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Unformatted text preview: 2 (P.305P.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) : nN(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 , lawt
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‘}?[ mnH
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 lEIX] 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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 Spring '11
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 Probability

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