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\title{Robust test statistics}
%-------------------------------------------------------------------
\author{Chanseok Park\footnote{Applied Statistics Laboratory,
Department of Industrial Engineering, Pusan National University, Busan 46241, Korea.
His work was partially supported by the National Research Foundation of Korea (NRF)
grant funded by the Korea government (No. 2022R1A2C1091319).}
}
\date{September 2026}
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\begin{document}
\maketitle
\begin{abstract}
In this note, we provide three robust test statistics which are
based on median and median absolute deviation (MAD) estimators,
Hodges-Lehmann and Shamos estimators, and Huber location and RC estimators.
\end{abstract}
%%

%====================================================================
\section{Introduction}
One of the widely-used conventional methods to test $H_0: \mu = \mu_0$
is to use the $z$-test statistic (with $n \ge 30$),
\[
T= \frac{\bar{X}-\mu}{S/\sqrt{n}},
\]
where $\bar{X}$ is the sample mean and $S$ is the sample standard deviation.
However, this method is very sensitive to data contamination.
In this note, we provide three robustified $z$-test statistics that can be used when
the sample size is large. However, when the sample size is small, it may not be appropriate to use
the standard normal distribution. In this note, we provide a method for obtaining the empirical distributions
of these test statistics. These results are implemented in the \texttt{rt.test} package.

%====================================================================
\section{Robustified test statistics}
For the sake of completeness, we briefly review the test statistics
proposed by \cite{Park:2018a}, \cite{Jeong/Son/Lee/Kim:2018}, 
and \cite{Park/Wang/Hwang:2022}.
By replacing the mean and the standard deviation
with robust location and scale estimators,
they proposed the following robustified $z$-test statistics.
These are given by
%-------------------------------
\begin{align} 
T_A &= \sqrt{\frac{2n}{\pi}} {\Phi^{-1}\Big(\frac{3}{4}\Big)} \cdot
\frac{\displaystyle\mathop{\mathrm{median}}_{1\le i\le n}X_i-\mu}%
{\displaystyle\mathop{\mathrm{median}}_{1\le i\le n}
  \big| X_i - \mathop{\mathrm{median}}_{1\le i\le n} X_i  \big|}
\stackrel{d}{\longrightarrow} N(0,1),   \label{EQ:TA} \\
%-------------------------------
T_{B} &= \sqrt{\frac{6n}{\pi}} {\Phi^{-1}\Big( \frac{3}{4} \Big)}
  \frac{\displaystyle\mathop{\mathrm{median}}_{i \le j} \Big(\frac{X_i+X_j}{2}\Big) - \mu }%
{\displaystyle\mathop{\mathrm{median}}_{i \le j}\big(|X_i-X_j|\big)}
\stackrel{d}{\longrightarrow} N(0,1), \label{EQ:TB}
\end{align} 
{where $\Phi^{-1}(\cdot)$ is the inverse of the standard normal cumulative distribution function and $\stackrel{d}{\longrightarrow}$ denotes convergence in distribution.}
%----------------------------------------------------------------

%% Recently, En Xie et al.\ obtained the following statistic based 
%% on the Huber and RC estimators.
%% It should be noted that this test statistic has a higher breakdown point of 50\%, while
%% $T_B$ has a breakdown point of only 29\%. In Table~\ref{TABLE}, we provide a summary of the
%% asymptotic relative efficiencies (AREs) and breakdown points of various estimators.
%% For more details, see \cite{Rousseeuw/Croux:1993}, \cite{Serfling:2011}, and \cite{Staudte/Sheather:1990}.
%% \begin{equation} \label{EQ:TC}
%% T_C = \frac{\sqrt{2n}\;\Phi^{-1}(5/8)\, \big\{2\Phi(\kappa)-1\big\}\cdot(\hat{\mu}_M-\mu)}%
%% {\sqrt{2(\kappa^2-1)\big\{1-\Phi(\kappa)\big\} - 2\kappa \phi(\kappa)+1}\cdot\big\{ |X_i-X_j| : i<j \big\}_{(s)}}
%% \stackrel{d}{\longrightarrow} N(0,1).
%% \end{equation}


%% %--------------------------------------------------
%% \begin{table}[t!]
%% \begin{center}
%% %% \begin{color}{red}
%% \caption{Breakdown Points and AREs of the estimators under consideration.}
%% \label{TABLE:ARL}
%% \begin{tabular}{clccccl@{\quad}ccccc} %% \\[-0.5ex]
%% \hline
%%         && \multicolumn{4}{c}{Location} && \multicolumn{5}{c}{Scale}  \\
%%        \cline{3-6} \cline{8-12}   \\[-1.5ex]
%%         && Mean & Median & HL & Huber && SD & IQR & MAD & Shamos & RC \\
%%        \cline{3-6} \cline{8-12}   \\[-1.5ex]
%% Breakdown&&  0\% & 50\% & 29\% & 50\%  &&   0\% & 25\% & 50\% & 29\% & 50\%  \\
%% ARE      &&100\% & 64\% & 96\% & 95\%  && 100\% & 38\% & 37\% & 86\% & 82\%  \\
%% \hline
%% \end{tabular}
%% %% \end{color}
%% \end{center}
%% \end{table}
%% %--------------------------------------------------


%--------------------------------------------------
\begin{table}[t!]
\begin{center}
\caption{Breakdown Points and AREs of the estimators under consideration.}
\label{TABLE:ARL}
\medskip
\begin{tabular}{clcccl@{\quad}cccc} %% \\[-0.5ex]
\hline
         && \multicolumn{3}{c}{Location} && \multicolumn{4}{c}{Scale}  \\
        \cline{3-5} \cline{8-10}   \\[-1.5ex]
         && Mean & Median & HL   && SD & IQR & MAD & Shamos  \\
        \cline{3-5} \cline{8-10}      \\[-1.5ex]
Breakdown&&  0\% & 50\% & 29\%   &&   0\% & 25\% & 50\% & 29\%   \\
ARE      &&100\% & 64\% & 96\%   && 100\% & 38\% & 37\% & 86\%   \\
\hline
\end{tabular}
\end{center}
\end{table}
%--------------------------------------------------









%===================================================================
\section{Empirical distributions}
%-------------------------------------------------------------------
%% \begin{color}{red}
As mentioned above, the robustified statistics, $T_A$ in (\ref{EQ:TA})
and $T_B$ in (\ref{EQ:TB}), converge to the standard normal distribution.
However, when the sample size is small,
it is not appropriate to use the standard normal distribution.

It may be impossible to obtain the theoretical distributions of $T_A$
and $T_B$. 
Thus, we obtain the empirical distributions of $T_A$ and $T_B$ 
using extensive Monte Carlo simulations
and calculate their empirical quantiles, which are useful for estimating critical values, 
confidence intervals, $p$-values,
etc. We used the R language to conduct the simulations, which are summarized as follows.
We generated one hundred million ($N=10^8$) samples of size $n$
from the standard normal distribution to obtain the empirical distributions of
$T_A$ and $T_B$, where $n=4,5,\ldots,50$.
Using these samples, we can obtain the empirical distribution of $T_A$ or $T_B$
for each sample size $n$.
Then, by inverting the empirical distribution, we obtained the empirical quantiles
of $p$.

It is worthwhile to discuss the accuracy of the empirical quantiles obtained above.
Let $F^{-1}_N(p)$ be the
empirical quantile of $p$ obtained from the $N$ replications and
$F^{-1}(p)$ be the true quantile of $p$. Then it is easily seen from
Corollary~21.5 of \cite{Vaart:1998} that the sequence $\sqrt{N}\big(
F_N^{-1} - F^{-1}(p) \big)$ is asymptotically normal with mean zero and
variance $p/(1-p)/f^2\big(F^{-1}(p) \big)$. Thus, the standard deviation
of the empirical quantile of $p$ is approximately proportional to
$\sqrt{p(1-p)/N}$, which has its maximum value at $p=0.5$. Consequently,
the empirical quantile values are computed with an approximate accuracy of
$0.5/\sqrt{N}$. With $N=10^8$, we have $0.5/\sqrt{N}=0.00005$, which
roughly indicates that the empirical quantile values are accurate
up to the fourth decimal point.

Given that the probability density functions of $T_A$ and $T_B$ are symmetric about zero,
we have $F(-x) = 1 - F(x)$ and $F^{-1}(1/2)=0$.
Letting $q_p$ be the $p$th lower quantile so that $F(q_p)=p$,
we have $q_{1-p} = -q_{p}$.
Thus, it is enough to find the $p$th quantile only when $p>1/2$.
Let $G(\cdot)$ be the cumulative distribution function of $|X|$. Then we have
\[
G(x) = P[ |X| \le x ] = P[ -x \le X \le x ] = F(x) - F(-x)
= 2F(x) -1.
\]
Substituting $x=q_p$ into the above, we have $G(q_p) = 2p-1$.
Thus, we have
\[
q_p = G^{-1}(2p-1),
\]
which is more efficient than using $q_p = F^{-1}(p)$ when obtaining empirical quantile values.
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