and Dixie cup problems under fixed and random sample size sampling schemes Suppose an urn contains m distinct coupons, labeled from 1 to m. A random sample of k coupons is drawn without replacement from the urn, numbers are recorded and the coupons are then returned to the urn. This procedure is done repeatedly and the sample sizes are independently identically distributed. Let W be the total number of random samples needed to see all coupons at least l times $$(l \ge 1)$$ ( l ≥ 1 ) . Recently, for $$l=1$$ l = 1 , the approximation for the first moment of the random variable W has been studied under random sample size sampling scheme by Sellke (Ann Appl Probab, 5:294–309, 1995). In this manuscript, we focus on studying the exact distributions of waiting times W for both fixed and random sample size sampling schemes given $$l \ge 1$$ l ≥ 1 . The results are further extended to a combination of fixed and random sample size sampling procedures. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of the Institute of Statistical Mathematics Springer Journals

On coupon collector’s and Dixie cup problems under fixed and random sample size sampling schemes

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Publisher
Springer Japan
Copyright
Copyright © 2016 by The Institute of Statistical Mathematics, Tokyo
Subject
Statistics; Statistics, general; Statistics for Business/Economics/Mathematical Finance/Insurance
ISSN
0020-3157
eISSN
1572-9052
D.O.I.
10.1007/s10463-016-0578-5
Publisher site
See Article on Publisher Site

Abstract

Suppose an urn contains m distinct coupons, labeled from 1 to m. A random sample of k coupons is drawn without replacement from the urn, numbers are recorded and the coupons are then returned to the urn. This procedure is done repeatedly and the sample sizes are independently identically distributed. Let W be the total number of random samples needed to see all coupons at least l times $$(l \ge 1)$$ ( l ≥ 1 ) . Recently, for $$l=1$$ l = 1 , the approximation for the first moment of the random variable W has been studied under random sample size sampling scheme by Sellke (Ann Appl Probab, 5:294–309, 1995). In this manuscript, we focus on studying the exact distributions of waiting times W for both fixed and random sample size sampling schemes given $$l \ge 1$$ l ≥ 1 . The results are further extended to a combination of fixed and random sample size sampling procedures.

Journal

Annals of the Institute of Statistical MathematicsSpringer Journals

Published: Aug 9, 2016

References

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