1.017/1.010 Class 18
Small Sample Statistics
Small Samples
When sample size
N
is small the estimator of a distributional property
a
(mean, variance, 90 percentile, etc.) is generally not normal.
In this case, the CDF’s of the estimate
a
and standardized statistic
z
(used to derive confidence intervals and hypothesis tests) can be
approximated with
stochastic simulation
.
ˆ
In order to generate random replicates in the stochastic simulation
we need to specify the property
a
(or parameters that are related to
it):
For estimating confidence intervals we assume
a
a
ˆ
=
(the
estimate computed from the actual data).
For testing hypotheses we assume
0
a
a
=
(the hypothesized
parameter value).
The stochastic simulation uses many
N
rep
random sample replicates, each
of length
N
, to generate
N
rep
estimates.
The desired estimate and
standardized statistic CDFs are derived from this ensemble of estimates.
Example – Smallsample twosided confidence Intervals for the
mean of an exponential distribution
Consider a small sample that is thought to be drawn from an
exponential distribution with unknown parameter
a
:
[
x
1
,
x
2
,
x
3
,
x
4
,
x
5
]
= [0.05
1.46
0.50
0.72
0.11 ]
The sample mean is an unbiased estimator of
a
:
57
.
0
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 Spring '05
 GeorgeKocur
 Normal Distribution

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