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{smcl}
{* 23oct2020}{...}
{cmd:help mata mm_mloc()}
{hline}
{title:Title}
{p 4 17 2}
{bf:mm_mloc() -- Robust M estimation of location and scale}
{title:Syntax}
{dlgtab:M estimate of location}
{pstd}
Optimize
{p 8 24 2}
{it:S} =
{cmd:mm_mloc(}{it:X} [{cmd:,} {it:w}{cmd:,} {it:eff}{cmd:,} {it:obj}{cmd:,}
{it:b0}{cmd:,} {it:s}{cmd:,} {it:log}{cmd:,}
{it:tol}{cmd:,} {it:maxiter}]{cmd:)}
{p 8 12 2}where{p_end}
{p 7 20 2}{bind: }{it:X}: {it:real colvector} containing data{p_end}
{p 7 20 2}{bind: }{it:w}: {it:real colvector} containing weights; specify {cmd:1} for unweighted results{p_end}
{p 7 20 2}{bind: }{it:eff}: {it:real scalar} setting gaussian efficiency; default is {cmd:95}; {it:eff} must be in [63.7,99.9] for {cmd:"huber"} and in [0.1,99.9] for {cmd:"biweight"}{p_end}
{p 7 20 2}{bind: }{it:obj}: {it:string scalar} specifying the objective function, either {cmd:"huber"} (default) or {cmd:"biweight"}{p_end}
{p 7 20 2}{bind: }{it:b0}: {it:real scalar} specifying the staring value; default is the median of {it:X}{p_end}
{p 7 20 2}{bind: }{it:s}: {it:real scalar} specifying the scale; default is the normalized median absolute deviation from {it:b0}{p_end}
{p 7 20 2}{bind: }{it:log}: {it:real scalar} requesting an iteration log; default is {cmd:0} (no log); {it:log}!=0 displays the log{p_end}
{p 7 20 2}{bind: }{it:tol}: {it:real scalar} specifying the convergence tolerance; default is {cmd:1e-10}{p_end}
{p 7 20 2}{bind: }{it:maxiter}: {it:real scalar} specifying the maximum number of iterations; default is as set by {helpb maxiter:set maxiter}{p_end}
{pstd}
Retrieve results
{p2colset 9 34 36 2}{...}
{p2col:{bind: }{it:b} = {cmd:mm_mloc_b(}{it:S}{cmd:)}}location estimate{p_end}
{p2col:{bind: }{it:conv} = {cmd:mm_mloc_conv(}{it:S}{cmd:)}}1 if converged, 0 else{p_end}
{p2col:{bind: }{it:d} = {cmd:mm_mloc_d(}{it:S}{cmd:)}}value of convergence criterion{p_end}
{p2col:{bind: }{it:iter} = {cmd:mm_mloc_iter(}{it:S}{cmd:)}}number of iterations{p_end}
{p2col:{bind: }{it:k} = {cmd:mm_mloc_k(}{it:S}{cmd:)}}tuning constant{p_end}
{p2col:{bind: }{it:eff} = {cmd:mm_mloc_eff(}{it:S}{cmd:)}}gaussian efficiency (in percent){p_end}
{p2col:{bind: }{it:b0} = {cmd:mm_mloc_b0(}{it:S}{cmd:)}}starting value{p_end}
{p2col:{bind: }{it:s} = {cmd:mm_mloc_s(}{it:S}{cmd:)}}scale{p_end}
{pstd}
Specifying an optional argument as missing (either {cmd:.} or {cmd:""} depending on type
of arguments) selects the default. An exception is argument {it:w} for which missing is
not allowed. {it:S} is a structure holding results and settings; declare {it:S} as {it:transmorphic}.
{dlgtab:M estimate of scale}
{pstd}
Optimize
{p 8 24 2}
{it:S} =
{cmd:mm_mscale(}{it:X} [{cmd:,} {it:w}{cmd:,} {it:bp}{cmd:,}
{it:b0}{cmd:,} {it:loc}{cmd:,} {it:log}{cmd:,}
{it:tol}{cmd:,} {it:maxiter}]{cmd:)}
{p 8 12 2}where{p_end}
{p 7 20 2}{bind: }{it:X}: {it:real colvector} containing data{p_end}
{p 7 20 2}{bind: }{it:w}: {it:real colvector} containing weights; specify {cmd:1} for unweighted results{p_end}
{p 7 20 2}{bind: }{it:bp}: {it:real scalar} setting the breakdown point in [1,50]; default is {cmd:50}{p_end}
{p 7 20 2}{bind: }{it:b0}: {it:real scalar} specifying the starting value; default is the normalized median absolute deviation from {it:l}{p_end}
{p 7 20 2}{bind: }{it:l}: {it:real scalar} specifying the location; default is the median of {it:X}{p_end}
{p 7 20 2}{bind: }{it:log}: {it:real scalar} requesting an iteration log; default is {cmd:0} (no log); {it:log}!=0 displays the log{p_end}
{p 7 20 2}{bind: }{it:tol}: {it:real scalar} specifying the convergence tolerance; default is {cmd:1e-10}{p_end}
{p 7 20 2}{bind: }{it:maxiter}: {it:real scalar} specifying the maximum number of iterations; default is as set by {helpb maxiter:set maxiter}{p_end}
{pstd}
Retrieve results
{p2colset 9 34 36 2}{...}
{p2col:{bind: }{it:b} = {cmd:mm_mloc_b(}{it:S}{cmd:)}}scale estimate{p_end}
{p2col:{bind: }{it:conv} = {cmd:mm_mloc_conv(}{it:S}{cmd:)}}1 if converged, 0 else{p_end}
{p2col:{bind: }{it:d} = {cmd:mm_mloc_d(}{it:S}{cmd:)}}value of convergence criterion{p_end}
{p2col:{bind: }{it:iter} = {cmd:mm_mloc_iter(}{it:S}{cmd:)}}number of iterations{p_end}
{p2col:{bind: }{it:k} = {cmd:mm_mloc_k(}{it:S}{cmd:)}}tuning constant{p_end}
{p2col:{bind: }{it:bp} = {cmd:mm_mloc_eff(}{it:S}{cmd:)}}breakdown point (in percent){p_end}
{p2col:{bind: }{it:b0} = {cmd:mm_mloc_b0(}{it:S}{cmd:)}}starting value{p_end}
{p2col:{bind: }{it:l} = {cmd:mm_mloc_s(}{it:S}{cmd:)}}location{p_end}
{pstd}
Specifying an optional argument as missing (either {cmd:.} or {cmd:""} depending on type
of arguments) selects the default. An exception is argument {it:w} for which missing is not
allowed. {it:S} is a structure holding results and settings; declare {it:S} as {it:transmorphic}.
{dlgtab:Objective functions}
{pstd}
Obtain tuning constant from efficiency, efficiency from tuning constant, tuning constant from breakdown point, or breakdown point
from tuning constant. {it:eff} must be in
[63.7,99.9] for the Huber function and in [0.1,99.9] for the biweight function. {it:bp} must be in
[1,50].
{it:real scalar} {cmd:mm_huber_k(}{it:real scalar eff}{cmd:)}
{it:real scalar} {cmd:mm_huber_eff(}{it:real scalar k}{cmd:)}
{it:real scalar} {cmd:mm_biweight_k(}{it:real scalar eff}{cmd:)}
{it:real scalar} {cmd:mm_biweight_eff(}{it:real scalar k}{cmd:)}
{it:real scalar} {cmd:mm_biweight_k_bp(}{it:real scalar bp}{cmd:)}
{it:real scalar} {cmd:mm_biweight_bp(}{it:real scalar k}{cmd:)}
{pstd}
Apply objective functions to (typically standardized) data given tuning constant {it:k}.
{it:real colvector} {cmd:mm_huber_psi(}{it:real colvector X}{cmd:,} {it:real scalar k}{cmd:)}
{it:real colvector} {cmd:mm_huber_phi(}{it:real colvector X}{cmd:,} {it:real scalar k}{cmd:)}
{it:real colvector} {cmd:mm_huber_w(}{it:real colvector X}{cmd:,} {it:real scalar k}{cmd:)}
{it:real colvector} {cmd:mm_biweight_rho(}{it:real colvector X}{cmd:,} {it:real scalar k}{cmd:)}
{it:real colvector} {cmd:mm_biweight_psi(}{it:real colvector X}{cmd:,} {it:real scalar k}{cmd:)}
{it:real colvector} {cmd:mm_biweight_phi(}{it:real colvector X}{cmd:,} {it:real scalar k}{cmd:)}
{it:real colvector} {cmd:mm_biweight_w(}{it:real colvector X}{cmd:,} {it:real scalar k}{cmd:)}
{title:Description}
{pstd}
{cmd:mm_mloc()} computes a fixed-scale M estimate of location using iterative
reweighting, either based on the Huber objective function or the biweight
objective function.
{pstd}
{cmd:mm_mscale()} computes a fixed-location M estimate of scale using iterative
reweighting, based on the biweight objective function.
{pstd}
For methodological background and details on algorithms see Maronna et al. (2006),
in particular chapter 2.
{pstd}
Stata 11 or newer is required.
{title:Examples}
{pstd}
Generate standard normally distributed data:
{com}: x = rnormal(10000, 1, 0, 1){txt}
{pstd}
Classical mean and standard deviation:
{com}: mean(x), sqrt(variance(x))
{res} {txt} 1 2
{c TLC}{hline 31}{c TRC}
1 {c |} {res}-.0091553032 .9988726582{txt} {c |}
{c BLC}{hline 31}{c BRC}{txt}
{pstd}
Robust M estimates of location and scale:
{com}: mm_mloc_b(mm_mloc(x)), mm_mscale_b(mm_mscale(x))
{res} {txt} 1 2
{c TLC}{hline 31}{c TRC}
1 {c |} {res}-.0076506477 .9844972461{txt} {c |}
{c BLC}{hline 31}{c BRC}{txt}
{pstd}
Add 5% contamination at x = 10:
{com}: x[|1\500|] = J(500, 1, 10){txt}
{pstd}
Classical mean and standard deviation:
{com}: mean(x), sqrt(variance(x))
{res} {txt} 1 2
{c TLC}{hline 29}{c TRC}
1 {c |} {res}.4884230517 2.389203199{txt} {c |}
{c BLC}{hline 29}{c BRC}{txt}
{pstd}
Robust M estimates of location and scale:
{com}: mm_mloc_b(mm_mloc(x)), mm_mscale_b(mm_mscale(x))
{res} {txt} 1 2
{c TLC}{hline 29}{c TRC}
1 {c |} {res}.0781973876 1.055197844{txt} {c |}
{c BLC}{hline 29}{c BRC}{txt}
{title:Diagnostics}
{pstd}
The functions return {cmd:.} (missing) if {it:X} is void.
{pstd}
The functions return error if {it:X} or {it:w} contain missing.
{title:Source code}
{pstd}
{help moremata11_source##mm_mloc:mm_mloc.mata}
{title:References}
{phang}
Maronna, R. A., D. R. Martin, V. J. Yohai (2006). Robust Statistics. Theory
and Methods. Chichester: John Wiley & Sons.
{p_end}
{title:Author}
{pstd}
Ben Jann, University of Bern, ben.jann@soz.unibe.ch
{title:Also see}
{p 4 13 2}
Online: help for
{helpb moremata}
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