*6.1. Telephone-Fault Data*

We consider the data on telephone line faults presented and analyzed by Welch [27] and Simpson [4]. The dataset consist of *n* = 14 ordered differences between the inverse test rates and the inverse control rates in matched pairs of areas,

−988, −135, −78, 3, 59, 83, 93, 110, 189, 197, 204, 229, 289, 310.

Basu et al. ([24,28]) modeled these differences as a normal random variable and pointed out that the first observation is a clear outlier, as its value is distant from the rest of the data. They tested simple and composite null hypotheses for the mean under the normal model, as well as a simple null hypothesis assuming a known mean. Here, we propose to test for the standard deviation of the normal distribution. Note that, computing the MLE of the sample with full and clean data (after removing the outlying observation), we obtain (*μ*4, <sup>4</sup>*σ*)=(40.36, 323.08), and (*μ*4, <sup>4</sup>*σ*)=(119.46, 134.82), respectively. Accordingly, the outlier clearly influences the model parameter estimates, playing a crucial role on the rejection of any null hypothesis. We consider the composite null hypothesis

$$\text{H}\_0: \sigma = 135 \text{ vs. H}\_1: \sigma \neq 135,\tag{34}$$

where the value *σ* = 135 has been chosen according to the estimation with clean data.

Figure 4 presents the RPTS (top) and Rao (bottom) test statistics (left) and *p*-values (right) for the telephone data against increasing tuning parameters. While it is clearly seen that both classical tests fail to not reject the null hypothesis when fitting the model with the original data, the decision turns around sharply as the tuning parameter *τ* crosses and goes beyond 0.2 for the RPTS and 0.15 for Rao-type test statistics based on MRPEs. On the other hand, the decision of not rejecting is agreed by all statistics when fitting the model with clean data. This example illustrates the great applicability of the robust methods, which are not too affected by a such outlying observation, and the good performance of the proposed statistics under contaminated observations, which stay stable.

**Figure 4.** *Cont*.

**Figure 4.** RPTS (**top**) and Rao-type test statistics (**bottom**), jointly with their associated *p*-valuess (right), for testing (34) with original and cleaned (after outliers removal) telephone-fault data.
