Limiting behavior of sampling proportional to random size

Conference · Mon Jan 01 04:00:00 EST 1979

OSTI ID:5499779

Uppuluri, V R.R.; Patil, S A

Let X/sub 1/,X/sub 2/,...,X/sub N/ be N nonnegative i.i.d. random variables. Let Y/sub 1/ = X/sub ..cap alpha../ with probability X/sub ..cap alpha..//(X/sub 1/+...+X/sub N/), ..cap alpha.. = 1,2...,N. This is referred to as the first realization when sampling with probability proportional to random size. Next Y/sub 1/ is deleted from X/sub 1/,X/sub 2/,...,X/sub N/ and another observation Y/sub 2/ is made similarly. Thus, in sampling proportional to random size, without replacement, one can obtain the sequence Y/sub 1/,Y/sub 2/,...,Y/sub n/. Let f(x) and ..mu.. denote the probability density function (pdf) and mean of X/sub ..cap alpha../, respectively. The pdf of the associated length-biased variable Y of X is defined by yf(y)/..mu... It is shown that, as N tends to infinity, Y/sub 1/ behaves like the length-biased sample of X. In a second theorem it is shown that, as N tends to infinity, Y/sub 1/,Y/sub 2/,...,Y/sub n/ behaves like independent length-biased variables of X.

• Research Organization: Union Carbide Corp., Oak Ridge, TN (USA). Computer Sciences Div.; Tennessee Technological Univ., Cookeville (USA). Dept. of Mathematics
• DOE Contract Number: W-7405-ENG-26
• OSTI ID: 5499779
• Report Number(s): CONF-791241-1
• Country of Publication: United States
• Language: English

Related Subjects

02 PETROLEUM
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03 NATURAL GAS
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99 GENERAL AND MISCELLANEOUS
990200 -- Mathematics & Computers
ASYMPTOTIC SOLUTIONS
GEOLOGIC DEPOSITS
MATHEMATICAL MODELS
MATHEMATICS
NATURAL GAS DEPOSITS
PETROLEUM DEPOSITS
PROBABILITY
RANDOMNESS
RESOURCE POTENTIAL
RESOURCES
SAMPLING
SIZE
STATISTICS