Composition by DEA Distance to the Frontier - Probabilistic Composition of Preferences, Theory and Applications - page 57

The biggest change occurred in the evaluation of development in the educational
dimension, whose indicator is now the geometric mean between two measures: the
average number of years that a person 25 years old now attended school and the
expected value of the number of years that a person 6 years old now will attend
school. This second parameter is estimated from the present distribution of school
enrolment provided by censuses and national sample surveys and employing the
estimation methodology of Barro and Lee ( 2010 ).
Before calculating their geometric mean, these two indicators are standardized to
values between zero and one. The average number of school years of the adult
observed in the country is divided by the observed maximum of 13.3 and the
expected number of school years of the child by the allowed maximum of 18.
By using, instead of the arithmetic mean of the three indicators, the geometric
mean, the analyst considers that the in

uence of the three factors accumulates in a
multiplicative basis. Thinking of each of the partial indicators as a probability to
meet a standard of welfare which seems to be the real aim of building measures

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between zero and 1 one may think HDI as the joint probability of reaching

excellence in the three dimensions.
Another interesting aspect is the application in the algorithm of the logarithmic
transformation to the income value. The logarithmic transformation is used to
reduce the importance of variations in the upper end of the indicator. A similar
effect may be obtained by applying the transformation into the probability of
reaching the lowest extreme.
In fact, the indicators developed below avoid two ad hoc transformations: the
logarithmic transformation and the prior determination of absolute maximum and
minimum for each component. The global indicator is constructed by a combination
of the probabilities of reaching the frontier of worst performance in each dimension.
HDI was created to emphasize the shortcomings in the today prevailing concept
of development. Employing the standardization in terms of probabilities of being
the worst has the advantage of calling attention to where the unattended features of
development are located by reducing the differences between the relatively satis-
factory performances and extending the differences between those countries in need
of larger improvement.
Table 6.1 , presenting the probabilities of being the worst for 30 countries
selected among those of best and worst performance in each component of the
index, illustrates this feature of the probabilistic transformation. These scores were
obtained by calculating the probability of each country being the worst in the entire
population of 187 countries, employing the data of 2013 of the United Nations
Development Programme (UNDP 2014 ). This data set is available at http://hdr.
undp.org/en/reports .
The computation of the probabilities of being the worst is performed adopting
triangular distributions with equal amplitudes for all countries and extreme exten-
ded by 10 %, for all the criteria.
It can be seen in Table 6.1 that the

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first 17 countries of the list have equal scores
for longevity, if rounding to the third decimal place is employed. In the other