Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 11

(a) 1 (b) 1

eoretical CDF eoretical CDF

0.9 0.9

Empirical CDF ( Equation 1.11 ) Empirical CDF ( Equation 1.11 )

0.8 0.8

0.7 0.7

0.6 0.6

0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1
0 0

0.1
0 0

50 100 150 200 50 100 150 200

x x

n = 10 n = 100

Figure 1.7 Theoretical and ECDFs of Y.

from data almost uniquely. Different estimators do produce different curves, but the effect
is rather minor (Kimball 1960). In contrast, the histogram can vary significantly depending
on the number of bins as shown in Figure 1.4 . It is noteworthy that goodness-of-fit tests are
based on the CDF.

1.2.3 estimation of normal parameters
There are two parameters in the normal distribution model: μ and σ. There are various
ways of estimating these two parameters based on the observed data. Here, we review point
estimators for μ and σ using the (a) method of moments, (b) percentile method, and (3)
maximum likelihood method.
The calculation procedure and behavior of each estimator are illustrated using simulated
normal data of size n = 10, μ = 100 kPa, and σ = 20 kPa. We illustrate all three methods
using the same simulated set [initialized by randn('state', 13) before executing normrnd(100,
20, 10, 1)]. The sample values are shown in Table 1.2 .

Table 1.2 Sample values of Y ( n = 10)
Simulated data Y (k) Index Sorted Y data k Rank

124.06 1 71.43 1

113.35 2 74.40 2

134.75 3 81.15 3

90.52 4 89.81 4

133.54 5 90.52 5

74.40 6 105.41 6

81.15 7 113.35 7

105.41 8 124.06 8

71.43 9 133.54 9

89.81 10 134.75 10