Robustified Ud-plot for Assessing Normality

Robustified Ud-plot for Assessing Normality Uditha Amarananda Wijesuriya* Abstract The Ad-plot developed from the cumulative average deviation function efficiently detects numerous distributional characteristics, including symmetry, skewness, and outliers analogous to sample variance plots that outperform histograms. In the meantime, the Ud-plot derived from a slight modification to the Ad-plot is outstanding in assessing normality, surpassing normal QQ-plot, normal PP-plot, and their derivations. In this work, the robustness of the Ud-plot is explored while employing the trimmed average in the cumulative average deviation function. From the standpoint of assessing normality, the robust version is as exceptional as the Ud-plot. For actual and simulated data, the performance of the novel substitute is compared with the Ud-plot. Markedly, this version is extremely competitive in assessing normality and capturing vital distribution properties. Thus, the innovative statistical plot is a noteworthy addition to data visualization implements, delivering insightful illustrations while enhancing perception. Key Words: Sv-plot1, Sv-plot2, Sv-plot3, Ad-plot, Cumulative average deviation function, dvalue 1. Introduction Assessing normality is an important step in many statistical inferences. Graphical methods and normality tests are two methods to assess normality. The most popular graphical methods include histograms, boxplots, normal QQ-plot, and normal PP-plot. The sample variance plots, Sv-plot1, Sv-plot2, and Sv-plot3, are capable of identifying distributional attributes (Wijesuriya 2020, 2021, 2023). Notably, Sv-plot2 and Sv-plot3 can be used for illustrating testing hypotheses over single and two population means. The average deviation plot, Ad-plot derived from cumulative average deviation function (cadf) identifies many of the distributional features, including symmetry, skewness, and outliers (Wijesuriya 2025a). A slight modification of Ad-plot leads to creating the Ud-plot, which is prominent in assessing normality (Wijesuriya 2025a, b). Its perceptive-added features include the most desirable normal curve and the d-value that quantifies how close the Ud-plot is to the graph of the estimated normal density function. None of the popular graphical tools includes these value-added features. The robustified versions of QQ-plot and PP-plot are found in the literature on assessing normality. Gel et al. used median and median absolute deviations as robust measures of central tendency and dispersion in QQ-plot to create RQQ-plot (Gel et al. 2005). Stehl´ ık et al. employed 5% trimmed mean and trim-trim standard deviation in its standardization as measures of location and scale to develop the robust version of QQ-plot, RTQ-plot (Stehl´ ık et al. 2014). Both RQQand RTQ-plots are insensitive to extreme values in the data. The stabilized PP-plot, SP-plot, and its modified version, MSP-plots assess normality including null bands (Michael 1983). The variance of empirical cumulative distribution function is stabilized by an effective transformation, illustrating pointwise null bands with a simulation study (Paolella 2015). The cadf is sensitive to extreme values, and hence Ad-plot and Ud-plot can deviate from the parent distribution. As a robustification of these statistical plots, the author proposes the *Department of Mathematical Sciences, University of Southern Indiana, 8600 University Blvd, Evansville, IN 47712 elimination of extreme data using 5% trimming prior to creating two visualizations. The investigation of the performance from a simulation study and actual data strongly supports for the proposal. 2. Methodology Suppose that X1, ..., Xnis a random sample from a distribution with mean µ, standard deviation σ, and the density function f. Then, for any t∈R, the cumulative average deviation function (cadf) and its empirical version (ecadf) are defined by U(t)=− t Z −∞ (x−µ)f(x)dx and Un(t)=−1 nX X≤t (X−¯ X), respectively (Wijesuriya 2025a, b). The Ad-plot and Ud-plot are the graphs of the two sets of ordered pairs defined by {(t, Un(t)) : t∈ {X1, ..., Xn}} and {(t, Vn(t)) : t∈ {X1, ..., Xn}}, respectively, where Vn(t) = [(1 −n−1)s2]−1Un(t) The d-value is computed from d-value =1 n n X i=1 min{fn(xi|¯ X, s2), Vn(xi)} max{fn(xi|¯ X, s2), Vn(xi)}, where fn(xi|¯ X, s2) = 1 √2πs2e−1 2s2(xi−¯ X)2 , for i= 1, ..., n, estimating µand σin the normal density fby sample average ¯ Xand the sample variance s2, respectively (Wijesuriya 2025a, b). In Ud-plot, the d-value that quantifies the proximity of the functional estimate and the most desirable normal density function is embedded as a unique value-added feature. This valuation is absent in the normal PP-plot, QQ-plot, and their derivations. Figure 1 displays Ud-plots for 50 (top) and 5000 (bottom) observations simulated from standard the normal distribution, estimated normal density N(¯ X, s2)in red, and the dvalue. First, Ud-plot is nearly symmetric, and hence it detects symmetric property. Second, the points in the Ud-plot are closely situated by the most desirable normal density functions N(¯ X=−0.03, s2= 0.892)and N(¯ X=−0.01, s2= 1.012)with sample sizes 50 and d−value =0.93 0.0 0.1 0.2 0.3 0.4 −2 −1 0 1 2 x V50(x) & N(−0.03,0.892) d−value =0.98 0.0 0.1 0.2 0.3 0.4 −4 −2 0 2 x V5000(x) & N(−0.01,1.012) Figure 1: Ud-plots for 50 (top) and 5000 (bottom) observations simulated from standard normal distribution, estimated normal density N(¯ X, s2)in red, and the d-value. 5000, respectively. Thus, Ud-plot fits adequately and is an informative graphical estimator for the normal density. Further, higher d-values 0.93 and 0.98 indicate that the Ud-plot is in a close proximity of the estimated normal density for each sample, providing a notable degree of confirmation. Indeed, for the large sample (bottom), Ud-plot firmly follows the most desirable normal density. The empirical cadf consists of the sample average that is sensitive to outliers. Thus robustifiying average deviation plots is of herein interest. To robustify the plots, the author proposes to use trimmed data and hence replace the sample average ¯ Xby the trimmed average defined by ¯ XTrimmed =1 n−2⌈αn⌉ n−⌈αn⌉ X i=1+⌈αn⌉ X(i), where X(i)is the ith ordered observation and α= 0.05. With trimming, the effect of extreme values is minimized, enhancing the identification of the distributional characteristics. The proposed robustification is explored by a simulation study and used to assess normality for real data sets. 3. Results and Discussion The summary of the results from the simulation study with Ud-plot is first presented while using normally distributed data with contamination. Herein, the most suitable 5% is chosen for trimming by minimizing loss of information. Finally, normality is assessed for two actual data sets. Table 1. The d-values for 10 random samples & ¯ d, each of size n, from N(µ, σ2)with two sets of (µ, σ), contaminated with five observations from each of the Model 2 and Model 3. n+ 10 (µ, σ)d-value ¯ d 15 (0,1) 0.74 0.74 0.77 0.76 0.74 0.75 0.73 0.73 0.75 0.76 0.75 15 (1,2) 0.77 0.75 0.76 0.72 0.76 0.75 0.73 0.75 0.73 0.78 0.75 20 (0,1) 0.81 0.84 0.84 0.83 0.82 0.81 0.83 0.80 0.85 0.81 0.82 20 (1,2) 0.83 0.81 0.79 0.84 0.80 0.83 0.82 0.81 0.81 0.83 0.82 30 (0,1) 0.81 0.83 0.84 0.80 0.79 0.84 0.72 0.73 0.77 0.78 0.79 30 (1,2) 0.79 0.76 0.80 0.82 0.80 0.77 0.80 0.80 0.78 0.79 0.79 50 (0,1) 0.73 0.78 0.71 0.74 0.72 0.73 0.76 0.74 0.72 0.79 0.74 50 (1,2) 0.71 0.74 0.77 0.70 0.76 0.70 0.75 0.73 0.77 0.83 0.75 60 (0,1) 0.72 0.73 0.75 0.67 0.71 0.71 0.67 0.67 0.73 0.70 0.71 60 (1,2) 0.72 0.72 0.69 0.71 0.68 0.74 0.76 0.73 0.70 0.71 0.72 110 (0,1) 0.71 0.71 0.73 0.70 0.74 0.72 0.69 0.64 0.71 0.73 0.71 110 (1,2) 0.73 0.66 0.70 0.78 0.77 0.65 0.69 0.74 0.71 0.66 0.71 510 (0,1) 0.86 0.80 0.83 0.83 0.84 0.80 0.84 0.82 0.86 0.85 0.83 510 (1,2) 0.87 0.82 0.86 0.80 0.82 0.87 0.84 0.86 0.86 0.85 0.84 1010 (0,1) 0.87 0.91 0.86 0.90 0.91 0.93 0.90 0.89 0.91 0.90 0.90 1010 (1,2) 0.91 0.89 0.92 0.91 0.90 0.91 0.89 0.89 0.91 0.88 0.90 3010 (0,1) 0.94 0.96 0.97 0.95 0.97 0.95 0.95 0.96 0.95 0.96 0.96 3010 (1,2) 0.96 0.96 0.95 0.95 0.92 0.95 0.95 0.95 0.95 0.96 0.95 5010 (0,1) 0.96 0.97 0.97 0.96 0.97 0.97 0.97 0.98 0.96 0.98 0.97 5010 (1,2) 0.98 0.96 0.97 0.96 0.97 0.97 0.96 0.98 0.98 0.97 0.97 Table 2. The d-values and their average for 10 contaminated samples with 5% trimming, each of size m=n+10−2⌈αn⌉for n= 5,10,20,30,50,100,500,1000,3000, and 5000. m(µ, σ)d-value ¯ d 13 (0,1) 0.75 0.76 0.78 0.75 0.76 0.77 0.75 0.75 0.77 0.75 0.76 13 (1,2) 0.75 0.75 0.78 0.72 0.76 0.75 0.73 0.74 0.72 0.76 0.75 18 (0,1) 0.78 0.83 0.82 0.79 0.81 0.81 0.82 0.80 0.84 0.80 0.81 18 (1,2) 0.81 0.81 0.79 0.83 0.80 0.82 0.84 0.81 0.80 0.79 0.81 28 (0,1) 0.79 0.82 0.82 0.78 0.77 0.83 0.70 0.71 0.76 0.76 0.77 28 (1,2) 0.76 0.73 0.77 0.80 0.81 0.74 0.78 0.78 0.75 0.78 0.77 46 (0,1) 0.73 0.79 0.71 0.73 0.71 0.72 0.77 0.73 0.71 0.78 0.74 46 (1,2) 0.70 0.76 0.80 0.70 0.74 0.69 0.75 0.77 0.81 0.84 0.76 54 (0,1) 0.77 0.78 0.84 0.70 0.76 0.77 0.71 0.78 0.76 0.77 0.76 54 (1,2) 0.73 0.79 0.74 0.76 0.71 0.81 0.81 0.79 0.77 0.73 0.76 100 (0,1) 0.93 0.88 0.93 0.87 0.92 0.88 0.89 0.94 0.93 0.90 0.91 100 (1,2) 0.93 0.94 0.90 0.88 0.95 0.94 0.94 0.90 0.93 0.93 0.92 460 (0,1) 0.90 0.93 0.90 0.92 0.88 0.93 0.91 0.89 0.89 0.91 0.91 460 (1,2) 0.91 0.92 0.91 0.91 0.90 0.91 0.89 0.91 0.92 0.91 0.91 910 (0,1) 0.90 0.90 0.90 0.90 0.91 0.92 0.92 0.90 0.92 0.90 0.91 910 (1,2) 0.91 0.91 0.91 0.89 0.92 0.92 0.90 0.93 0.92 0.91 0.91 2710 (0,1) 0.90 0.91 0.91 0.91 0.90 0.90 0.91 0.90 0.90 0.90 0.90 2710 (1,2) 0.91 0.91 0.92 0.90 0.90 0.90 0.90 0.89 0.91 0.91 0.90 4510 (0,1) 0.90 0.91 0.91 0.90 0.91 0.90 0.90 0.91 0.91 0.90 0.90 4510 (1,2) 0.91 0.91 0.90 0.90 0.89 0.91 0.90 0.90 0.90 0.90 0.90 3.1 Simulation Study In the simulation study, nnumber of observations from Model 1 is contaminated with 5 observations from each of the Model 2 and Model 3, where Model 1:N(µ, σ) Model 2:N(µ−5σ, σ) Model 3:N(µ+ 5σ, σ) The average of d-values from 10 samples, ¯ dis found for two parameter combinations (µ, σ),{(0,1),(1,2)}and, for 10 sample sizes n= 5,10,20,30,50,100,500,1000,3000, and 5000. This study is completed before trimming and reported in Table 1. After 5% trimming, the results from the study are listed in Table 2. Regardless of trimming, it is noticeable that first, for a particular sample size, the average value of d-values, ¯ dremains similar for both parameter combinations. Second, it is approaching its maximum value as the sample size is large. That is, for large n, the Ud-plot is an adequate fit for the estimated normal density. With 5% trimming, the d-values and ¯ d in Table 2 have slightly changed for n≤50, indicating less effect from trimming. As n increases, ¯ dincreases, providing an improved fit. However, the average d-values are still closer to one without trimming compared to with trimming. 3.2 With Two Actual Data Sets In Figure 2, the Ud-plot of the average price of avocados sold across 169 weeks from 2015 to 2018 in California found in the causaldata R package, nearly situates at the desirable normal curve f169(1.11,0.232), providing a high d-value of 0.80. However, it’s noticeable that it overestimates for data in the upper end, the top panel in Figure 2. With 5% trimming, it illuminates 18 data values from both ends. Thus, the Ud-plot for the trimmed data closely aligns with the estimated normal density curve, leaving an improved d-value of 0.86. That is, the functional estimate for the trimmed data adequately fits with the most desirable normal model f151(1.09,0.172), strongly agreeing with the proposal. In the tseries R package, the MarkDollar data set consists of 518 observations of intradayreturns of Deutschemark/US dollar (DEM/US) exchange rate from October 1, 1992, to September 29, 1993, excluding weekends. Figure 3 includes Ud-plot for MarkDollar data without (top) and with (bottom) 5% trimming. The functional estimate firmly fits into the estimated normal curve despite having no trimming, with a high d-value of 0.91. It notes that the 5% trimming procedure employed before creation of average deviation plots is vastly productive for identifying distributional features and assessing normality. Thus, the experimenter could follow the 2-step procedure to obtain the robust version of average deviation plots. •Step 1. Eliminate extreme data values by applying 5% trimming. •Step 2. Use adplot() and udplot() functions in ’svplots’ R package for trimmed data to obtain robust versions of average deviation plots (Wijesuriya 2025b). With the 2-step procedure, one can employ Ad-plot for visualizing characteristics of the distribution and Ud-plot for assessing normality. In conclusion, the robust versions of two statistical graphs are extremely beneficial in exploratory data analysis. d−value =0.8 0.0 0.5 1.0 1.5 0.9 1.2 1.5 1.8 x V169(x) & N(1.11,0.232) d−value =0.86 0.0 0.5 1.0 1.5 2.0 1.00 1.25 1.50 x V151(x) & N(1.09,0.172) Figure 2: Ud-plot for average price of avocado data before (top) and after (bottom) 5% trimming. d−value =0.91 0.0 0.2 0.4 0.6 0.8 −1 0 1 2 x U518(x)[(1−518−1)0.512] & N(0.02,0.512) d−value =0.92 0.0 0.3 0.6 0.9 −0.5 0.0 0.5 x U466(x)[(1−466−1)0.372] & N(0.01,0.372) Figure 3: Ud-plot for MarkDollar data before (top) and after (bottom) 5% trimming. 4. Conclusion In this article, the author proposes a procedure to obtain robust versions of Ad-plot and Ud-plot. A simulation study is conducted to explore the proposal. Further, 5% trimming is employed for two publicly available actual data sets. As a graphical tool for assessing normality, Ud-plot is among the best. With trimming, it is as extreme or more extreme than the existing robust versions derived from normal QQ-plot and PP-plot, such as RQQ-plot and RTQ-plots. Further, the value-added feature, dvalue, helps users to make a decision on assessing normality, being a competitive visual aid. In statistical inference, robust visualization tools for assessing the normality assumption are indeed critical. Thus, the Ud-plot with the trimming proposal is noteworthy. In sum, the trimmed version of the Ud-plot is an exceptional visual alternative for assessing normality, expanding the graphical aids literature. REFERENCES Gel, Y. R., Miao, W. & Gastwirth, J. L. (2005), ‘The importance of checking the assumption underlying statistical analysis: graphical methods for assessing normality,’ Jurimetrics 46(1), 3–29. Michael, J. R. (1983), ‘The stabilized probability plot,’ Biometrika 70, 11–17. Paolella, M. S. 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