*4.2. Rao's-Type Tests Based on RMRPE*

Rao test statistics are one of the most popular score test statistics for testing a simple and composite null hypothesis in general statistical models. For the simple null hypothesis testing, it requires no parameter estimation, but for composite ones, the classical Rao test is based on the likelihood score function associated with the restricted MLE (see Rao [23]). Basu et al. [24] generalized Rao's procedure by using score functions associated with RMDPDEs, bringing in a considerable gain of robustness of the Rao-type test obtained. In this section, we develop Rao-type test statistics based on the score function associated to RMRPEs.

Let us consider the *τ*−score function associated to the RMRPE,

$$
\psi\_{\mathsf{T}}(\mathsf{x};\mathsf{\theta}) = f\_{\mathsf{\theta}}(\mathsf{x})^{\mathsf{\mathsf{T}}} (\mathsf{u}\_{\mathsf{\theta}}(\mathsf{x}) - \mathsf{c}\_{\mathsf{T}}(\mathsf{\theta})),
$$

so the estimating equations for the MRPE are given by

$$\sum\_{i=1}^{n} \boldsymbol{\upmu}\_{\tau} (\boldsymbol{\uppi}\_{i}; \boldsymbol{\uptheta}) = \mathbf{0}\_{\mathcal{P}}.$$

Then, the *τ*-score statistic can be defined as

$$\Psi\_{\tau}(\boldsymbol{\theta}) = \sum\_{i=1}^{n} \psi\_{\tau}(\boldsymbol{x}\_{i}; \boldsymbol{\theta}) = \left(\sum\_{i=1}^{n} \psi\_{\tau}^{1}(\boldsymbol{x}\_{i}; \boldsymbol{\theta}), \dots, \sum\_{i=1}^{n} \psi\_{\tau}^{k}(\boldsymbol{x}\_{i}; \boldsymbol{\theta})\right)^{T}.$$

However, taking expectations in the corresponding quantities, it is not difficult to show that

$$E\left[\left(\frac{\tau}{\mathbb{C}\_{\tau}(\boldsymbol{\theta})} f\_{\boldsymbol{\theta}}(\boldsymbol{X})^{\tau} (\boldsymbol{u}\_{\boldsymbol{\theta}}(\boldsymbol{X}) - \mathbf{c}\_{\tau}(\boldsymbol{\theta}))\right)\_{\boldsymbol{\theta} = \boldsymbol{\theta}\_{0}}\right] = \mathbf{0}\_{p}$$

$$E\left[\left(f\_{\boldsymbol{\theta}}(\boldsymbol{X})^{2\tau} (\boldsymbol{u}\_{\boldsymbol{\theta}}(\boldsymbol{X}) - \mathbf{c}\_{\tau}(\boldsymbol{\theta})) (\boldsymbol{u}\_{\boldsymbol{\theta}}(\boldsymbol{X}) - \mathbf{c}\_{\tau}(\boldsymbol{\theta}))^{T}\right)\_{\boldsymbol{\theta} = \boldsymbol{\theta}\_{0}}\right] = \mathbf{K}\_{\tau}(\boldsymbol{\theta}\_{0})\_{\boldsymbol{\theta}}$$

where *Kτ*(*θ*) is defined in (16), and so, by the central limit theorem, the *τ*-score statistic is asymptotically normal,

$$\frac{1}{\sqrt{n}}\Psi\_{\tau}(\theta) \underset{n\to\infty}{\underset{n\to\infty}{\to}} \mathcal{N}\left(\mathfrak{d}\_{\mathcal{V}^{\prime}}\mathcal{K}\_{\tau}(\theta)\right). \tag{29}$$

The previous convergence motivates the definition of the Rao-type test statistics.

4.2.1. Rao-Type Test Statistics for Testing Simple Null Hypothesis

We first consider the simple null hypothesis test

$$H\_0: \theta = \theta\_0 \text{ vs. } H\_1: \theta \neq \theta\_0. \tag{30}$$

Then, the Rao-type test statistics *Rτ*(*θ*0) for testing (30) is defined as

$$R\_{\tau}(\theta\_0) = \frac{1}{n} \mathbf{Y}\_{\tau}(\theta\_0)^T \mathbf{K}\_{\tau}(\theta\_0)^{-1} \mathbf{Y}\_{\tau}(\theta\_0).$$

Note that here the last test statistics depend on *τ* through the matrices **Ψ***τ*(*θ*0) and *Kτ*(*θ*0) involved in the definition, and again, the robustness of the statistics increases with *τ*. Moreover, the last matrix may have an explicit expression for certain statistical models, but otherwise it would have to be estimated from the sample.

Further, from (29), we have that, under the null hypothesis,

$$R\_{\tau}(\boldsymbol{\theta}\_{0}) \underset{\boldsymbol{\pi} \to \infty}{\overset{L}{\underset{\boldsymbol{\pi} \to \infty}{\rightleftharpoons}} \boldsymbol{\chi}\_{P}^{2}$$

with *p* being the dimension of the parameter space. Then, the null hypothesis is rejected if *Rτ*(*θ*0) > *χ*<sup>2</sup> *<sup>p</sup>*,*α*, where *χ*<sup>2</sup> *<sup>p</sup>*,*<sup>α</sup>* denotes the upper *α*-quantile of a chi-square distribution with *p* degrees of freedom.

4.2.2. Rao-Type Test Statistics for Testing Composite Null Hypothesis

Next, let us consider composite null hypothesis of the form

$$H\_0: \mathbf{g}(\theta) = \mathbf{0}\_{\mathsf{T}} \text{ vs. } H\_1: \mathbf{g}(\theta) \neq \mathbf{0}\_{\mathsf{T}} \tag{31}$$

where the function **<sup>g</sup>** : <sup>R</sup>*<sup>p</sup>* <sup>→</sup> <sup>R</sup>*<sup>r</sup>* is a differentiable vector-valued function. Then, any vector *θ* satisfying the null hypothesis belongs to a restricted parameter space given in (3). The generalized Rao-type test statistic associated to the RMRPE with tuning parameter *τ*, \* *θτ*, for testing (31) is given by

$$R\_{\tau}\left(\tilde{\boldsymbol{\theta}}\_{\tau}\right) = \frac{1}{n} \boldsymbol{\Psi}\_{\tau}(\tilde{\boldsymbol{\theta}}\_{\tau})^{T} \boldsymbol{Q}\_{\tau}(\tilde{\boldsymbol{\theta}}\_{\tau}) \left[\boldsymbol{Q}\_{\tau}(\tilde{\boldsymbol{\theta}}\_{\tau})^{T} \boldsymbol{K}\_{\tau}(\tilde{\boldsymbol{\theta}}\_{\tau}) \boldsymbol{Q}\_{\tau}(\tilde{\boldsymbol{\theta}}\_{\tau})\right]^{-1} \boldsymbol{Q}\_{\tau}(\tilde{\boldsymbol{\theta}}\_{\tau})^{T} \boldsymbol{\Psi}\_{\tau}(\tilde{\boldsymbol{\theta}}\_{\tau}).\tag{32}$$

Using similar arguments to Basu et al. [24], it is possible to show that, under general regularity conditions, the Rao-type test statistics *R<sup>τ</sup>* - \* *θτ* have an asymptotic chi-square distribution with *r* degrees of freedom under the null hypothesis given in (31). Therefore, the rejection region of the test is given by

$$\{X\_{1\prime}, \ldots, X\_{\hbar} : \mathcal{R}\_{\mathbf{r}}(\tilde{\theta}\_{\mathbf{r}}) > \chi^2\_{\mathbf{r}, \hbar}\}.$$

Again, the tuning parameter *τ* controls the trade-off between efficiency and robustness of the test. Indeed, for *τ* = 0, the generalized Rao type test statistic *Rτ*=<sup>0</sup> - \* *θ*0 coincides with the classical Rao test for composite null hypothesis.

4.2.3. Rao Test for Normal Populations

Consider the test defined in (26) for testing the standard deviation value of a normal population with unknown mean. The explicit expression of the main matrices involved in the definition (32) for such testing procedure and assumed parametric model is given by

$$\begin{split} \Psi\_{\mathsf{T}}(X; (\mu, \sigma)) &= \left( \frac{X - \mu}{\sigma^2} \frac{1}{\left( \sigma \sqrt{2\pi} \right)^{\mathsf{T}}} e^{-\frac{\mathsf{T}}{2} \left( \frac{X - \mu}{\sigma} \right)^2}, \left( \left( \frac{X - \mu}{\sigma} \right)^2 - \frac{1}{1 + \pi} \right) \frac{1}{\sigma} \frac{1}{\left( \sigma \sqrt{2\pi} \right)} e^{-\frac{\mathsf{T}}{2} \left( \frac{X - \mu}{\sigma} \right)^2} \right)^T, \\ \mathsf{K}\_{\mathsf{T}}((\mu, \sigma)) &= \frac{1}{\sigma^2} \frac{1}{\left( \sigma \sqrt{2\pi} \right)^{2\pi} (1 + 2\pi)^{3/2}} \begin{pmatrix} 1 & 0 \\ 0 & \frac{3\pi^2 + 2 + 4\pi}{(1 + \pi)^2 (1 + 2\pi)} \end{pmatrix}, \\ \mathsf{Q}\_{\mathsf{T}}((\mu, \sigma)) &= \begin{pmatrix} 0 \\ 1 \end{pmatrix}. \end{split}$$

The step-by-step calculation of such values are detailed in the Appendix A. Then, the Rao-type test for composite null hypothesis of the form (31) is given by

$$R\_{\tau}(\tilde{\mu}) = \frac{1}{n} \frac{(1+2\tau)^{3/2} (1+\tau)^2 (1+2\tau)}{3\tau^2 + 4\tau + 2} \left[ \sum\_{i=1}^{n} \left( \left( \frac{x\_i - \tilde{\mu}}{\sigma\_0} \right)^2 - \frac{1}{\tau+1} \right) e^{-\frac{\tau}{2} \left( \frac{x\_i - \tilde{\mu}}{\sigma\_0} \right)^2} \right]^2$$

where (*μ*\**τ*, *<sup>σ</sup>*0) denotes the RMRPE with tuning parameter *<sup>τ</sup>*. Note that, for *<sup>τ</sup>* <sup>=</sup> 0, *<sup>μ</sup>*\**τ*=<sup>0</sup> <sup>=</sup> *<sup>X</sup>*. Then, the Rao-type test statistic based on RMRPE with *τ* = 0 (the restricted MLE) coincides with the classical Rao test.

## **5. Simulation Study: Application to Normal Populations**

In this section, we empirically analyze the performance of the proposed estimators under the normal parametric model and RPTS and Rao-type test statistics for the problem of testing (26) in terms of efficiency and robustness. We examine the accuracy of the RMRPEs, and we further examine the robustness properties of both families of estimators under different contamination scenarios. Further, we investigate the empirical level and power of the proposed test statistics under different sample sizes and contamination scenarios.

Let us consider a univariate normal model with true parameter value *θ*<sup>0</sup> = (*μ* = 0, *σ* = 1), and the problem of testing

$$\mathbb{H}\_0: \sigma = 1 \text{ vs. } \mathbb{H}\_1: \sigma \neq 1. \tag{33}$$

The restricted parameter space is then given by

$$\Theta\_0 = \{ (\mu, 1) : \mu \in \mathbb{R} \} .$$

In order to evaluate the robustness properties of the estimators and test statistics, we introduce contamination in data by replacing a *ε*% of the observations by a contaminated sample, where *ε* denotes the contamination level. We generate five different scenarios of contamination:


Further, in order to evaluate the power of the test, we consider an alternative true parameter value *θ*<sup>1</sup> = (0, 0.7) which does not satisfy the null hypothesis (33) (or equivalently the restrictions of the parameter space). In this scenario, contaminated parameters are set *θ*<sup>1</sup> = (0, 1.2) for slightly and *θ*<sup>1</sup> = (0, 1.5) for heavily contamination.

Figure 1 shows the root mean square error (RMSE) of the RMRPE of the scale parameter *σ*, for different values of the tuning parameter *τ* = 0, 0.2, 0.4, 0.6 and *τ* = 0.8 over *R* = 10,000 replications. As expected, large values of the tuning parameter produce more robust estimators, which is particularly advantageous for the heavily contaminated scenario. Furthermore, even when introducing very low levels of contamination in data, *ε* = 5%, the RMRPE with moderate value of the tuning parameter outperforms the classical MLE, without a significant loss of efficiency in the absence of contamination.

**Figure 1.** RMSE of the RMRPE under increasing contamination levels (slightly contaminated at left and heavily contaminated at right) for different values of the tuning parameter *τ* over *R* = 10,000 replications. (**a**) Scenario 1, (**b**) Scenario 2.

On the other hand, Figure 2 presents the empirical level and power of both RPTS and Rao-type test statistics based on RMRPEs for different values of the tuning parameter, *τ* = 0, 0.2, 0.4, 0.6, 0.8, under increasing contamination levels. The empirical level and power are computed as the mean number of rejections over *R* = 10,000 replications. The empirical level produced by the classical ratio and Rao-type tests rapidly increases and separates from levels obtained with any robust test. Regarding the empirical power, all robust tests with moderate and large values of the tuning parameter outperform the classical estimator within their family under contaminated scenarios, but Rao-type test statistics based on RMRPEs are more conservative than RPTSs, thus exhibiting lower levels and powers. Then, the proposed test statistics provides an appealing alternative to classical likelihood ratio and Rao tests, with a small loss of efficiency in favor of a clear gain in terms of robustness.

**Figure 2.** *Cont*.

**Figure 2.** Empirical level and power under increasing contamination (slightly contaminated at left and heavily contaminated at right) over *R* = 10,000 repetitions. (**a**) Scenario 1, (**b**) Scenario 2.

On the other hand, the sample size could play a crucial role in the performance of the tests, even more accentuated when there exists data contamination. Figure 3 shows the sample size effect on the performance of the tests in terms of empirical level, under a 10% of contamination level in data. As discussed, Rao-type test statistics based on RMRPEs is more conservative and so tests based on RMRPEs with positive values of the tuning parameter produce lower empirical levels. Here, it outperforms the poor performance of the classical Rao-type test statistics with respect to any other. Moreover, when the sample size increases, the performance gap between non-robust and robust methods is widening.

**Figure 3.** Empirical level under increasing sample sizes for 10% of contamination level (slightly contaminated at left and heavily contaminated at right) over *R* = 10,000 repetitions. (**a**) Scenario 1, (**b**) Scenario 2.

Following the discussions in the preceding sections, larger values of the tuning parameter produce more robust but less efficient estimators. Therefore, the optimal value of *τ* should obtain the best trade-off between efficiency and robustness. Warwick and Jones [25] first introduced a useful data-based procedure for the choice of the tuning parameter for the MDPDE based on minimizing the asymptotic MSE of the estimator. However, this method depends on the choice of a pilot estimator, and Basak et al. [26] improved the method by removing the dependency on an initial estimator. The proposed algorithm was developed

ad hoc for the MDPDE, but it can be easily adapted to the MRPE and RMRPE by simply substituting the expression of the variance of the MDPDE by the variance of the MRPPE or the RMRPE, respectively.
