[R] lmom - Estimating Normal Distribution Parameters using lmom package

[J. R. M. Hosking]

Maithili Shiva wrote:
> Dear R helpers,
>  
> I have a data of 2102 observations (consisting of 0's also), to which I am trying to fit Normal distribution using "lmom" pacakage. If I use Excel, its easy to estimate the parameters of Normal distribution as simple mean and standard devaition. The results I get if I use teh excel are as 
>  
> Parameters of Normal distribution :-
>  
> Mean = 22986.44 and standard deviation = 223452.88
>  
> However, if I use the R code using the lmom package, I get the mean as 22986.39 and standard deviation as 39029.79.
>  
> Regards
>  
> Maithili
>  
> My R code is as follows. (Actually its a two line code, but since I am representing data using 'c', the code looks too big.)
>  
>  
> library(lmom)
>  
> amounts <-
>  c(0,0,18561.9,0,0,0,34400,0,0,0,0,2190,0,0,0,0,60000,0,0,0,19583,0,0,0,109872.87,0,0,0,0,0,0,1244,0,0,25150,0,500,0,0,0,0,0,0,0,100,0,0,0,0,0,0,500,41533.94,1365,0,0,11400,0,0,0,0,0,10000,0,0,11000,0,0,0,0,0,0,11600,0,0,0,21530,0,2000,0,10100,4500,5000,0,0,10000,0,28667,0,0,0,45000,0,0,0,0,0,100,0,0,2100,0,0,0,1000,0,0,0,0,17000,0,0,0,0,0,0,0,0,140270,2000,0,1900678.25,19450,0,0,4400,0,0,0,6136,0,0,0,0,0,0,0,0,0,0,0,0,0,20900,0,0,525,8306,0,0,0,0,0,9497,0,291264,0,0,0,0,0,2825,0,0,0,0,0,75000,0,0,0,0,0,6000,4300,3062,0,0,159649,0,0,61329,0,0,0,0,0,0,0,0,0,214816,0,0,0,0,0,0,0,1200,0,0,10364,0,0,0,300,0,0,0,0,0,156888,0,0,0,0,0,0,0,0,0,0,0,0,0,200,0,1164.55,0,0,0,0,0,0,0,0,540,0,0,0,460.52,0,0,0,0,0,0,0,0,0,0,0,500,0,0,0,0,540,500,15000,0,0,0,0,6400,0,0,0,2900,7200,0,0,0,0,400,0,800,0,500,0,0,0,0,0,0,13550,0,0,40410,100,0,0,0,0,5818,50700,0,0,0,0,0,0,0,0,4800,0,0,0,0,0,0,0,0,0,0,0,500,0,0,0,0,0,0,0,0,0,0,0,0,0,133.29,0,5750,100,0,0,0,0,0,0,0,2116,3165.7
> 4,0,0,14554,2700,0,151869,0,0,0,0,6400,0,0,15827.73,0,0,0,0,0,0,0,235607.56,0,0,0,0,225.65,0,0,0,0,725.04,0,0,0,0,151869,0,0,0,0,0,0,0,46800,0,0,0,0,0,3520,50000,0,0,0,0,2790,0,0,800,0,0,0,0,0,0,0,0,156.66,0,0,0,0,0,0,2200,0,357,0,283205,3466.26,0,503875,0,328681.27,0,0,0,0,1000,3600,12050,1000,0,0,0,0,0,0,0,0,0,0,0,0,121.44,0,1485,0,0,5100,0,937675,0,0,0,0,0,356.87,0,12923.56,9576,0,0,207879,0,0,0,1989,0,0,10233,207.55,1322,0,0,0,0,2320.38,0,0,6440,6111,82463,0,0,0,132.84,0,0,0,0,0,0,96161.74,0,0,0,0,0,0,271.16,0,0,0,0,225.83,0,0,0,0,0,0,0,0,0,17398,0,0,0,0,0,0,0,0,0,0,0,260.61,0,0,0,0,0,0,0,0,412.14,0,0,0,0,102.21,170420,29465,0,0,0,0,0,20819.21,0,10056,26200,0,2975.81,7199.83,0,0,0,2650,0,0,0,0,0,0,0,0,101.57,9000,0,0,105,0,774099.69,0,235.28,0,247.63,0,0,0,0,25761.56,13483,0,0,170.72,0,0,137.3,180.02,0,17555,0,0,0,468.29,0,0,0,154.51,0,0,11200,0,0,0,0,0,130.89,0,3927.38,0,0,0,0,0,1307.78,0,0,2869.32,1642.74,0,0,0,0,401.41,0,0,0,0,12503.86,10366.19,0
> ,124358,0,0,37953.17,0,1009.74,0,12110,1046.9,0,5610,3118.39,0,5682.04,0,0,0,1905.77,7707.59,0,3264.68,0,797.7,0,2371.42,0,0,7279.16,1093.15,0,0,1066510.66,8979.86,2989.93,129.92,0,1095.61,0,1125.01,20499.51,2240.99,0,0,3353.65,0,0,1129.23,0,2155.72,4000,800.56,0,0,0,1736.44,5584.1,1899.55,1334.25,239925,200768.55,560.61,11037,4739.74,1953.09,174.18,0,112.53,0,0,831.75,0,800.78,0,23877.68,0,1235.74,950,73796.05,9065.76,0,828.12,0,112.12,94637.19,0,1565.34,0,0,27121.17,53940,84872.23,0,0,0,0,0,0,116.83,0,0,0,0,114.17,0,0,886.28,5820,0,0,0,0,4888,0,0,0,0,0,1138.89,0,621.47,177513.55,0,531.32,50000,0,0,897,0,0,0,0,0,20000,0,0,0,188474,0,743.24,0,0,958.16,0,0,12321,561.36,0,0,1947.76,0,4262.85,2478632.75,0,0,0,4671.33,0,0,2985.59,0,0,0,0,0,0,0,0,0,0,10952.18,0,0,0,63505.07,5656.89,0,1609.95,0,0,1267,0,6355.59,1350,1708.71,0,951.67,0,0,0,0,908.81,0,0,0,0,0,0,1130.1,0,0,0,4453.55,0,0,12394,0,0,0,0,2886.8,0,0,81147,0,0,0,0,13958.46,1440,112453.28,11800,0,0,0,5
> 00,2399.96,0,0,2953,0,2000,0,0,0,6288.88,1375.95,13093,38726.88,0,122.8,0,2455.27,0,233549.36,0,0,0,0,0,0,0,0,4906.52,0,0,0,26160,0,309.22,0,0,0,0,0,21500,7257.68,0,0,0,18069,0,23625,0,28673.9,0,0,200000,0,0,5031.4,1096.59,0,0,836.39,8818,0,0,0,0,0,227.65,48775.83,0,0,124.8,3870.2,0,2596.91,203.44,0,0,0,0,0,352643.58,0,792.92,5000,0,417.93,0,0,0,1900,0,0,309.41,0,0,0,214.71,0,405.84,0,0,0,0,0,324.63,133.96,0,0,806.64,500,245258.43,210,0,0,0,0,0,2638.06,0,0,0,0,34914,32571,0,0,0,0,0,42925.37,1979.87,1667,0,7546.64,0,0,
> 0,0,0,500,1667,3079877.63,0,0,0,186500,0,201.18,0,0,392.88,0,0,8780.5,20424..14,272284.9,0,10822,2871.27,0,0,0,1450,0,0,28,4164,0,1571.74,0,0,839.56,259278.95,12880,0,0,319,0,7287,0,0,0,0,0,0,0,110,0,40968.56,442,0,0,0,255,0,168.57,0,2500,0,0,375652.22,2222.12,0,0,0,0,9485,1332,0,116.02,16552.68,0,0,8700.94,0,0,0,2587.5,50445.97,0,257.09,0,0,0,0,2118.77,0,0,0,3239.59,0,7122756.78,0,0,0,0,1300,0,0,0,0,0,0,600,1250,0,0,0,0,400,180,0,0,0,0,3592,0,0,0,0,0,20350,0,5030,203371,1850,0,1000,0,0,100,4290.71,87400,3400,0,0,200,0,0,600,5000,1050,0,0,0,0,1500,0,11838,0,700,18005,229183.22,0,9753,200,0,5059..74,3784,10638,1850,0,500,0,0,3000,8330,600,1480,0,0,0,38508,500,0,0,1000,0,0,0,46527,700,979470.35,500,0,2500,0,100,6964.04,0,500,0,0,100,0,650,12150,0,0,2643,100,1230,0,10000,2523,0,0,400,182279.03,0,0,0,0,37678.09,900,0,1600,600,500,0,1310,500,100,2400,0,700,0,0,0,0,500,1382.5,0,0,230,0,0,0,500,0,0,1200,100,0,134173,19550,3300,0,0,59928.59,9900,0,7100.13,8600,0,
> 3658,100,1000,3000,0,200,0,2750,27611,26700,500,0,1267,0,26818.54,0,4625,1020,0,0,0,0,200,6726,982,2500,200,0,1200,1600,29552,1000,16500,0,0,0,1000,1130,600,0,0,0,2300,0,1000,2050,0,18000,5900,716.53,0,0,0,4875.79,2130,0,3000,9675,9351,0,0,141550,0,17885,4052.74,17893.81,2325,47308,16034,0,0,11626,310,450,2300,300,0,100,0,203383.5,118.11,5500,100,0,4100,25100,3215.39,2650,2200,11702,0,500,98911.38,6450,203,500,3563,0,118110,8975,53173.2,5080,161.51,100,0,0,500,1226,100,1000,0,0,7100,844.32,4390,0,900,3057,0,0,1100,3561,19263,1150,0,0,0,0,1000,0,138905,0,0,41921,1502.13,26990,11484.89,0,1000,0,2146,0,2500,1017,500,6500,200,50100,1428,1100,730,19080,1000,0,239257.68,22546.07,5814.02,0,0,2000,0,300,18936.16,3656.1,4934167.89,77926.55,0,0,600,303210,6445,550,16550.75,4339.1,0,57115.03,0,3773.12,2173.38,5029.74,32822.68,0,500,12140.41,12890,20000,2500,900,0,0,19475,3924,13706.82,18574,500,0,1162361.91,0,1000,4416.24,11629.2,500,0,0,100,2980,6475.6,0,802573.7
> ,0,3550,30377.5,12810,500,0,0,0,2800,12490,834,1000,300,0,0,2290,0,9689.7,0,300,22260,0,0,20450,0,100000,100,5217.7,0,766.37,2200,523.9,1000,0,0,0,0,5469.94,0,5800,1750,500,0,0,3000,1100,0,5400,107.3,0,1294.34,20684,0,14740,589726.02,0,0,1800,3200,3700,0,20000,8000,0,1000,794.1,120,31214.16,7505.69,7070.71,0,9787.46,27858.04,3400,106.3,1426.97,2500,0,2634.91,2500,2500,8570..85,41168.6,262.63,0,0,61052,500,331306,231505,2327.06,0,1500,1993.87,0,23000,17494.64,715.11,0,3650,1500,0,24375.77,0,1102.18,0,612.28,123902.53,0,5220,0,0,0,17170.19,500,12378.1,2325.21,127.77,7310.31,0,197044.03,211.28,600..15,500,12832.09,1021.41,0,3275.95,0,0,204200,3500,717.33,0,0,283.66,379.78,900,5453.36,4906.52,0,5100,1774.33,2600,30046.35,25344.1,347.26,0,600,18500,6070.8,27200,9237.34,2000,0,0,0,2449.83,1314.1,2190.26,900,0,0,1105.31,55008.81,699.68,16248.34,3814.63,0,2050,1786.64,3090.3,7020.92,1494.15,170.14,0,20100,0,169.36,200741.43,103825.98,11100,0,501.45,46993.69,3403.
> 67,1727.96,2627.81,1001.9,0,2903.86,117.06,0,12002.84,2169.79,700,0,16006.42,1152.92,1316.44,824.22,502300,902.73,0,11547.47,142.01,205,3547,175.69,7000,0,2008.14,0,5456.13,500,156.97,12500.32,0,7852,12200,0,4954.39,1246.83,116.41,0,787.95,1611.2,212.82,794.99,39054.92,1417.08,0,630.93,500,115.7,2699.66,389.24,100,0,5793.59,2000,193.98,14100,10436.69,320,0,2173.94,6500,0,3698.53,610,0,0,167.69,2586.19,26080,1044,0,10370.71,0,1000,0,3002.48,368,26522.27,1860.7,0,0,4000,12383.9,3100,1145.22,176.12,0,7900,5178.7,1643.57,2505.25,267.16,11116.5,0,1988.38,2100,0,1577.42,401.46,0,0,0,3815.61,117.9,1041.99,249.13,0,0,454.89,54482.18,193352.4,0,14775.57,12000,0,605.14,158.57,3450,734816.68,200,33780.05,0,837.08,2980,4063.54,400.18,1761.91,0,0,300,1882.82,3601.6,51741.94,668.69,1126.85,0,100,25552.11,109081.43,16951.1,2653.36,373.18,0,3800,2024.48,8259,107.07,4190,183.6,0,0,13290.22,3258.9,4480,300,0,0,1948.43,85932.57,5361.34,200,0,208.32,0,2400,19229,138067,248
> 499.2,1500.49,34514,0,557345.18,0,1716.28,1686.29,667.35,1400,1000,1000,3754.95,1001.19,752.45,4762.14,1000,0,0,270.11,639.87,3245.39,400,924.08,0,1500,226.21,3281.6,1169.6,6466,0,0,1808.94,320.79,0,0,2089,100,1400,400,625.03,111.2,668.76,1144.52,274.06,0,380.15,3529,0,2626.79,3498.75,829.76,0,36884..17,44650,84357.17,15417.36,39612.48,9297.96,0,8716.95,7428.41,38019.06,12908.65,139072.48,20049.35,500000,2654.55,371.62,1718,5000,1000,403.28,0,500,0,125.34,8841,14543.77,151.03,0,105.95,30293.83,100,1995.35,131.66,0,0,15121.11,10146.3,2740.96,1342.92,1302.66,314.91,0,280.25,0,2934,387.61,152.48,5378.71,1350,1853.93,158.98,1390.74,5328.39,1770,12165,0,500,500,1960,9500,4425,212172.95,0,0,3137.88,0,510,1000,788.84,0,7628,1218,6500,8913.28,0,0,0,3500,21850,0,3700,0,300,0,0,0,7107.32,1800,0,16951.55,0,6244.62,250,2340,8104,1400,2500,0,1630.99,24239.29,25060.16,4100,4820,250,0,72579,0,0,18409.85,250,6498,0,548,1100,0,500,617,700,0,11921.75,200,113528.31,5500,0,
> 163813,
> 0,15990,6933.03,2500,100,200,1233542.42,0,400,0,0,0,0,0,0,0,0,0,300,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,400,0,33928,2360,2510,0,177000,0,12490,500,0,1750,0,2100,600,0,13139.58,115777,1400,0,5800,4000,0,1500,1608,11199,14050,42295.49,10900,0,76688,204400,8947.2,4900,4850,3300,0,2000,0,3000,1650,5400,4354,0,0,10070,22457.95,0,100,0,0,4000,1100,1500,0,3992.83,19100,0,4626.65,500,17332)
>  
>  lmom                          <- samlmu(amounts); lmom
>  
>  parameters_of_Normal  <- pelnor(lmom); 
>  
> parameters_of_Normal
>  
>  
>  
> Please guide as o why there is so much of a difference in standard deviation i.e. .

The sample standard deviation is one estimate of the sigma parameter
of the normal distribution.  The value returned by pelnor() is another.
There is no guarantee that the two estimates will be the same.  For
data that are so skew that fitting a normal distribution is grossly
inappropriate, they can be very far apart.  Your data, which have
sample skewness 22 and sample L-skewness 0.93, fall into this category.


J. R. M. Hosking