The HodgesLehmann function computes a robust measure of the location of the specified data set or random variable, as introduced by Hodges and Lehmann in [2] and independently by Sen in [3]. This statistic is variously called the Hodges-Lehmann-Sen estimator, the Hodges-Lehmann estimator, the Hodges-Lehmann-Sen statistic, or the Hodges-Lehmann statistic.

The Hodges-Lehmann statistic, referred to as $\mathrm{HLE}$ in the remainder of this help page, is defined for a data set $A_1,A_2,\mathrm{...},A_n$ as:

The Hodges-Lehmann statistic is a reasonably robust statistic: it has a fairly high breakdown point (the proportion of arbitrarily large observations it can handle before giving an arbitrarily large result). The breakdown point of $\mathrm{HLE}$ is $1-\frac{\sqrt{2}}{2}$ or about $0.29$.

The Hodges-Lehmann statistic is a measure of location: if $\mathrm{HodgesLehmann}\left(X\right)=a$, then for all real constants $\mathrm{\alpha }$ and $\mathrm{\beta }$, we have $\mathrm{HodgesLehmann}\left(\mathrm{\alpha }X+\mathrm{\beta }\right)=\mathrm{\alpha }a+\mathrm{\beta }$.

The first parameter can be a data set, a distribution (see Statistics[Distribution]), a random variable, or an algebraic expression involving random variables (see Statistics[RandomVariable]). For a data set $A$, HodgesLehmann computes the Hodges-Lehmann statistic as defined above. For a distribution or random variable $X$, HodgesLehmann computes the asymptotic equivalent - the value that the Hodges-Lehmann statistic converges to for ever larger samples of $X$.

By default, all computations involving random variables are performed symbolically (see option numeric below).

All computations involving data are performed in floating-point; therefore, all data provided must have type/realcons and all returned solutions are floating-point, even if the problem is specified with exact values.

For more information about computation in the Statistics package, see the Statistics[Computation] help page.

The ds_options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[DescriptiveStatistics] help page.

ignore=truefalse -- This option controls how missing data is handled by the HodgesLehmann command. Missing items are represented by undefined or Float(undefined). So, if ignore=false and A contains missing data, the HodgesLehmann command may return undefined. If ignore=true all missing items in A will be ignored. The default value is false.

weights=Vector -- Data weights. The number of elements in the weights array must be equal to the number of elements in the original data sample. By default all elements in A are assigned weight $1$.

The rv_options argument can contain one or more of the options shown below. More information for some options is available in the Statistics[RandomVariables] help page.

numeric=truefalse -- By default, the Hodges-Lehmann statistic is computed using exact arithmetic. To compute the Hodges-Lehmann statistic numerically, specify the numeric or numeric = true option.

$\mathrm{with}\left(\mathrm{Statistics}\right)\:$

$s≔⟨1\,5\,2\,2\,7\,4\,1\,6⟩$

$s≔\left[\begin{array}{c}1\\ 5\\ 2\\ 2\\ 7\\ 4\\ 1\\ 6\end{array}\right]$

$\mathrm{HodgesLehmann}\left(s\right)$

$3.50000000000000$

$t≔\mathrm{copy}\left(s\right)\:$

$t\left[1..2\right]≔{10}^{100}\:$

$t$

$\left[\begin{array}{c}10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000\\ 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000\\ 2\\ 2\\ 7\\ 4\\ 1\\ 6\end{array}\right]$

$\mathrm{HodgesLehmann}\left(t\right)$

$5.75000000000000$

$\mathrm{HodgesLehmann}\left('\mathrm{Exponential}'\left(1\right)\,'\mathrm{numeric}'\right)$

$0.839173495008330$

$\mathrm{HodgesLehmann}\left('\mathrm{Exponential}'\left(b\right)\right)$

$-\frac{\left(\mathrm{LambertW}\left(\mathrm{-1}\,-\frac{{\ⅇ}^{\mathrm{-1}}}{2}\right)+1\right)b}{2}$

$\mathrm{evalf}\left(\mathrm{eval}\left(\,b=1\right)\right)$

$0.8391734950$

$A≔\mathrm{Sample}\left('\mathrm{Exponential}'\left(1\right)\,1000000\right)\:$

$\mathrm{HodgesLehmann}\left(A\right)$

$0.840272671959114$

$M≔\mathrm{Matrix}\left(\left[\left[3\,1130\,114694\right]\,\left[4\,1527\,127368\right]\,\left[3\,907\,88464\right]\,\left[2\,878\,96484\right]\,\left[4\,995\,128007\right]\right]\right)$

$M≔\left[\begin{array}{ccc}3& 1130& 114694\\ 4& 1527& 127368\\ 3& 907& 88464\\ 2& 878& 96484\\ 4& 995& 128007\end{array}\right]$

$\mathrm{HodgesLehmann}\left(M\right)$

$\left[\begin{array}{ccc}3.& 1018.50000000000& 111926.\end{array}\right]$

[1] Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.

[2] Hodges, Joseph L.; Lehmann, Erich L. Estimation of location based on ranks. Annals of Mathematical Statistics 34 (2), 1963, pp.598-611.

[3] Sen, Pranab K. On the estimation of relative potency in dilution(-direct) assays by distribution-free methods. Biometrics 19(4), 1963, pp.532-552.

The Statistics[HodgesLehmann] command was introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.