# New Graphical Methods and Test Statistics for Testing Composite Normality

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## Abstract

**:**

## 1. Introduction

## 2. Review of Relevant Material

**Figure 1.**(Top) Power of the KD test for normality, using size $\mathsf{\alpha}=0.05$, for three different sample sizes, and the Student’s t alternative (left) and skew normal alternative (right), based on one million replications. (Middle) The same, but for the AD test. (Bottom) The same, but power of the ${W}^{2}$ (lines without circles) and ${U}^{2}$ (lines with circles) test for normality. The ${W}^{2}$ and ${U}^{2}$ power curves for the Student’s t alternative are graphically indistinguishable.

## 3. Null Bands

#### 3.1. Mapping Pointwise and Simultaneous Significance Levels

**Figure 2.**Q-Q plot for a random $N(10,2)$ sample of size $n=50$ with 10% and 5% pointwise null bands obtained via simulation (top panels), using the estimated parameters (left) and the true parameters (right) of the data. The bottom ones are similar, but based on the asymptotic distribution in (2).

**Figure 3.**The mapping between pointwise and simultaneous significance levels, for normal data (left) and Weibull data (right) using sample size n.

#### 3.2. Q-Q Test

**Figure 4.**Power of Q-Q test for normality, for three different sample sizes, and the Student’s t alternative (left) and skew normal alternative (right), based on simulation with 1000 replications.

## 4. Further P-P and Q-Q Type Plots

#### 4.1. (Horizontal) Stabilized P-P Plots

**Figure 5.**S-P plot for i.i.d. Cauchy data of sample size 1000, with null bands, obtained via simulation, using a pointwise significance level of 0.01. The right panel is the same, but using the horizontal format.

**Figure 6.**(Top) Horizontal S-P plot using same random $N(10,2)$ sample of size $n=50$ as used in Figure 2 with 10% and 5% pointwise null bands obtained via simulation, using the estimated parameters (left) and the true parameters (right) of the data. (Bottom) The same as the top, but with constant width null bands.

#### 4.2. Modified S-P (MSP) Plots

n \ coef | ${\mathbf{b}}_{\mathbf{1}}$ | ${\mathbf{b}}_{\mathbf{2}}$ | ${\mathbf{b}}_{\mathbf{3}}$ | ${\mathbf{b}}_{\mathbf{4}}$ |
---|---|---|---|---|

20 | $0.53830$ | $-0.41414$ | $1.25704$ | $-1.43734$ |

50 | $0.35515$ | $-0.33988$ | $0.95000$ | $-1.02478$ |

100 | $0.26000$ | $-0.24000$ | $0.70463$ | $-0.76308$ |

**Figure 7.**(Left) The solid, dashed and dash-dot lines are the widths for the pointwise null bands of the normal MSP plot, as a function of the pointwise significance level p, computed using simulation with 50,000 replications. The overlaid dotted curves are the same, but having used the instantaneously-computed approximation from (5) and (6). There is no optical difference between the simulated and the approximation. (Right) For the normal MSP plot, the mapping between pointwise and simultaneous significance levels using sample size n.

#### 4.3. MSP Test for Normality

**h**and

**g**are the vectors formed from values in (3).

**Figure 8.**Power of the MSP test for normality, for three different sample sizes, and the Student’s t alternative (left) and skew normal alternative (right), based on one million replications.

**X**. Fortunately, there is a better way: Figure 9 shows the kernel density (solid line) of $n\times {T}_{\mathrm{MSP}}$, for two sample sizes, $n=10$ and $n=50$. Remarkably, and as an amusing coincidence, the distribution strongly resembles that of a location-scale skew normal. The dashed lines in the plots show the best fitted location-scale skew normal densities; the match is striking. In particular, using the MLE, we obtain asymmetry parameter $\widehat{\mathsf{\lambda}}=2.6031$, location parameter $\widehat{\mathsf{\mu}}=0.6988$ and scale parameter $\widehat{c}=0.3783$ for $n=10$ and $\widehat{\mathsf{\lambda}}=2.7962$, $\widehat{\mathsf{\mu}}=2.2282$ and $\widehat{c}=1.0378$ for $n=50$. By using other sample sizes, up to $n=500$, we confirm that the skew normal yields an extremely accurate approximation to the true distribution of ${T}_{\mathrm{MSP}}$ under the null for all sample sizes between 10 and (at least) 500.

**Figure 9.**Kernel density and fitted skew normal distribution of sample size n times the MSP test statistic (7), computed under the null, and based on one million replications.

**Figure 10.**One million p-values from the MSP test with $n=50$, under the null (left) and for a $t(8)$ alternative (right).

#### 4.4. Modified Percentile (Fowlkes-MP) Plots

**Figure 11.**Normal Fowlkes-MP (left) and normal MSP (right) plots, with simultaneous null bands, for normal data (top) and mixed normal data (bottom).

**Figure 12.**Power of Fowlkes-MP test for normality, for three different sample sizes and the Student’s t alternative (left) and skew normal alternative (right), based onone million replications.

#### 4.5. Power Comparisons Against Two-Component Mixed Normal Alternative

**Table 2.**Comparison of power for various normal tests of size 0.05, using the two-component mixed normal distribution as the alternative, obtained via simulation with one million replications for each model and based on two sample sizes, $n=100$ and $n=200$. Model No. 0 is the normal distribution, used to serve as a check on the size. The entry with the highest power for each alternative model and sample size n is marked in boldface. Entries with a power lower than the nominal/actual value of 0.05, indicating a biased test, are in italics.

Model \ Test | n | KD | AD | ${\mathbf{W}}^{\mathbf{2}}$ | ${\mathbf{U}}^{\mathbf{2}}$ | MSP | F–MP | JB | ${\mathbf{X}}_{\mathbf{P}}^{\mathbf{2}}$ |
---|---|---|---|---|---|---|---|---|---|

No. 0 (Normal) | 100 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 |

No. 1 (Finance) | 100 | 0.799 | 0.712 | 0.916 | 0.924 | 0.799 | 0.798 | 0.890 | 0.635 |

No. 2 (Equal Means) | 100 | 0.198 | 0.322 | 0.298 | 0.309 | 0.216 | 0.197 | 0.417 | 0.127 |

No. 3 (Equal Vars) | 100 | 0.303 | 0.001 | 0.400 | 0.439 | 0.250 | 0.302 | 0.039 | 0.191 |

No. 0 (Normal) | 200 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 |

No. 1 (Finance) | 200 | 0.983 | 0.881 | 0.997 | 0.998 | 0.973 | 0.983 | 0.994 | 0.940 |

No. 2 (Equal Means) | 200 | 0.358 | 0.430 | 0.523 | 0.545 | 0.328 | 0.358 | 0.642 | 0.225 |

No. 3 (Equal Vars) | 200 | 0.593 | 0.000 | 0.756 | 0.789 | 0.492 | 0.594 | 0.428 | 0.394 |

## 5. Further Tests for Composite Normality

#### 5.1. Jarque-Bera Test

**Figure 13.**Kernel density estimate (solid) of the log of the JB test statistic, under the null of normality and using a sample size of $n=50$, based on 10 million replications (and having used MATLAB’s ksdensity function with 300 equally spaced points). (Left) The fitted generalized asymmetric t (GAt) density (dashed); (right) fitted noncentral t (dashed) and asymmetric stable (dash-dot).

**Figure 14.**Simulated p-values of the JB test statistic, based on one million replications, using the GAt approximation (left) and the two-component GAt mixture (right).

**Figure 15.**Power of JB test for normality, for three different sample sizes, and Student’s t alternative (left) and skew normal alternative (right), based on 100,000 replications.

#### 5.2. Ghosh Graphical Test

**Figure 16.**Power of the test of [15] for normality, for three different sample sizes, and the Student’s t alternative (left) and skew normal alternative (right), based on 100,000 replications.

#### 5.3. Information-Theoretic Distribution Test

**Figure 17.**Power of the KL1 test for normality, for three different sample sizes, and the Student’s t alternative (left) and skew normal alternative (right), based on 100,000 replications.

## 6. Combining Tests and Power Envelopes

#### 6.1. Combining Tests

**Figure 18.**Power of the MSP, JB and joint tests for composite normality, using $n=50$ and based on 100,000 replications.

#### 6.2. Power Comparisons for Testing Composite Normality

#### 6.3. Most Powerful Tests and Power Envelopes

**Figure 19.**(Left) The power of the JB test (8) against the alternative of a Student’s $t(v)$ (lines with circles; same power curves as given in the left panel of Figure 15), along with the power of the likelihood ratio test (using Student’s t as the specific alternative), based on 10,000 replications. (Right) The power of the MSP test for normality against the alternative of skew normal (lines with circles; same power curves as in the right panel of Figure 8), along with the power of the likelihood ratio test (using the skew normal as the specific alternative), based on 10,000 replications.

## 7. Conclusions

## Acknowledgments

## Conflicts of Interest

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Paolella, M.S.
New Graphical Methods and Test Statistics for Testing Composite Normality. *Econometrics* **2015**, *3*, 532-560.
https://doi.org/10.3390/econometrics3030532

**AMA Style**

Paolella MS.
New Graphical Methods and Test Statistics for Testing Composite Normality. *Econometrics*. 2015; 3(3):532-560.
https://doi.org/10.3390/econometrics3030532

**Chicago/Turabian Style**

Paolella, Marc S.
2015. "New Graphical Methods and Test Statistics for Testing Composite Normality" *Econometrics* 3, no. 3: 532-560.
https://doi.org/10.3390/econometrics3030532