Question

Exercise 2 Consider the following observations. 1261,11972,3664,1474, 3590, 11171,25,1697,3571,2324, 367,1747,377,1640, 1543,2305,935,6854,1391,3854 11875,1268,444,933,47076,120,381587,356,988,136 8999,7543,2893,562,10729,4879,2264,5852,646,3571, 3405,501,2699,228,5319,488,411,99543,150,2775, 6098,269, 113,1342,346,797,70,727,4018,4788, 438,7563,992,370,46,327,7098,1073,2397,1134, 2358,32561,10984,38,4863,476,1630,173,17534,908, 316,101,430,11916,140,774,6620,1019,53,31, 902,116,1075,16513,143,56,2005,8971,217,4854 1. Take the logarithms of the data and give the corresponding histogram. It looks like the logarithm of the data follows a normal distribution which implies that the original data follow a lognormal distribution. Estimate the parameters of the lognormal distribution and perform goodness-of-fit tests 102 Exercise 2 Consider the following observations. 1261,11972,3664,1474, 3590, 11171,25,1697,3571,2324, 367,1747,377,1640, 1543,2305,935,6854,1391,3854 11875,1268,444,933,47076,120,381587,356,988,136 8999,7543,2893,562,10729,4879,2264,5852,646,3571, 3405,501,2699,228,5319,488,411,99543,150,2775, 6098,269, 113,1342,346,797,70,727,4018,4788, 438,7563,992,370,46,327,7098,1073,2397,1134, 2358,32561,10984,38,4863,476,1630,173,17534,908, 316,101,430,11916,140,774,6620,1019,53,31, 902,116,1075,16513,143,56,2005,8971,217,4854 1. Take the logarithms of the data and give the corresponding histogram. It looks like the logarithm of the data follows a normal distribution which implies that the original data follow a lognormal distribution. Estimate the parameters of the lognormal distribution and perform goodness-of-fit tests 102

Answer 1

Take the log of all the data points to obtain the following table:

Logs
4.074634	3.954194	3.532117	3.785187	2.641474	3.372544	2.499687	2.955207
3.103119	3.877544	2.699838	2.429752	3.878694	4.512698	2.004321	2.064458
2.647383	3.461348	3.431203	2.053078	2.996512	4.040761	2.633468	3.031408
2.969882	2.749736	2.357935	3.127753	2.568202	1.579784	4.07613	4.217826
4.6728	4.030559	3.72583	2.539076	1.662758	3.686904	2.146128	2.155336
2.079181	3.688331	2.68842	2.901458	2.514548	2.677607	2.888741	1.748188
4.581574	3.354876	2.613842	1.845098	3.851136	3.212188	3.820858	3.302114
2.55145	3.767304	4.998011	2.861534	3.0306	2.238046	3.008174	3.952841
2.994757	2.810233	2.176091	3.60401	3.379668	4.243881	1.724276	2.33646
2.133539	3.55279	3.443263	3.680154	3.054613	2.958086	1.491362	3.6861

Now, using bins of size 0.5 from 0 to 5, we get the following histogram for the data in Excel (Data -> Data Analysis -> Histogram, choose the above table data range as input and these bins as the bin range):

Histogranm 25 20 2 15 Frequency Bin

The logarithm of the data indeed looks like following a normal distribution as is evident from the shape of the curve above.

This implies that the original data given follows a lognormal distribution.

Let's call the random variable generating the log data above Y and the original data X.

Calculating the mean of X from data, we get Mean ^μ = 3.067, and Std Dev σ = 0.796

Hence, X ~ N(μ, σ²) ~ N(3.067, 0.796²)

$\because$ Y is lognormal, Y has Mean μ_Y = e⁽
u 1/202
⁾ = e^{(3.067 + 1/2*0.796^2)} = 29.48

And, Variance, σ_Y² = (e^{($\sigma^2$)} - 1) * e
^{1 2}

= (1.884 - 1) * 869 = 768.25

$\therefore$ Std Dev, σ_Y = 27.72