A Unified Approach to Measuring Poverty and Inequality
• Finally, another well-known mean is the Euclidean mean (WE), obtained when a = 2. The Euclidean mean formula is 1
⎛ x12 + x 22 + L + x 2N ⎞ 2 WE (x) = ⎜ ⎟⎠ . N ⎝
(2.8)
Example 2.2 shows the results of calculating these means for a given income vector. Example 2.2: Consider the income vector x = ($2k, $4k, $8k, $10k). • • • •
The arithmetic mean of x is ($2k + $4k + $8k + $10k)/4 = $6k. The geometric mean of x is ($2k × $4k × $8k × $10k)1/4 = $5.03k. The harmonic mean of x is [($2k−1 + $4k−1 + $8k−1 + $10k−1)/4]−1 = $4.10k. The Euclidean mean of x is [($2k2 + $4k2 + $8k2 + $10k2)/4]1/2 = $6.78k.
Having been introduced to the family, one can now understand the properties of general means and the way they depend on parameter a. All means in this family satisfy symmetry, normalization, population invariance, linear homogeneity, monotonicity, and subgroup consistency. Furthermore, for a < 1, general means satisfy the transfer principle. Thus, the general means satisfy all the dominance properties introduced earlier. One reason is that, unlike the quantile means and the partial means, general means consider all incomes in the distribution. It is straightforward to show that general means satisfy symmetry, normalization, population invariance, linear homogeneity, and monotonicity. That general means satisfy subgroup consistency may be verified as follows: if vector x is divided into subgroup vectors x' and x", then the general mean of x can be expressed as WGM(x; a) = WGM((WGM(x'; a), WGM(x"; a)); a).
(2.9)
In other words, the general mean of x is the general mean of the general means of x' and x". Then the monotonicity property ensures that subgroup consistency is satisfied.
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