Asymptotic Methods in Statistical Inference - Statistics Centre

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imagine ‘large’ but remaining fixed; in this
case the CLT gives
−
q
(1 − )
→ (0 1) as →∞
This results in a normal approximation to the distribution
of and in the use of ˆ = as
a consistent estimate of (i.e. ˆ → ). On the
other hand, if is the frequency of a rare type
of X-ray out of a large number of emissions, we
might imagine ‘ large and small with the mean
number of emissions constant’; a way to formalize
this is that → 0as →∞. In this
case → P(), the Poisson distribution with
mean . Thus we get two possible but very different
limit distributions, depending on the sequence
within which is embedded.