The name is absent

The formulas used to derive the distributional summary statistics presented (optionally) are as follows. The rth moment
about the origin is given by

b^rB() + r∣a,q-r∣a)∣B(1,q)

where B(u,v) is the Beta distribution = G(u)G(y)∕G(u + an and G is the gamma function (exp(Ingamma(.)) in Stata),
which by substitution and using the result that G(1) = 1, implies that the moments can be written

b^,Gi∖ + r∣a)G(q - r∣a)∣G(q)

and hence

E(X) = 1G(1 + 1∕a)G(q - 1∕a)∕G(q)

Var(X) = δ²G^,(1 + 2∕a)G(q - 2∕a)∕G(q) - (E(X))²

from which the standard deviation and half the squared coefficient of variation can be derived. The percentiles are derived by
inverting the distribution function

_ir_p = 0[(1-_p)(-¹∕9)-1](¹∕^α)

for each p = F(x_p).

The Gini coefficient of inequality, Gini, is given by

1 — Gini = 2(q)G(2q — 1∕a)∕[G(q — 1∕α)G(2g)]

The Lorenz curve ordinates L(p) at each p = F(x_p) use the Beta cdf, ibeta(.) in Stata:

L(p) = ibeta(1 + 1∕a,q — 1∕α, 1-(1 — p)^^q^)

Syntax

smfit incvar Weeghlp [if exp∖ [in range] [, stats cdf (cdfname) pdf (pdfname)
level(#) nolog trace aO(#) bO(#) qO(#)]

fweights and aweights are allowed.

To reset problem-size limits, see help matsize.

Options

stats displays selected distributional statistics implied by the Singh-Maddala parameter estimates; percentiles, cumulative shares
of total income at percentiles (i.e., the Lorenz curve ordinates), the mean, standard deviation, variance, half the coefficient
of variation squared, Gini coefficient, and percentile ratios p90∕p10, p15∕p25.

cdf Cdffnmne) creates a new variable cdfname containing the estimated Singh-Maddala cdf value F (ж) for each x in the dataset.

pdf Pfffamee) creates a new variable pdfname containing the estimated Singh-Maddala pdf value f(x) for each x in the dataset.

level(#) specifies the confidence level, in percent, for confidence intervals. The default is level(95) or as set by set level;
see [U] 26.4 Specifying the width of confidence intervals.

nolog suppresses the iteration logs.

trace reports the current value of the estimated parameters at each iteration; see [R] maximize.

aO(#), b0(#), q0(#) allow the user to specify starting values for the Singh-Maddala parameters. Default starting values are
2 = 2, q = 2, and b = sample mean of incvar.

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