45     Distribution and Lorenz Functions

45.1  Formula for Break Points

pi = V (a) + b [ V (a+ 1) - V (a) ]

where V is an ordered data vector, e.g. V (3) is the third item in the vector,

a = entier é ê ë i ( N + 1) s ù ú û

b = i ( N + 1) s - a

and entier(x) is the greatest integer not exceeding x.

45.2  Distribution Function Break Points

• If a break point falls exactly on a value and the value is not tied with any other value, then the value itself is the break point.
• If a break point falls between two values and the two values are not the same, then the break point is determined using ordinary linear interpolation.
• If a break point falls exactly on a value and the value is tied with one or more other values, then the procedure involves computing new midpoints. Let k be the value, m be the frequency with which it occurs and d be the minimum distance between items in the vector V. The interval k±min(d,1)/2 is divided into m parts and midpoints are computed for these new intervals. The break point is then the appropriate midpoint.
• If a break point falls between two values which are identical, the procedure involves both the calculation of new midpoints and ordinary linear interpolation. Let k be the value, m be the frequency with which it occurs and d be the minimum distance between items in the vector V. The interval k±min(d,1)/2 is divided into m parts and midpoints are computed for these new intervals. Then linear interpolation is performed between the two appropriate new midpoints.

45.3  Lorenz Function Break Points

45.4  Lorenz Curve

Note that Lorenz function values are called "Fraction of wealth" on the printout.

45.5  The Gini Coefficient

The program uses an approximation:

Gini coefficient = 1 - 1 s - 2 s s-1 å i = 1  li

where l_i is the i^th Lorenz function break point.

This approximation becomes more accurate as the number of break points is increased; it is recommended that at least ten be used.

45.6  Kolmogorov-Smirnov D Statistic

Let V₁ and V₂ be the ordered data vectors for the first and the second variable respectively, and X the vector of codes which appear in either distribution. The program creates the two cumulative step functions F₁ (x) and F₂ (x) respectively. Then it looks for maximum absolute difference between the distributions,

D = max ( | F1 (x) - F2 (x) | )

and prints:

x : the value where the first maximum absolute difference occurs f1 : the value of F1 associated with the x f2 : the value of F2 associated with the x.

If the N's for V₁ and V₂ are equal and less than 40, the program prints K statistic equal to the difference in frequencies associated with the maximum difference. A table of critical values of K statistic, denoted K_D, can be consulted to determine the significance of the observed difference.

If the N's for V₁ and V₂ are unequal or larger than 40, the program prints the following statistics:

Unadjusted deviation = D = | f1 - f2 |

Adjusted deviation = D     æ  ú Ö N1  N2 N1 + N2

where N₁ and N₂ are equal to the number of cases in V₁ and V₂ respectively.

Chi-squared approximation = 4 D2  ×  N1  N2 N1 + N2

Note: The significance of the maximum directional deviation can be found by referring this chi-square value to a chi-square distribution with two degrees of freedom.

45.7  Note on Weights