−67, −48, 6, 8, 14, 16, 23, 24, 28, 29, 41, 49, 56, 60, 75

A parametric approach to analyze the data as a random sample from a normal distribution with unknown mean and standard deviation was developed by Basu et al. [24]. Here, there is not any huge outlying observation, but the first two observations seem to be distant from the rest of the sample, influencing the model parameter estimates and test decisions. Indeed, the MLE, computing with original data, is (*μ*4, <sup>4</sup>*σ*)=(20.93, 37.74), while the MLE, when removing the two first observations, switches to (*μ*4, <sup>4</sup>*σ*)=(33, 21.54). Therefore, removing influential observations may alter the decision of a test. According to these results, we consider the testing problem

$$\text{H}\_0: \sigma = 23 \text{ vs. } \text{H}\_1: \sigma \neq 23. \tag{35}$$

Figure 5 shows the test statistics (left) and corresponding *p*-values (right) for the two families of statistics considered, the RPTS (top) and Rao-type test statistics (bottom) against the tuning parameter value *τ*. Again, test statistics based on RMRPE with large enough tuning parameters do not reject the null hypothesis, unlike tests based on low values of *τ* = 0, including the RMLE. The disagreement departs when using the clean data, as all tests agree on not rejecting the null hypothesis.

**Figure 5.** *Cont*.

**Figure 5.** RPTS (**top**) and Rao-type test statistics (**bottom**), jointly with their associated *p*-values (right), for testing (35) with original and cleaned (after outliers removal) Darwing data.
