Constructing multivariate distributions for soil parameters - Risk and Reliability in Geotechnical Engineering - page 14

It is easy to verify that the best estimates are

n n

1 1

∑ ∑

( ) =

2

() k () k

µ = Y = 101 84 . σ = Y − µ 22 60 . (1.17)

MLE MLE MLE

n n

k = 1 k = 1

Note that m = μ MLE and ( n − 1) s 2 = n σ MLE 2 . This relation is not true for other distribution
types. In fact, in many cases, there are no closed-form solutions to Equation 1.17 and opti-
mization is needed.

1.2.3.4 Normal probability plot
The method of moments, percentile method, and maximum likelihood method provide only
point estimates of μ and σ. They do not address the more basic question as to whether Y
is normally distributed in the first place. The normal probability plot compares the ECDF
F n ( y ) of the data with the theoretical normal CDF, F( y ). The concept is relatively simple as
explained below. Let us say you suspect Y to be normally distributed with the unknown
mean, μ, and standard deviation, σ. Then the CDF of Y must be

y − µ

F( y = Φ (1.18)

σ

and

y − µ

()

Φ −

1

= (1.19)

F y

σ

If Y is indeed normal, it is clear from Equation 1.19 that a plot with y values on the
vertical axis and Φ −1 [F( y )] on the horizontal axis will produce a straight line with the ver-
tical intercept, μ, and gradient, σ. In summary, linearity of the plot implies that the nor-
mal hypothesis is reasonable. Equation 1.19 also provides another method to estimate μ
and σ!
Table 1.4 illustrates how this method can be applied to simulated normal data of size = 10,
μ = 100 kPa, and σ = 20 kPa. Again, we initialized by randn('state', 13) before executing
normrnd(100, 20, 10, 1). The first column contains the simulated data sorted in an ascend-
ing order. The second column contains the rank. The third column computes the ECDF
using Equation 1.12 . The fourth column is obtained by applying the inverse of the stan-
dard normal CDF to the third column. Finally, the normal probability plot is obtained by
drawing the first column on the vertical axis and the fourth column on the horizontal axis.
Linear regression can be used to fit a straight line to the normal probability plot as shown in
Figure 1.9 to get estimates for μ and σ from the vertical intercept and gradient, respectively.

1.2.3.5 Statistical uncertainties in the μ and σ estimators
1.2.3.5.1 Sampling distributions and confidence intervals
Let us denote the sample mean and sample standard deviation in Equation 1.13 by m and
s, respectively. For the sample mean m, its sampling distribution is a normal distribution
with mean = μ and standard deviation = σ/ n 0.5 , where n is the number of data points.