... code

A full-code demo program is available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/477/967

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...layer

In applications the coarsened version of this rule is of particular interest. The difference is only in the first stage of this procedure: instead of removing a point with $d_i=d_{\rm max}$, we remove all points such that their distances $d_i\ge r d_{\rm max}(s)$, where $0<r\le 1$ is a removal parameter. For r=1 we are back at the original rule, while for r< 1 we remove more points in a single step.

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... holds

More precisely, for subsets s such that $d_{\rm max}(s)=d$, the probability that the opposite inequality $n_0\bar{D}_k(d)<D_k(s)$ holds tends to zero when n₀ approaches infinity.

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... DLS maximum

Let $s=\{(x_{i_1},y_{i_1}),\ldots,(x_{i_n},y_{i_n})\}$ is a subset of s₀ having $d_{\rm max}(s)=d$. Its best horizontal line y(x)=a is obtained for $a=(1/n)\sum_{k=1}^ny_{i_k}$. If s is not a subset of s_d, we translate it by amount a and obtain a set $s'\equiv \{(x_{i_1},y_{i_1}-a),\ldots,(x_{i_n},y_{i_n}-a)\}$ contained in s_d. Its best horizontal line is y(x)=0, $d_{\rm max}(s')=d_{\rm max}(s)$, so that D_k(s)=D_k(s'), because the least-square sums for subsets s and s' are equal as well. As subset s_d contains all points from s₀ inside boundaries $y=\pm d$, and $d_{\rm max}(s_d)=d$, it follows that $D_k(s_d)\ge D_k(s')=D_k(s)$.

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... sub-distribution

For simplicity, here we take $d_{\rm max}=d_n$ and $d_i<d_{\rm max}$ for all i<n.

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... technique

The Boltzmann-plot technique is a common method for estimating electron temperatures in plasmas at local thermodynamic equilibrium; for some typical applications in astrophysics see (Popovic 2003).

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... estimators

In this subsection we closely follow the terminology and notations used in Press et al. (2001)

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...)

The corresponding distribution is $P'(z)=K{\rm e}^{-z^2/2}$ for |z|<c, and $P'(z)=K{\rm e}^{-c^2/2}={\rm const.}$ for $\vert z\vert\ge c$, where K is the normalization constant. Though the particular value of K has no consequence for the analysis that follows note, however, that K depends on c and on the available interval of z, which is finite due to a finite experimental range in which the data were collected.

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... points

Note that the form of $\rho_c$ is unique, if we want to relate it with the DLS method. Indeed, in the DLS method we treat close points like in the OLS method, whereas distant points are discarded. Therefore, in a related local-M approach, one has $\rho(z)=z^2/2$ for close points and $\psi(z)=0$for distant points, implying that $\rho(z)={\rm const.}$ for distant points. The only continuous $\rho(z)$ satisfying both conditions is (16).

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... vanishes

For a given c, let a(c)>0 be the abscissa of the right-hand side maximum of $\mu '_c(a)$ and let a=a(c) is the line of right-hand side maxima. Along this line $\partial \mu'_c/\partial a=0$, hence the first derivative of (22) satisfies $0=c^2{\rm e}^{-c^2/2}[a\cosh(ac)-x\sinh(ac)]+\frac{{\rm d}a}{{\rm d}c}\int_0^cz^2 {\rm e}^{-z^2/2}[(1-x^2)\cosh(az)+a\sinh(az)]{\rm d}z$. As $a(c)\to 0$ when $c\to c_c$, the previous equation simplifies to $0=\int_0^cz^2{\rm e}^{-z^2/2}[1-z^2]{\rm d}z$. Finally, taking into account $\int_0^cz^4{\rm e}^{-z^2/2}{\rm d}z=3\int_0^zz^2{\rm e}^{-z^2/2}{\rm d}z-c^3{\rm e}^{-c^2/2}$, we retrieve (6) for k=2.

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... points

As in the DLS method we adopt $2\le k<3$, and $c=d_b/\sigma$ is obtained from (6), it follows that $c\le c_c$.

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... margins

Note that the bigger the value of the removal parameter, the bigger the number of subsets found by DLSFIT, and the longer the execution time, which is of particular importance in the case of large data sets.

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