------------------------------------------------------------------------------- help forsmgfitAustin Nichols (June 2009) -------------------------------------------------------------------------------

Program to fit a Singh-Maddala distribution to grouped data via ML

smgfitnvar[ifexp] [inrange] [,z1(z1var)z2(z2var)avar(var)bvar(var)qvar(var)sva(var)svb(var)svq(var)replacedoublefrom(string)level(#)maximize_options]

by...:may be used withsmgfit; see help by.

Description

smgfitfits by ML the 3 parameter Singh-Maddala (1976) distribution to a sample of counts or frequencies innvarby income category. (For an estimator designed for unit record data, see smfit). Optionally specified variablesz1varandz2varencode the lower and upper limits of each interval; if they are not specified, variables serving that role and called z1 and z2 are assumed to exist.Otherwise known as the Burr Type 12 distribution or as a Beta-P distribution (Cronin 1979 and Johnson and Kotz 1970), the Singh-Maddala distribution seems to provide a good fit to empirical income data relative to other parametric functional forms; see e.g. McDonald (1984). The Singh-Maddala distribution is closely related to the Dagum (Burr Type 3) distribution of Dagum (1977,1980) ; see dagfit (or see dagumfit for an estimator designed for unit record data). Both are special cases of the Generalized Beta of the Second Kind distribution (see gb2fit for an estimator designed for unit record data). For a comprehensive review of these and other related distributions, see Kleiber and Kotz (2003). For derivation of Lorenz orderings of pairs of income distributions in terms of their Singh-Maddala parameters, see Wifling and Kraemer (1993) and Kleiber (1996). The Singh-Maddala distribution may be useful for describing any skewed positive variable.

The likelihood function for a sample of observations on

nvaris specified as the product of the integrated densities (between z1 and z2) to thenvarpower, and is maximized usingml model lf. See McDonald (1984) for more information and references.

Example from McDonald (1984)clear all input z1 f70 f75 f80 0 66 35 21 2.5 125 85 41 5 152 106 62 7.5 166 106 65 10 158 114 73 12.5 110 109 69 15 131 188 140 20 46 116 137 25 30 95 198 35 11 32 128 50 05 14 67 end g z2=z1[_n+1] g mid=(z1+z2)/2 replace mid=100 if mid==. set obs 200 g x=_n/2 la var x "Income" dagfit f70 g dg70=e(ba)*e(bp)*(e(bb)/x)^e(ba)/x*(1+(e(bb)/x)^e(ba))^(-e(bp)-1) la var dg70 "PDF for grouped Dagum MLE 1970" tw hist mid [fw=f70]||line dg70 x, name(d) smgfit f70 g sg70=(e(ba)*e(bq)/e(bb))*((1+(x/e(bb))^e(ba))^-(e(bq)+1))*((x/e(bb))^(e(ba)-1 > )) la var sg70 "PDF for grouped Singh-Maddala MLE 1970" tw hist mid [fw=f70]||line sg70 x, name(sm) dagumfit mid [fw=f70] g di70=e(ba)*e(bp)*(e(bb)/x)^e(ba)/x*(1+(e(bb)/x)^e(ba))^(-e(bp)-1) la var di70 "PDF for individual Dagum MLE applied to group data from 1970" smfit mid [fw=f70] g si70=(e(ba)*e(bq)/e(bb))*((1+(x/e(bb))^e(ba))^-(e(bq)+1))*((x/e(bb))^(e(ba)-1 > )) la var si70 "PDF for individual Singh-Maddala MLE applied to group data from 19 > 70" line dg70 sg70 di70 si70 x, leg(col(1)) scale(.8)

Options

z1(z1var),z2(z2var)are the lower and upper limits of each interval; if they are not specified, variables serving that role and called z1 and z2 are assumed to exist. It should always be true that the upper bound of one category is the same as the lower bound of the next highest category.

?var(var)options specify varlists that are assumed to have a linear effect on the parameter specified. By default, each varlist contains only a constant, so only the parameter itself is estimated.

sv?(var)options specify newvars in which to store the estimated parameters. If?var(var)options have been specified, the prediction assumes all explanatory variables are zero, i.e. the prediction is only for the constant.

replaceallowssv?(var)options to specify existing variables, whose values are replaced in the estimation sample.

doublerequests thatsv?(var)create doubles.

from(string)specifies initial values for the Dagum parameters, and is likely to be used only rarely. You can specify the initial values in one of three ways: the name of a vector containing the initial values (e.g., from(b0) where b0 is a properly labeled vector); by specifying coefficient names with the values (e.g., from(a:_cons=1 b:_cons=5 p:_cons = 0); or by specifying an ordered list of values (e.g., from(1 5 0 .16, copy)). Poor values in from() may lead to convergence problems. For more details, including the use of copy and skip, see {help:maximize}.

level(#)specifies the confidence level, in percent, for the confidence intervals of the coefficients; see help level.

nologsuppresses the iteration log.

maximize_optionscontrol the maximization process. The options available are those shown by maximize, with the exception offrom(). If you are seeing many "(not concave)" messages in the iteration log, using thedifficultortechniqueoptions may help convergence.

Saved resultsIn addition to the usual results saved after

ml,dagfitalso saves the following:

e(a),e(b), ande(p)are the estimated Singh-Maddala parameters.

e(mode),e(mean),e(var),e(sd),e(i2), ande(gini)are the estimated mode, mean, variance, standard deviation, half coefficient of variation squared, Gini coefficient.e(pX), ande(LpX)are the quantiles, and Lorenz ordinates, where X = {1, 5, 10, 20, 25, 30, 40, 50, 60, 70, 75, 80, 90, 95, 99}.

FormulaeThe Singh-Maddala distribution has distribution function (c.d.f.)

F(x) = 1-(1+(x/b)^a)^(-q)

where a, b, q, are parameters, each positive, for random variable x > 0. Parameters a and q are the key distributional 'shape' parameters; b is a scale parameter.

The probability density function (p.d.f.) is

f(x) = (a*q/b)*((1+(x/b)^a)^-(q+1))*((x/b)^(a-1))

The likelihood function for a sample of observations on

nvaris specified as the product of the density integrated from z1 to z2 and raised to the powernvar, the count of observations in the category, and is maximized usingml model lf.The formulae used to derive the distributional summary statistics are as follows. The r-th moment about the origin is given by

b^r*B(1+r/a,q-r/a)/B(1,q)

where B(u,v) is the Beta function = G(u)*G(v)/G(u+v) and G(.)=exp(lngamma(.)) is the gamma function, which by substitution and using G(1) = 1, implies the moments can be written

b^r*G(1+r/a)*G(q-r/a)/G(q)

and hence

mean = b*G(1+1/a)*G(q-1/a)/G(q)

variance = b*b*G(1+2/a)*G(q-2/a)/G(q) - (mean^2)

from which the standard deviation and half the squared coefficient of variation can be derived. The mode is

mode = b*((a-1)/(aq+1))^(1/a) if a > 1, and 0 otherwise.

The quantiles are derived by inverting the distribution function:

x_s = b*((1-s)^(-1/q) - 1)^(1/a) for each s = F(x_s).

The Gini coefficient of inequality is given by

Gini = 1 - G(q)*G(2q - 1/a) / ( G(q-1/a)*G(2q) ).

The Lorenz curve ordinates at each s = F(x_s) use the incomplete Beta function:

L(s) =

ibeta(1+1/a, q- 1/a, 1-(1-s)^(1/q) ).

AuthorAustin Nichols <austinnichols@gmail.com>

AcknowledgementsStephen P. Jenkins has made available commands for fitting various distributions (including the Singh-Maddala) to individual record data; see Jenkins (2004). This package draws liberally from that work; note in particular the similarity of verbiage under headings Formulae and Description.

ReferencesCronin, D. C. (1979). A Function for the Size Distribution of Income: A Further Comment.

Econometrica47: 773-774.Dagum, C. (1977). A new model of personal income distribution: specification and estimation.

Economie Appliquée30: 413-437.Dagum, C. (1980). The generation and distribution of income, the Lorenz curve and the Gini ratio.

Economie Appliquée33: 327-367.Jenkins, S.P. (2004). Fitting functional forms to distributions, using

ml. Presentation at Second German Stata Users Group Meeting, Berlin. http://www.stata.com/meeting/2german/Jenkins.pdfJohnson, N. L., and S. Kotz (1970).

Continuous Univariate Distributions1. New York: John Wiley and Sons.Kleiber, C. (1996). Dagum vs. Singh-Maddala income distributions.

Economics Letters53: 265-268.Kleiber, C. and Kotz, S. (2003).

Statistical Size Distributions inEconomics and Actuarial Sciences. Hoboken, NJ: John Wiley.McDonald, J.B. (1984). Some generalized functions for the size distribution of income.

Econometrica52: 647-663.Singh, S.K. and G.S. Maddala (1976). A function for the size distribution of income.

Econometrica44: 963-970.Tadikamalla, P. R. (1980). A Look at the Burr and Related Distributions.

International Statistical Review48: 337-349.Wifling, B. and W. Kraemer (1993). The Lorenz-ordering of Singh- Maddala income distributions.

Economics Letters43: 53-57.

Also seeOnline: help for dagfit, gbgfit, dagumfit, smfit, gb2fit, lognfit, if installed, or acquire them from ssc.