{smcl}
{* 29 Dec 2007}{...}
{hline}
help for {hi:fiskfit}{right:Maarten L. Buis and}
{right:Stephen P. Jenkins (December 2007)}
{hline}
{title:Fitting a Fisk distribution by ML to unit record data}
{p 8 17 2}{cmd:fiskfit} {varname} {weight} {ifin} [{cmd:,}
{opt a:var(varlist1)} {opt b:var(varlist2)}
{opt ab(varlist)}
{cmdab:st:ats} {opt f:rom(string)} {opt poor:frac(#)}
{opt cdf(cdfname)} {opt pdf(pdfname)}
{cmdab:r:obust} {opt cl:uster(varname)} {cmd:svy}
{opt l:evel(#)} {it:maximize_options} {it:svy_options} ]
{pstd}{cmd:by} {it:...} {cmd::} may be used with {cmd:fiskfit}; see help
{help by}.
{pstd}{cmd:pweight}s, {cmd:aweight}s, {cmd:fweight}s, and {cmd:iweight}s
are allowed; see help {help weights}. To use {cmd:pweight}s, you must first
{cmd:svyset} your data and then use the {cmd:svy} option.
{title:Description}
{pstd}
{cmd:fiskfit} fits by ML the 2 parameter Fisk (1961) or log-logistic distribution
to sample observations on a random variable {it:var}. Unit record data are
assumed (rather than grouped data). It is closely related to the
Singh-Maddala (Burr Type 12) distribution (Singh and Maddala, 1976) and the
Dagum (Dagum, 1977,1980) distribution. All are special cases of the Generalized Beta
of the Second Kind distribution (see {help gb2fit}). For a comprehensive
review of these and other related distributions, see Kleiber and Kotz (2003).
{title:Options}
{phang}
{opt avar(varlist1)}, and {opt bvar(varlist2)}, allow the user to specify each
parameter as a function of the covariates specified in the respective
variable list. A constant term is always included in each equation.
{phang}
{opt ab(varlist)} can be used instead of the previous option if the same
covariates are to appear in each parameter equation.
{phang}
{opt from(string)} specifies initial values for the Fisk 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}.
{phang}
If covariates are specified, the next four options are not available.
{phang}
{cmd:stats} displays selected distributional statistics implied by the
Fisk parameter estimates: quantiles, cumulative
shares of total {it:var} at quantiles (i.e. the Lorenz curve
ordinates), the mode, mean, standard deviation, variance, half the
coefficient of variation squared, Gini coefficient, and
quantile ratios p90/p10, p75/p25.
{phang}
{opt poor:frac(#)} displays the estimated proportion with values of {it:var}
less than the cut-off specified by {it:#}. This option may be specified when replaying
results.
{phang}
{opt cdf(cdfname)} creates a new variable {it:cdfname} containing the
estimated Fisk c.d.f. value F(x) for each x.
{phang}
{opt pdf(pdfname)} creates a new variable {it:pdfname} containing the
estimated Fisk p.d.f. value f(x) for each x.
{phang}
{cmd:robust} specifies that the Huber/White/sandwich estimator of
variance is to be used in place of the traditional calculation; see
{hi:[U] 23.14 Obtaining robust variance estimates}. {cmd:robust} combined
with {cmd:cluster()} allows observations which are not independent within
cluster (although they must be independent between clusters). If you
specify {help pweight}s, {cmd:robust} is implied.
{phang}
{cmd:cluster(}{it:varname}{cmd:)} specifies that the observations are
independent across groups (clusters) but not necessarily within groups.
{it:varname} specifies to which group each observation belongs; e.g.,
{cmd:cluster(personid)} in data with repeated observations on individuals.
See {hi:[U] 23.14 Obtaining robust variance estimates}. {cmd:cluster()} can be
used with {help pweight}s to produce estimates for unstratified
cluster-sampled data. Specifying {cmd:cluster()} implies {cmd:robust}.
{phang}
{cmd:svy} indicates that {cmd:ml} is to pick up the {cmd:svy} settings
set by {cmd:svyset} and use the robust variance estimator. Thus, this option
requires the data to be {cmd:svyset}; see help {help svyset}. {cmd:svy} may not be
combined with weights or the {cmd:strata()}, {cmd:psu()}, {cmd:fpc()}, or
{cmd:cluster()} options.
{phang}
{cmd:level(}{it:#}{cmd:)} specifies the confidence level, in percent,
for the confidence intervals of the coefficients; see help {help level}.
{phang}
{cmd:nolog} suppresses the iteration log.
{phang}
{it:maximize_options} control the maximization process. The options
available are those shown by {help maximize}, with the exception of {cmd:from()}.
If you are seeing many "(not concave)" messages in the iteration
log, using the {cmd:difficult} or {cmd:technique} options may help convergence.
{phang}
{it:svy_options} specify the options used together with the {cmd:svy} option.
{title:Saved results}
{p 4 4 2}In addition to the usual results saved after {cmd:ml}, {cmd:fiskfit} also
saves the following, if no covariates have been specified and the relevant options used:
{p 4 4 2}{cmd:e(ba)}, and {cmd:e(bb)} are the estimated Fisk parameters.
{p 4 4 2}{cmd:e(cdfvar)} and {cmd:e(pdfvar)} are the variable names specified for the
c.d.f. and the p.d.f.
{p 4 4 2}
{cmd:e(mode)}, {cmd:e(mean)}, {cmd:e(var)}, {cmd:e(sd)}, {cmd:e(i2)}, and {cmd:e(gini)}
are the estimated mode, mean, variance, standard deviation, half coefficient of
variation squared, Gini coefficient. {cmd:e(pX)}, and {cmd:e(LpX)} are the
quantiles, and Lorenz ordinates, where X = {1, 5, 10, 20, 25, 30, 40, 50,
60, 70, 75, 80, 90, 95, 99}.
{p 4 4 2}The following results are saved regardless of whether covariates have been
specified or not.
{p 4 4 2}{cmd:e(b_a)}, and {cmd:e(b_b)} are row vectors containing the
parameter estimates from each equation.
{p 4 4 2}{cmd:e(length_b_a)},and {cmd:e(length_b_b)} contain
the lengths of these vectors. If no covariates have been specified in an equation,
the corresponding vector has length equal to 1 (the constant term);
otherwise, the length is one plus the number of covariates.
{title:Formulae}
{p 4 4 2}
The Fisk distribution has distribution function (c.d.f.)
{p 8 8 2}
F(x) = 1/[ 1 + (b/x)^a ]
{p 4 4 2}
where a, b, are parameters, each positive, for random variable x > 0.
Parameters a is the key distributional 'shape' parameters; b is a scale parameter.
{p 4 4 2}
Letting z = 1 + (b/x)^a, then F(x) = 1/z, and the probability
density function (p.d.f.) is
{p 8 8 2}
f(x) = a(b/x)^a*(1/x)/z^2
{p 4 4 2}
The likelihood function for a sample of observations on {it:var} is specified
as the product of the densities for each observation (weighted where relevant), and
is maximized using {cmd:ml model lf}.
{p 4 4 2}
The formulae used to derive the distributional summary statistics
presented (optionally) are as follows. The r-th moment about the origin
is given by
{p 8 8 2}
b^r*B(1+r/a,1-r/a)
{p 4 4 2}
where B(u,v) is the Beta function = G(u).G(v)/G(u+v) and G(.) is the
gamma function [exp({cmd:lngamma}(.)], which by substitution and using G(1) = 1,
implies the moments can be written
{p 8 8 2}
b^r*G(1-r/a)*G(1+r/a)
{p 4 4 2}
and hence
{p 8 8 2}
mean = b*G(1-1/a)*G(1+1/a)
{p 8 8 2}
variance = (b^2)*G(1-2/a)*G(1+2/a) - mean^2
{p 4 4 2}
from which the standard deviation and half the squared coefficient of
variation can be derived. The mode is
{p 4 4 2}
mode = b*((a-1)/(a+1))^(1/a) if a > 1, and 0 otherwise.
{p 4 4 2}
The quantiles are derived by inverting the distribution function:
{p 8 8 2}
x_s = b*( 1/s - 1)^(1/a) for each s = F(x_s).
{p 4 4 2}
The Gini coefficient of inequality is given by
{p 8 8 2}
Gini = -1 + G(2 + 1/a) / { G(1+1/a)*G(2) }.
{p 4 4 2}
The Lorenz curve ordinates at each s = F(x_s) use the incomplete Beta function:
{p 8 8 2}
L(s) = {cmd:ibeta}(1+1/a, 1-1/a, s ).
{title:Examples}
{p 4 8 2}{inp:. fiskfit x [w=wgt] }
{p 4 8 2}{inp:. fiskfit }
{p 4 8 2}{inp:. fiskfitfit, stats poorfrac(100) }
{p 4 8 2}{inp:. fiskfitmfit x, a(age sex) b(age sex) }
{p 4 8 2}{inp:. fiskfit x, ab(age sex) }
{p 4 8 2}{inp:. predict double a_i, eq(a) xb }
{p 4 8 2}{inp:. predict double b_i, eq(b) xb }
{p 4 8 2}{inp:. predict double p_i, eq(p) xb }
{title:Authors}
{p 4 4 2}Maarten L. Buis , Department of Social Research Methodology,
Vrije Universiteit Amsterdam, Boelelaan 1081, 1081 HV Amsterdam, The Netherlands.
{p 4 4 2}Stephen P. Jenkins , Institute for Social
and Economic Research, University of Essex, Colchester CO4 3SQ, U.K.
{title:References}
{p 4 8 2}Dagum, C. (1977). A new model of personal income distribution:
specification and estimation. {it:Economie Appliqu{c e'}e} 30: 413-437.
{p 4 8 2}Dagum, C. (1980). The generation and distribution of income, the
Lorenz curve and the Gini ratio. {it:Economie Appliqu{c e'}e} 33: 327-367.
{p 4 8 2}Fisk, P.R. (1961). The Graduation of Income Distributions.
{it:Econometrica} 29: 171-185.
{p 4 8 2}Jenkins, S.P. (2004). Fitting functional forms to distributions, using {cmd:ml}. Presentation
at Second German Stata Users Group Meeting, Berlin. {browse "http://www.stata.com/meeting/2german/Jenkins.pdf"}
{p 4 8 2}Kleiber, C. (1996). Dagum vs. Singh-Maddala income distributions.
{it: Economics Letters} 53: 265-268.
{p 4 8 2}Kleiber, C. and Kotz, S. (2003). {it:Statistical Size Distributions in Economics and Actuarial Sciences}.
Hoboken, NJ: John Wiley.
{p 4 8 2}McDonald, J.B. (1984). Some generalized functions for the size
distribution of income. {it:Econometrica} 52: 647-663.
{p 4 8 2}Singh, S.K. and G.S. Maddala (1976). A function for the size
distribution of income. {it:Econometrica} 44: 963-970.
{title:Also see}
{p 4 13 2}
Online: help for {help hangroot} {help dagumfit}, {help smfit},
{help gb2fit}, {help lognfit}, if installed.