# Bias, Precision, and Accuracy of Skewness and Kurtosis Estimators for Frequently Used Continuous Distributions

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Review of Kurtosis and Skewness Estimators

_{1}, which in this paper are referred to as Kr2 and Kr3. The Kr2 estimator is as follows:

_{0.05}is the mean of the upper 5% of the order statistics of the sample, L

_{0.05}is the mean of the lower 5% of the order statistics of the sample, and U

_{0.5}and L

_{0.5}are, respectively, the means of the upper and lower 50% of the order statistics. Thus, Kr2 is a ratio of two linear functions of the order statistics.

_{0.2}and L

_{0.2}are, respectively, the means of the upper and lower 20% of the order statistics of the sample, and U

_{0.5}and L

_{0.5}are defined as in Equation (2).

_{0.05}and L

_{0.05}are the means of the largest and smallest 5% of the order statistics of the sample, and m

_{0.25}is the 25-trimmed mean, that is, the mean of the ordered observations trimmed by 25% at both the upper and lower ends. Keselman et al. [13] and Reed and Stark [15] referred to the 25-trimmed mean as T

_{0.25}. The calculation of m

_{0.25}or T

_{0.25}coincides with the mean of the middle (MID) 50% of the sample. Hence, the mean of the middle 50% is referred to by Hogg et al. [14] as M

_{0.5}and by Othman et al. [23] as MID.

## 3. Theoretical Values of Kurtosis and Skewness

## 4. Method

#### 4.1. Study Variables

#### 4.2. Evaluation Criteria

## 5. Results

#### 5.1. Exponential Distribution

#### 5.2. Lognormal Distribution

#### 5.3. Gamma Distributions

#### 5.4. Summary of the Performance of Hogg’s and Conventional Estimators

## 6. Discussion

#### 6.1. Utility for Researchers

#### 6.2. Limitations and Future Directions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

- /*SAS macro to calculate Hogg’s estimators of kurtosis and skewness (Kr2 = Q, Kr3 = Q1 and Sk2 = Q2) and the conventional estimators of kurtosis and skewness (Kr1 and Sk1) from a data file*/
- %MACRO KURTSKEW (N = 50); /*Specify N*/
- %let k = (0.2*&n) + 1;
- %let k = round(&k);
- %let k1 = (0.5*&n) + 1;
- %let k1 = round(&k1);
- %let k2 = (0.05*&n) + 1;
- %let k2 = round(&k2);
- %let k3 = (0.25*&n) + 1;
- %let k3 = round(&k3);
- /*Load a data file*/
- proc iml;
- data bias;
- infile ‘c:\name.dat’; /*Specify the path and name of the data file*/
- input id x; /*id = subject identifier; x = data*/
- run;
- /*Sort the data from least to greatest*/
- proc sort data = bias;
- key x/ascending;
- run;
- quit:
- /*Obtaining the data matrix for the analysis*/
- data model;
- set bias;
- counter = _n_;
- if counter >= &k3 + 1 and counter <= &n − &k3 then do;
- y = x; end;
- if counter = &k3 or counter = &n − &k3 + 1 then do;
- y1 = x; end;
- /*Calculation of L2 U2 L5 U5 L05 U05 & T25*/
- proc iml;
- use model;
- j1 = 0.2*&n;j2 = 0.5*&n; j3 = 0.05*&n;
- k1 = &k-1; k11 = &k; k12 = &n-&k+2; k13 = &n-&k + 1;
- k2 = &k1-1; k21 = &k1; k23 = &n-&k1+2;k24 = &n-&k1 + 1;
- k3 = &k2-1; k31 = &k2; k32 = &n-&k2+2;k33 = &n-&k2 + 1;
- m = 1/(0.2*&n);
- m1 = 1/(0.5*&n);
- m2 = 1/(0.05*&n);
- m3 = 1/((1-2*0.25)*&n);
- r = 1 − (&k − 0.2*&n);
- r1 = 1 − (&k1 − 0.5*&n);
- r2 = 1 − (&k2 − 0.05*&n);
- r3 = &k3 − (0.25*&n);
- read all variables (‘x’) into table1;
- if j1 <= 1 then do; L2 = table1[1];
- end;
- else do;
- p1 = table1[1:k1];
- p1 = sum(p1);
- p2 = table1[k11];
- L2 = m*(p1+(r*p2));
- end;
- read all variables (‘x’) into table2;
- if j1 <= 1 then do; U2 = table1[1];
- end;
- else do;
- p3 = table2[k12:&n];
- p3 = sum(p3);
- p4 = table2[k13];
- U2 = m*(p3+(r*p4));
- end;
- read all variables (‘x’) into table3;
- if j2 <= 1 then do; L5 = table1[1]; end;
- else do;
- p5 = table3[1:k2];
- p5 = sum(p5);
- p6 = table3[k21];
- L5 = m1*(p5 + (r1*p6));
- end;
- read all variables (‘x’) into table4;
- if j2 <= 1 then do; U5 = table4[&n]; end;
- else do;
- p7 = table4[k23:&n];
- p7 = sum(p7);
- p8 = table4[k24];
- U5 = m1*(p7 + (r1*p8));
- end;
- read all variables (‘x’) into table5;
- if j3 <= 1 then do; L05 = table5[1]; end;
- else do;
- p9 = table5[1:k3];
- p9 = sum(p9);
- p10 = table5[k31];
- L05 = m2*(p9 + (r2*p10));
- end;
- read all variables (‘x’) into table6;
- if j3 <= 1 then do; U05 = table6[&n]; end;
- else do;
- p11 = table6[k32:&n];
- p11 = sum(p11);
- p12 = table6[k33];
- U05 = m2*(p11+(r2*p12));
- end;
- read all variables (‘y’) into table7;
- p13 = table7;
- p13 = sum(p13);
- read all variables (‘y1’) into table8;
- p14 = table8;
- p14 = sum(p14);
- p14 = r3*p14;
- p14 = p13+p14;
- T25 = m3*p14;
- H= L2||U2||L5||U5||L05||U05||T25;
- varnames= ‘L2’||’U2’||’L5’||’U5’||’L05’||’U05’||’T25’;
- create table from H [colname=varnames];
- append from H;
- close table;
- ods listing close;
- proc means data = model kurtosis skewness;
- ods output ‘Summary statistics’ = data;
- ods results off;
- run;
- proc iml;
- varNames = {“x_Kurt” “x_Skew”};
- use data;
- read all var varNames into view;
- view1 = view;
- Kr = view1[1];
- Sk = view1[2];
- Qc = Kr||Sk;
- Varnames = ‘Qc1’:’Qc2’;
- create new from Qc [colname=varnames];
- append from Qc;
- close new;
- proc append base = firstdata;
- run;
- ods listing;
- /*Calculation of kurtosis & skewness*/
- data table;
- set table;
- Q = (U05-L05)/(U5-L5);
- Q1 = (U2-L2)/(U5-L5);
- Q2 = (U05-T25)/(T25-L05);
- /*Q & Q1 = kurtosis, Q2 = skewness*/
- proc iml;
- use table;
- read all variables (‘Q’||’Q1’||’Q2’) into summary;
- q = summary[1];
- q1 = summary[2];
- q2 = summary[3];
- Q = q||q1||q2;
- varnames=‘Q1’:’Q3’;
- create summary1 from Q [colname = varnames];
- append from Q;
- close summary1;
- proc append base = seconddata;
- run;
- data seconddata;
- set seconddata;
- proc iml;
- use seconddata;
- read all var(‘Q1’:’Q3’)into seconddata;
- data seconddata1(rename = (Q1 = KR2 Q2 = KR3 Q3 = SK2));
- set seconddata; run;
- proc print data = seconddata1; run;
- data firstdata;
- set firstdata;
- proc iml;
- use firstdata;
- read all var (‘Qc1’:’Qc2’) into firstdata;
- data firstdata1(rename = (Qc1 = KR1 Qc2 = SK1));
- set firstdata; run;
- proc print data = firstdata1; run;
- %mend kurtskew;
- %kurtskew;

## References

- Blanca, M.J.; Arnau, J.; López-Montiel, D.; Bono, R.; Bendayan, R. Skewness and kurtosis in real data samples. Methodology
**2013**, 9, 78–84. [Google Scholar] [CrossRef] - Micceri, T. The unicorn, the normal curve, and other improbable creatures. Psychol. Bull.
**1989**, 105, 156–166. [Google Scholar] [CrossRef] - García, J.F.; Musitu, G.; Riquelme, E.; Riquelme, P. A confirmatory factor analysis of the “Autoconcepto Forma 5” questionnaire in young adults from Spain and Chile. Span. J. Psychol.
**2011**, 14, 648–658. [Google Scholar] [CrossRef] [PubMed] [Green Version] - García, J.F.; Musitu, G.; Veiga, F. Autoconcepto en adultos de España y Portugal. Psicothema
**2006**, 18, 551–556. [Google Scholar] [PubMed] - Arnau, J.; Bendayan, R.; Blanca, M.J.; Bono, R. Should we rely on the Kenward–Roger approximation when using linear mixed models if the groups have different distributions? Br. J. Math. Stat. Psychol.
**2014**, 67, 408–429. [Google Scholar] [CrossRef] - Shang-Wen, Y.; Ming-Hua, H. Estimation of air traffic longitudinal conflict probability based on the reaction time of controllers. Saf. Sci.
**2010**, 48, 926–930. [Google Scholar] [CrossRef] - Deluchi, K.L.; Bostrom, A. Methods for analysis of skewed data distributions in psychiatric clinical studies: Working with many zero values. Am. J. Psychiat.
**2004**, 161, 1159–1168. [Google Scholar] [CrossRef] [Green Version] - Soler, H.; Vinayak, P.; Quadagno, D. Biosocial aspects of domestic violence. Psychoneuroendocrinology
**2000**, 25, 721–739. [Google Scholar] [CrossRef] - Diaz-Serrano, L. Labor income uncertainty, skewness and homeownership: A panel data study for Germany and Spain. J. Urban Econ.
**2005**, 58, 156–176. [Google Scholar] [CrossRef] - Zhou, X.; Lin, H.; Johnson, E. Non-parametric heteroscedastic transformation regression models for skewed data with an application to health care costs. J. R. Stat. Soc. Ser. B
**2009**, 70, 1029–1047. [Google Scholar] [CrossRef] - Bono, R.; Blanca, M.J.; Arnau, J.; Gómez-Benito, J. Non-normal distributions commonly used in health, education, and social sciences: A systematic review. Front. Psychol.
**2017**, 8, 1602. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hogg, R.V. Adaptive robust procedures: A partial review and some suggestions for future applications and theory. J. Am. Stat. Assoc.
**1974**, 69, 909–923. [Google Scholar] [CrossRef] - Keselman, H.J.; Wilcox, R.R.; Lix, L.M.; Algina, J.; Fradette, K. Adaptive robust estimation and testing. Br. J. Math. Stat. Psychol.
**2007**, 60, 267–293. [Google Scholar] [CrossRef] [PubMed] - Hogg, R.V.; Fisher, D.M.; Randles, R.H. A two-sample adaptive distribution free test. J. Am. Stat. Assoc.
**1975**, 70, 656–661. [Google Scholar] [CrossRef] - Reed, J.F., III; Stark, D.B. Hinge estimators of location: Robust to asymmetry. Comput. Meth. Programs Biomed.
**1996**, 49, 11–17. [Google Scholar] [CrossRef] - Bonato, M. Robust estimation of skewness and kurtosis in distributions with infinite higher moments. Financ. Res. Lett.
**2011**, 8, 77–87. [Google Scholar] [CrossRef] - Kim, T.H.; White, H. On more robust estimation of skewness and kurtosis. Financ. Res. Lett.
**2004**, 1, 56–73. [Google Scholar] [CrossRef] - Bowley, A.L. Elements of Statistics; Scribner’s Sons: New York, NY, USA, 1920. [Google Scholar]
- Groeneveld, R.A.; Meeden, G. Measuring skewness and kurtosis. J. R. Stat. Soc. Ser. D
**1984**, 33, 391–399. [Google Scholar] [CrossRef] - Moors, J.J.A. A quantile alternative for kurtosis. J. R. Stat. Soc. Ser. D
**1988**, 37, 25–32. [Google Scholar] [CrossRef] [Green Version] - Aytaçoğlu, B.; Sazak, H.S. A comparative study on the estimators of skewness and kurtosis. Ege Univ. J. Fac. Sci.
**2017**, 41, 1–13. [Google Scholar] - Hogg, R.V. On adaptive statistical inferences. Commun. Stat. Theory Methods
**1982**, 11, 2531–2542. [Google Scholar] [CrossRef] - Othman, A.R.; Keselman, H.J.; Wilcox, R.R.; Fradette, K.; Padmanabhan, A.R. A test of symmetry. JMASM
**2002**, 1, 310–315. [Google Scholar] [CrossRef] [Green Version] - D’Agostino, R.B.; Cureton, E.E. A class of simple linear estimators of the standard deviation of the normal distribution. J. Am. Stat. Assoc.
**1973**, 68, 207–210. [Google Scholar] [CrossRef] - Hertsgaard, D.M. Distribution of asymmetric trimmed means. Commun. Stat. Simul. Comput.
**1979**, 8, 359–367. [Google Scholar] [CrossRef] - Wicklin, R. Simulating Data with SAS; SAS Institute Inc.: Cary, NC, USA, 2013. [Google Scholar]
- SAS Institute Inc. SAS/STAT
^{®}14.2 User’s Guide; SAS Institute Inc.: Cary, NC, USA, 2016. [Google Scholar] - Walther, B.A.; Moore, J.L. The concepts of bias, precision and accuracy, and their use in testing the performance of species richness estimators, with a literature review of estimator performance. Ecography
**2005**, 28, 815–829. [Google Scholar] [CrossRef] - Burton, A.; Altman, D.G.; Royston, P.; Holder, R.L. The design of simulation studies in medical statistics. Stat. Med.
**2006**, 25, 4279–4292. [Google Scholar] [CrossRef] - Jung, K.; Panko, P.; Lee, J.; Hwang, H. A comparative study on the performance of GSCA and CSA in parameter recovery for structural equation models with ordinal observed variables. Front. Psychol.
**2018**, 9, 2461. [Google Scholar] [CrossRef] - Koran, J.; Headrick, T.C.; Kuo, T.C. Simulating univariate and multivariate nonnormal distributions through the method percentiles. Multivar. Behav. Res.
**2015**, 50, 216–232. [Google Scholar] [CrossRef] - Sokal, R.R.; Rohlf, F.J. Biometry: The Principles and Practice of Statistics in Biological Research, 4th ed.; Freeman and Company: New York, NY, USA, 2012. [Google Scholar]
- Collins, L.M.; Schafer, J.L.; Kam, C.M. A comparison of inclusive and restrictive strategies in modern missing data procedures. Psychol. Methods
**2001**, 6, 330–351. [Google Scholar] [CrossRef] - Zu, J.; Yuan, K. Standard error of linear observed-score equating for the NEAT design with nonnormally distributed data. J. Educ. Meas.
**2012**, 49, 190–213. [Google Scholar] [CrossRef] - Joanes, D.N.; Gill, C.A. Comparing measures of sample skewness and kurtosis. J. R. Stat. Soc. Ser. D
**1998**, 47, 183–189. [Google Scholar] [CrossRef] - Islam, T.U. Ranking of normality tests: An appraisal through skewed alternative space. Symmetry
**2019**, 11, 872. [Google Scholar] [CrossRef] [Green Version] - Keselman, H.J.; Wilcox, R.R.; Othman, A.R.; Fradette, K. Trimming, transforming statistics, and bootstrapping: Circumventing the biasing effects of heteroscedasticity and nonnormality. JMASM
**2002**, 38, 288–309. [Google Scholar] [CrossRef] [Green Version] - Babu, G.J.; Padmanabhan, A.R.; Puri, M.L. Robust one-way ANOVA under possibly non-regular conditions. Biom. J.
**1999**, 41, 321–339. [Google Scholar] [CrossRef]

Estimators | |||||
---|---|---|---|---|---|

Distributions | Kr1 | Kr2 | Kr3 | Sk1 | Sk2 |

Exponential | 6 | 2.87 | 1.81 | 2 | 4.57 |

Lognormal (ζ = 1 and σ = 0.5) | 5.9 | 2.88 | 1.81 | 1.75 | 2.81 |

Gamma (α = 0.5) | 12 | 3.26 | 1.89 | 2.83 | 9.43 |

Gamma (α = 2) | 3 | 2.71 | 1.78 | 1.41 | 2.81 |

Gamma (α = 4) | 1.5 | 2.64 | 1.77 | 1 | 2.04 |

N = 50 | N = 100 | N = 400 | N = 1000 | N = 5000 | |
---|---|---|---|---|---|

Theoretical Values | Mean (SD) | Mean (SD) | Mean (SD) | Mean (SD) | Mean (SD) |

RB (%) | RB (%) | RB (%) | RB (%) | RB (%) | |

CV (%) | CV (%) | CV (%) | CV (%) | CV (%) | |

SRMSE (%) | SRMSE (%) | SRMSE (%) | SRMSE (%) | SRMSE (%) | |

Kr1 = 6 | 3.531 (3.686) | 4.418 (3.883) | 5.467 (3.277) | 5.759 (2.361) | 5.947 (1.259) |

−41.15 | −26.37 | −8.89 | −4.01 | −0.88 | |

104.37 | 87.88 | 59.95 | 40.99 | 21.18 | |

73.93 | 69.87 | 55.34 | 39.55 | 21.01 | |

Kr2 = 2.87 | 2.799 (0.418) | 2.811 (0.294) | 2.848 (0.151) | 2.86 (0.096) | 2.863 (0.043) |

−2.47 | −2.06 | −0.78 | −0.34 | −0.25 | |

14.94 | 10.45 | 5.31 | 3.36 | 1.51 | |

14.78 | 10.44 | 5.32 | 3.37 | 1.53 | |

Kr3 = 1.81 | 1.795 (0.092) | 1.799 (0.065) | 1.803 (0.033) | 1.805 (0.021) | 1.805 (0.009) |

−0.85 | −0.6 | −0.41 | −0.3 | −0.29 | |

5.14 | 3.63 | 1.85 | 1.17 | 0.52 | |

5.17 | 3.66 | 1.88 | 1.2 | 0.6 | |

Sk1 = 2 | 1.676 (0.614) | 1.811 (0.546) | 1.941 (0.364) | 1.977 (0.247) | 1.995 (0.12) |

−16.21 | −9.47 | −2.93 | −1.17 | −0.27 | |

36.61 | 30.17 | 18.76 | 12.49 | 5.99 | |

34.69 | 28.91 | 18.44 | 12.4 | 5.98 | |

Sk2 = 4.57 | 4.464 (1.408) | 4.48 (0.976) | 4.535 (0.494) | 4.565 (0.313) | 4.567 (0.142) |

−2.32 | −1.96 | −0.77 | −0.11 | −0.07 | |

31.53 | 21.78 | 10.89 | 6.86 | 3.11 | |

30.89 | 21.44 | 10.84 | 6.85 | 3.11 |

N = 50 | N = 100 | N = 400 | N = 1000 | N = 5000 | |
---|---|---|---|---|---|

Theoretical Values | Mean (SD) | Mean (SD) | Mean (SD) | Mean (SD) | Mean (SD) |

RB (%) | RB (%) | RB (%) | RB (%) | RB (%) | |

CV (%) | CV (%) | CV (%) | CV (%) | CV (%) | |

SRMSE (%) | SRMSE (%) | SRMSE (%) | SRMSE (%) | SRMSE (%) | |

Kr1 = 5.9 | 2.815 (3.624) | 3.737 (4.304) | 4.997 (4.263) | 5.475 (3.884) | 5.765 (2.182) |

−52.28 | −36.66 | −15.3 | −7.21 | −2.29 | |

128.7 | 115.16 | 85.31 | 70.93 | 37.86 | |

80.65 | 81.64 | 73.86 | 66.22 | 37.06 | |

Kr2 = 2.88 | 2.812 (0.398) | 2.822 (0.285) | 2.863 (0.146) | 2.872 (0.093) | 2.875 (0.041) |

−2.38 | −2 | −0.59 | −0.28 | −0.16 | |

14.14 | 10.08 | 5.09 | 3.24 | 1.43 | |

14.01 | 10.08 | 5.09 | 3.24 | 1.43 | |

Kr3 = 1.81 | 1.795 (0.088) | 1.799 (0.062) | 1.804 (0.031) | 1.805 (0.02) | 1.805 (0.009) |

−0.82 | −0.58 | −0.35 | −0.29 | −0.29 | |

4.89 | 3.46 | 1.73 | 1.09 | 0.49 | |

4.91 | 3.49 | 1.76 | 1.13 | 0.57 | |

Sk1 = 1.75 | 1.37 (0.654) | 1.513 (0.617) | 1.67 (0.446) | 1.715 (0.338) | 1.74 (0.167) |

−21.73 | −13.54 | −4.58 | −2 | −0.57 | |

47.74 | 40.78 | 26.68 | 19.69 | 9.6 | |

43.23 | 37.77 | 25.87 | 19.39 | 9.57 | |

Sk2 = 2.81 | 2.78 (0.879) | 2.787 (0.613) | 2.801 (0.307) | 2.805 (0.195) | 2.806 (0.086) |

−1.06 | −0.8 | −0.33 | −0.17 | −0.14 | |

31.61 | 21.98 | 10.96 | 6.94 | 3.06 | |

31.29 | 21.82 | 10.93 | 6.93 | 3.06 |

N = 50 | N = 100 | N = 400 | N = 1000 | N = 5000 | |
---|---|---|---|---|---|

Theoretical Values | Mean (SD) | Mean (SD) | Mean (SD) | Mean (SD) | Mean (SD) |

RB (%) | RB (%) | RB (%) | RB (%) | RB (%) | |

CV (%) | CV (%) | CV (%) | CV (%) | CV (%) | |

SRMSE (%) | SRMSE (%) | SRMSE (%) | SRMSE (%) | SRMSE (%) | |

Kr1 = 12 | 6.428 (5.478) | 8.11 (6.264) | 10.556(6.376) | 11.327(5.148) | 11.846(2.751) |

−46.44 | −32.42 | −12.03 | −5.6 | −1.28 | |

85.23 | 77.24 | 60.4 | 45.44 | 23.22 | |

65.12 | 61.45 | 54.48 | 43.26 | 22.96 | |

Kr2 = 3.26 | 3.169 (0.566) | 3.183 (0.39) | 3.237 (0.202) | 3.249 (0.129) | 3.253 (0.057) |

−2.79 | −2.37 | −0.71 | −0.35 | −0.22 | |

17.84 | 12.26 | 6.24 | 3.97 | 1.76 | |

17.57 | 12.2 | 6.23 | 3.97 | 1.77 | |

Kr3 = 1.89 | 1.868 (0.108) | 1.875 (0.076) | 1.881 (0.038) | 1.882 (0.024) | 1.882 (0.011) |

−1.15 | −0.81 | −0.48 | −0.42 | −0.41 | |

5.78 | 4.08 | 2.04 | 1.29 | 0.57 | |

5.83 | 4.12 | 2.09 | 1.35 | 0.7 | |

Sk1 = 2.83 | 2.275 (0.766) | 2.48 (0.721) | 2.716 (0.546) | 2.779 (0.345) | 2.817 (0.193) |

−19.6 | −12.38 | −4.02 | −1.81 | −0.46 | |

33.67 | 29.07 | 20.1 | 12.4 | 6.86 | |

33.42 | 28.32 | 19.71 | 12.31 | 6.85 | |

Sk2 = 9.43 | 9.298 (3.092) | 9.301 (2.494) | 9.402 (1.26) | 9.427 (0.766) | 9.429 (0.351) |

−1.4 | −1.37 | −0.29 | −0.03 | −0.01 | |

33.26 | 26.81 | 13.4 | 8.12 | 3.72 | |

32.82 | 26.48 | 13.36 | 8.12 | 3.72 |

N = 50 | N = 100 | N = 400 | N = 1000 | N = 5000 | |
---|---|---|---|---|---|

Theoretical Values | Mean (SD) | Mean (SD) | Mean (SD) | Mean (SD) | Mean (SD) |

RB (%) | RB (%) | RB (%) | RB (%) | RB (%) | |

CV (%) | CV (%) | CV (%) | CV (%) | CV (%) | |

SRMSE (%) | SRMSE (%) | SRMSE (%) | SRMSE (%) | SRMSE (%) | |

Kr1 = 3 | 1.938 (2.64) | 2.348 (2.517) | 2.771 (1.77) | 2.903 (1.242) | 2.99 (0.613) |

−35.39 | −21.74 | −7.62 | −3.22 | −0.33 | |

136.18 | 107.21 | 63.86 | 42.78 | 20.48 | |

94.84 | 86.68 | 59.48 | 41.53 | 20.42 | |

Kr2 = 2.71 | 2.659 (0.348) | 2.664 (0.239) | 2.693 (0.123) | 2.699 (0.077) | 2.704 (0.035) |

−1.87 | −1.7 | −0.62 | −0.42 | −0.23 | |

13.07 | 8.98 | 4.56 | 2.84 | 1.28 | |

12.96 | 8.99 | 4.57 | 2.86 | 1.3 | |

Kr3 = 1.78 | 1.768 (0.086) | 1.771 (0.06) | 1.775 (0.03) | 1.775 (0.019) | 1.776 (0.009) |

−0.65 | −0.49 | −0.29 | −0.28 | −0.25 | |

4.83 | 3.39 | 1.71 | 1.08 | 0.48 | |

4.85 | 3.41 | 1.73 | 1.11 | 0.55 | |

Sk1 = 1.41 | 1.227 (0.521) | 1.309 (0.43) | 1.381 (0.256) | 1.4 (0.169) | 1.413 (0.079) |

−13.01 | −7.18 | −2.04 | −0.68 | 0.2 | |

42.51 | 32.85 | 18.52 | 12.08 | 5.61 | |

39.2 | 31.33 | 18.26 | 12.02 | 5.62 | |

Sk2 = 2.81 | 2.763 (0.781) | 2.771 (0.539) | 2.794 (0.268) | 2.8 (0.169) | 2.805 (0.077) |

−1.67 | −1.38 | −0.56 | −0.37 | −0.19 | |

28.26 | 19.44 | 9.6 | 6.04 | 2.74 | |

27.84 | 19.22 | 9.56 | 6.03 | 2.74 |

N = 50 | N = 100 | N = 400 | N = 1000 | N = 5000 | |
---|---|---|---|---|---|

Theoretical Values | Mean (SD) | Mean (SD) | Mean (SD) | Mean (SD) | Mean (SD) |

RB (%) | RB (%) | RB (%) | RB (%) | RB (%) | |

CV (%) | CV (%) | CV (%) | CV (%) | CV (%) | |

SRMSE (%) | SRMSE (%) | SRMSE (%) | SRMSE (%) | SRMSE (%) | |

Kr1 = 1.5 | 1.008 (1.836) | 1.226 (1.656) | 1.413 (1.027) | 1.467 (0.733) | 1.489 (0.338) |

−32.79 | −18.29 | −5.8 | −2.23 | −0.71 | |

182.15 | 135.09 | 72.71 | 49.98 | 22.69 | |

126.74 | 111.89 | 68.74 | 48.91 | 22.54 | |

Kr2 = 2.64 | 2.602 (0.301) | 2.606 (0.206) | 2.63 (0.105) | 2.635 (0.067) | 2.638 (0.03) |

−1.45 | −1.28 | −0.38 | −0.18 | −0.08 | |

11.56 | 7.91 | 3.98 | 2.53 | 1.13 | |

11.49 | 7.91 | 3.98 | 2.53 | 1.13 | |

Kr3 = 1.77 | 1.759 (0.081) | 1.762 (0.057) | 1.763 (0.029) | 1.764 (0.018) | 1.764 (0.008) |

−0.65 | −0.46 | −0.39 | −0.36 | −0.35 | |

4.58 | 3.22 | 1.62 | 1.03 | 0.46 | |

4.6 | 3.24 | 1.66 | 1.09 | 0.58 | |

Sk1 = 1 | 0.885 (0.44) | 0.942 (0.348) | 0.983 (0.193) | 0.993 (0.128) | 0.998 (0.058) |

−1.46 | −5.84 | −1.71 | −0.73 | −0.22 | |

49.74 | 36.97 | 19.67 | 12.84 | 5.79 | |

45.51 | 35.3 | 19.41 | 12.77 | 5.78 | |

Sk2 = 2.04 | 2.015 (0.547) | 2.03 (0.374) | 2.031 (0.185) | 2.031 (0.116) | 2.036 (0.052) |

−1.22 | −0.49 | −0.47 | −0.46 | −0.21 | |

27.12 | 18.4 | 9.12 | 5.73 | 2.55 | |

26.82 | 18.32 | 9.09 | 5.72 | 2.56 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bono, R.; Arnau, J.; Alarcón, R.; Blanca, M.J.
Bias, Precision, and Accuracy of Skewness and Kurtosis Estimators for Frequently Used Continuous Distributions. *Symmetry* **2020**, *12*, 19.
https://doi.org/10.3390/sym12010019

**AMA Style**

Bono R, Arnau J, Alarcón R, Blanca MJ.
Bias, Precision, and Accuracy of Skewness and Kurtosis Estimators for Frequently Used Continuous Distributions. *Symmetry*. 2020; 12(1):19.
https://doi.org/10.3390/sym12010019

**Chicago/Turabian Style**

Bono, Roser, Jaume Arnau, Rafael Alarcón, and Maria J. Blanca.
2020. "Bias, Precision, and Accuracy of Skewness and Kurtosis Estimators for Frequently Used Continuous Distributions" *Symmetry* 12, no. 1: 19.
https://doi.org/10.3390/sym12010019