Sampling Methods for Rare Events

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• Sampling Methods for Rare Events
•  Basic Ideas
•   When we survey rare events, the conventional
  methods of sampling and estimation may be
  unsatisfactory. Even a large sample may not
  provide enough rare events.

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• In such cases we may consider using the following
  methods
•   1. Inverse sampling
• 2. Network sampling
• 3. Snowball sampling
• 4. Dual sampling (capture-recapture methods)

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• Inverse Sampling
•  In this method the sample n is not fixed in
  advance. Instead, sampling is continued until a
  predetermined number of units of possessing the
  rare attribute have been drawn.

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• Let P denote the proportion of units in the
  population possessing the rare attribute.
  Sampling is continued n times until m units with
  rare attribute are selected from the population
  with N units. Then there are NP units with rare
  attribute in the population, m is fixed, and n
  is a random variable. Then the probability
  distribution of n is given by

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As N tends to be large, the probability
distribution of n can be given by the well-known
negative binomial distribution.
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• Unbiased estimate of P and its sampling
  variance is as follows

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• Network Sampling
• See Section 14.5 on page 439
• See Sudman (Medical Care, October 1988)

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• Snowball Sampling - Chain referral sampling
•   See (Goodman, Annals of Mathematical
  Statistics 32(1), March 1961)
• See (Biernacki and Waldorf, Sociological
  Methods Analysis 10(2), November 1981)

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• Dual Sampling and Capture-Recapture Methods
•  There are several variations in dual sampling.
• 1. See Section 14.6 on page 443
•  
• 2. Tagging model under simple random sampling
• Let t denote the number of units tagged. Then
  Pt/N and Nt/P in the initial capture. Let
  ps/n the proportion tagged in the recapture
  sample. (n is fixed)

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(It is a biased estimate random variable is in
the denominator)

(See Cochran, Chapter 3 )
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3. Tagging model under inverse sampling   In
this model s is fixed and n is random.
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4. Tagging model under cluster sampling   For
example, if fishes are recaptured at several
randomly selected locations of a lake, you will
have several sets of n and s. Then it is a ratio
estimation problem. Let the number of clusters
(sampling spots) be m.