More on Hypothesis Testing:An example of some empirical research: An analysis of
city size distributions in China introducing The Difference in Means and Associated
Tests
Sometimes rather than be concerned about the particular value of the mean or
variance of a population, or the nature of the underlying population distribution and we
may wish to examine whether two population distributions differ, for example we may be
interested in comparing the means or variances from two distinct populations. It turns out
that with some minor modifications we perform very similar tests to the ones we have
already discussed. To exemplify the tests discussed in the previous chapter and to
introduce these new tests we will use some recent research on the nature and progress of
city sizes in China over the period 19491999 (Anderson and Ge (2003)).
Some Background: A Theory of City Size Distributions.
There is a theory in urban economics (Gabaix (1999)), for which evidence has been
found in many countries, which says that when measured by the number of inhabitants
city sizes within a country are governed by Zipf’s law Zipf (1949) (the logarithm of the
rank of the city size
≈
 logarithm of the size relative to the minimum). This means that
the distribution of city sizes at any point in time is a Pareto Distribution with a parameter
θ
= 1. Letting x be the city size and x
min
be the minimum possible city size, the pdf and
cdf of this distribution are given by f(x) =
θ
(x
min
/x
θ
+1
) and F(x) = 1 (x
min
/x)
θ
respectively.
The law derives from two ideas, the first is that individual cities start at some
minimum size (call it Z
0
) which is subject to a sequence of mutually independent
multiplicative shocks (call the one in the i’th period (1+X
i
) where X
i
is small relative to 1
and to Z
0
). In essence the shock is related to the number of people that die in, the number
of people that are born in, the number of people that immigrate to and the number of
people that emigrate from the city in period i, all of which are random events. The
logarithm of city size in period I may be written as:
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0
0
1
1
ln(
)
ln
(1
)
ln
I
I
I
i
i
i
i
Z
Z
X
Z
=
=
⎛
⎞
⎛
⎞
=
+
=
+
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
e
∑
∏
where e
i
= ln(1+X
i
). For I sufficiently large (that is after a sufficiently long period of
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 Spring '11
 Kambourov
 Normal Distribution, city size

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