Example - 2(P.305-P.308 Sum Quota Sampling A common...

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Unformatted text preview: 2 (P.305-P.308) Sum Quota Sampling A common procedure in statistical inference is to observe a fixed number n of inde- pendent and identically distributed random variables X1, . . . ,Xn and use their sample mean - _ X1+---+ X" X"— n as an estimate of population mean or expected value E [X1]. Here we examine a situtation in which the sample size is not fixed in advance, but is determined by a preassigned quota t > 0. In sum quota sampling, a sequence of independent and iden- tically distributed nonnegative random variables X1, X2, . .. is observed sequentially, the sampling continuing as long as the sum of the observations is less than the quota t. Let this random sample size be denoted by N (t) Formally, N(t)=max{n20;X1+...+Xn <15}- The sample mean is WN(t) _ X1 + - ' ' +XN(t) N (t) N (0 Of 'course it is possible that X1 2 t, and then N (t) = 0 and the sample mean is unde- fined. Thus, we must assume, or condition on, the event that N (t) Z 1. An importamt question in statistical theory is whether or not this sample mean is unbiased. That is, how does the expected value of this sample mean relate to the expected value, say, X1? XNU) = Ems} EM Em] In general, the determination of the expected value of the sample mean under sum quota sampling is very difficult. It can be carried out, however, in the special case in which the individual X summands are expponentially distribution with common parameter A, so that N (t) is a Poisson process. One hopes that the results in the special case will shed some light on the behaviour of the sample mean under other distributions. E B323) [N(t) > 0] = :E [% We want To determined E[YNL+)1 Q N(t) = n] Pr{N(t) : n|N(t) > 0} The key is the use of the theorem to evaluate the conditional expectation Emotive) = n]: ELM {w.,... . .. ,wn} imam] : E[max‘{u;,, ,Unll , :4 -3‘. where, U:,ux,..,...)un~uniifom1[Qt] M“) t , law-t Let M:max{U1,...,Un} EMF Fri/Wm} 2 flful§m} gi‘ulém} WWW} : [1,2]” t 380"): 61/510") ,L— ’1’! M dn : in mm E[max{U1,...,Un}]= EL‘M] : J‘}?[ mn-H l : {ifmfimkln Note also that (At)"e‘)‘t Then We can perhaps more clearly see the effect of the sum quota sampling if we express the preceding calculation in terms of the ratio of the bias to the true mean E [X 1] = 1 / A. E[X l-EIX] T‘l' J—i—f )A’i’ 7”: We then have ———%i[)—X1]—’=____;__ ___. - T’— A e ‘ Fraction Bias E[N(t)] 0.58 0.31 0.16 0.17 0.03 0.015 0.0005 10 CBCflibOONl—I The left side is the fraction of bias, and the right side expresses this fraction bias as a function of the expected sample Size under sum quota sampling. We conclude that the bias due to sum quota sampling can be made acceptably small by choosing the quota t sufficiently large so that, on average, the sample Size so selected is reasonably large. If the individual observations are eXponentially distributed, the bias can be kept within .05% of the true value provided the quota t is large enough to give an average sample size of 10 or more. a ...
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