# Composite Tests under Corrupted Data

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## Abstract

**:**

## 1. Introduction

#### Statement of the Test Problem

## 2. An Extension of the Likelihood Ratio Test

**Remark**

**1.**

**Proposition**

**2.**

## 3. Minimax Tests under Noisy Data, Least-Favorable Hypotheses

#### 3.1. An Asymptotic Definition for the Least-Favorable Hypotheses

**Theorem**

**3.**

#### 3.2. Identifying the Least-Favorable Hypotheses

**Proposition**

**4.**

#### 3.3. Numerical Performances of the Minimax Test

- In the first case, ${f}_{0}$ is a normal density with expectation 0 and variance 1, whereas ${g}_{0}$ is a normal density with expectation $0.3$ and variance $1$.
- The second case handles a situation where ${f}_{0}$ and ${g}_{0}$ belong to different models: ${f}_{0}$ is a log-normal density with location parameter 1 and scale parameter $0.2$, whereas ${g}_{0}$ is a Weibull density on ${\mathbb{R}}^{+}$ with shape parameter 5 and scale parameter $3$. Those two densities differ strongly, in terms of asymptotic decay. They are, however, very close to one another in terms of their symmetrized Kullback-Leibler divergence (the so-called Jeffrey distance). Indeed, centering on the log-normal distribution ${f}_{0}$, the closest among all Weibull densities is at distance $0.10$—the density ${g}_{0\phantom{\rule{4.pt}{0ex}}}$ is at distance $0.12$ from ${f}_{0}.$

- The noise ${h}_{\delta}$ is a centered normal density with variance ${\delta}^{2}$;
- the noise ${h}_{\delta}$ is a centered Laplace density with parameter $\lambda \left(\delta \right)$;
- the noise ${h}_{\delta}$ is a symmetrized Weibull density with shape parameter $1.5$ and variable scale parameter $\beta \left(\delta \right)$; and
- the noise ${h}_{\delta}$ is Cauchy with density ${h}_{\delta}\left(x\right)=\gamma \left(\delta \right)/\pi (\gamma {\left(\delta \right)}^{2}+{x}^{2})$.

#### 3.3.1. Case A: The Shift Problem

#### Theoretical Power Bound

#### Numerical Power Bound

#### Comparison of the Two Power Curves

#### 3.3.2. Case B: The Tail Thickness Problem

## 4. Some Alternative Statistics for Testing

#### 4.1. A Family of Composite Tests Based on Divergence Distances

#### 4.2. A Practical Choice for Composite Tests Based on Simulation

#### 4.2.1. Case A: The Shift Problem

- Under Gaussian noise, the power remains mostly stable over the values of ${\delta}_{\ast}$, as shown by Figure 5. The tests with statistics ${T}_{n}^{1}$ and ${T}_{n}^{2}$ are equivalently powerful for large values of ${\delta}_{\ast}$, while the first one achieves higher power when ${\delta}_{\ast}$ is small.
- When the noise follows a Laplace distribution, the three power curves overlap the NP power curve, and the different test statistics can be indifferently used. Under such a noise, the alternative hypotheses are extremely well distinguished by the class of tests considered, and this remains true as ${\delta}_{\ast}$ increases (cf. Figure 6).
- Under the Weibull hypothesis, ${T}_{n}^{1}$ and ${T}_{n}^{2}$ perform similarly well, and almost always as well as ${T}_{n,{\delta}_{\ast}}^{1}$, while the power curve associated to ${T}_{n}^{1/2}$ remains below. Figure 7 illustrates that, as ${\delta}_{max}$ increases, the power does not decrease much.
- Under a Cauchy assumption, the alternate hypotheses are less distinguishable than under any other parametric hypothesis on the noise, since the maximal power is about 0.84, while it exceeds 0.9 in cases a, b, and c (cf. Figure 5, Figure 6, Figure 7 and Figure 8). The capacity of the tests to discriminate between $\mathbf{H}\mathbf{0}\left({\delta}_{max}\right)$ and $\mathbf{H}\mathbf{1}\left({\delta}_{max}\right)$ is almost independent of the value of ${\delta}_{max}$, and the power curves are mainly flat.

#### 4.2.2. Case B: The Tail Thickness Problem

- With the noise defined by case A (Gaussian noise), for KL ($\gamma =1$), ${\delta}_{\ast}={\delta}_{max}$ due to Proposition 4 and statistics ${T}_{n}^{1}$ provides the best power uniformly upon ${\delta}_{max}.$ Figure 9 shows a net decrease of the power as ${\delta}_{max}$ increases (recall that the power is evaluated under the least favorable alternative ${G}_{{\delta}_{max}}$).
- When the noise follows a Laplace distribution, the situation is quite peculiar. For any value of $\delta $ in $\Delta $, the modes ${M}_{{G}_{{\delta}_{max}}}\phantom{\rule{4pt}{0ex}}$ and ${M}_{{F}_{{\delta}_{max}}}$ of the distributions of $\left({f}_{\delta}/{g}_{\delta}\right)\left(X\right)$ under ${G}_{{\delta}_{max}}$ and under ${F}_{{\delta}_{max}}$ are quite separated; both larger than $1.$ Also, for $\delta $ all the values of $\left|{\varphi}_{\gamma}\right({M}_{{G}_{{\delta}_{max}}})-{\varphi}_{\gamma}({M}_{{F}_{{\delta}_{max}}}\left)\right|$ are quite large for large values of $\gamma .$ We may infer that the distributions of ${\varphi}_{\gamma}\left(\left({f}_{\delta}/{g}_{\delta}\right)\left(X\right)\right)$ under ${G}_{{\delta}_{max}}$ and under ${F}_{{\delta}_{max}}$ are quite distinct for all $\delta $, which in turn implies that the same fact holds for the distributions of ${T}_{n}^{\gamma}$ for large $\gamma .$ Indeed, simulations presented in Figure 10 show that the maximal power of the test tends to be achieved when $\gamma =2.$
- When the noise follows a symmetric Weibull distribution, the power function when $\gamma =1$ is very close to the power of the LRT between ${F}_{{\delta}_{max}}$ and ${G}_{{\delta}_{max}}$ (cf. Figure 11). Indeed, uniformly on $\delta $, and on x, the ratio $\left({f}_{\delta}/{g}_{\delta}\right)\left(x\right)$ is close to 1. Therefore, the distribution of ${T}_{n}^{1}$ is close to that of ${T}_{n,{\delta}_{max}}^{1}$, which plays in favor of the KL composite test.
- Under a Cauchy distribution, similarly to case A, Figure 12 shows that ${T}_{n}^{\gamma}$ achieves the maximal power for $\gamma =1$ and 2, closely followed by $\gamma =0.5$.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Proposition 2

#### Appendix A.1. The Critical Region of the Test

#### Appendix A.2. The Power Function

#### Appendix A.3. A Synthetic Result

## Appendix B. Proof of Theorem 3

**Theorem**

**A1.**

**Remark**

**A2.**

## Appendix C. Proof of Proposition 4

**Lemma**

**A3.**

**Proof.**

**Lemma**

**A4.**

**Proof.**

**Lemma**

**A5.**

**Proof.**

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**Figure 1.**Theoretical and numerical power bound of the test of case A under Gaussian noise (with respect to n), for the first kind risk $\alpha =0.05$.

**Figure 2.**Theoretical and numerical power bound of the test of case A under symmetrized Weibull noise (with respect to n), for the first kind risk $\alpha =0.05$.

**Figure 3.**Theoretical and numerical power bound of the test of case A under a symmetrized Laplacian noise (with respect to n), for the first kind risk $\alpha =0.05$.

**Figure 5.**Power of the test of case A under Gaussian noise (with respect to ${\delta}_{max}$), for the first kind risk $\alpha =0.05$ and sample size $n=100$.

**Figure 6.**Power of the test of case A under Laplacian noise (with respect to ${\delta}_{max}$), for the first kind risk $\alpha =0.05$ and sample size $n=100$.

**Figure 7.**Power of the test of case A under symmetrized Weibull noise (with respect to ${\delta}_{max}$), for the first kind risk $\alpha =0.05$ and sample size $n=100$.

**Figure 8.**Power of the test of case A under noise following a Cauchy distribution (with respect to ${\delta}_{max}$), for the first kind risk $\alpha =0.05$ and sample size $n=100$.

**Figure 9.**Power of the test of case B under Gaussian noise (with respect to ${\delta}_{max}$), for the first kind risk $\alpha =0.05$ and sample size $n=100$. The NP curve corresponds to the optimal Neyman Pearson test under ${\delta}_{max}$. The KL, Hellinger, and $G=2$ curves stand respectively for $\gamma =1,\gamma =0.5$, and $\gamma =2$ cases.

**Figure 10.**Power of the test of case B under Laplacian noise (with respect to ${\delta}_{max}$), for the first kind risk $\alpha =0.05$ and sample size $n=100$.

**Figure 11.**Power of the test of case B under symmetrized Weibull noise (with respect to ${\delta}_{max}$), for the first kind risk $\alpha =0.05$ and sample size $n=100$.

**Figure 12.**Power of the test of case B under a noise following a Cauchy distribution (with respect to ${\delta}_{max}$), for the first kind risk $\alpha =0.05$ and sample size $n=100$.

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Broniatowski, M.; Jurečková, J.; Moses, A.K.; Miranda, E. Composite Tests under Corrupted Data. *Entropy* **2019**, *21*, 63.
https://doi.org/10.3390/e21010063

**AMA Style**

Broniatowski M, Jurečková J, Moses AK, Miranda E. Composite Tests under Corrupted Data. *Entropy*. 2019; 21(1):63.
https://doi.org/10.3390/e21010063

**Chicago/Turabian Style**

Broniatowski, Michel, Jana Jurečková, Ashok Kumar Moses, and Emilie Miranda. 2019. "Composite Tests under Corrupted Data" *Entropy* 21, no. 1: 63.
https://doi.org/10.3390/e21010063