# Probabilistic Analysis as a Method for Ground Freezing Depth Estimation
## 2 Material and Methodology
### Estimation of Gumbel Distribution Parameters
#### 2.3.4. Lieblein Method--BLUE
The first one assumes the division of the \(Z\) variable axis into one-element intervals, the limits of which are arithmetic means of two values adjacent in the order statistics. The only exception here are points \(g_{0}\), \(g_{r+1}\) equal to minus and plus infinity, respectively. In the second variant, the division into five intervals with equal probability \(p_{j}\) was used. Point \(g_{1}\) was set as the arithmetic mean of the fifth and sixth elements in a series of ascending order data: \(g_{1}=(Z_{5}+Z_{6})/2\). The value of the cumulative distribution function at point \(g_{1}\) is equal to the probability of the first interval: \(F(g_{1})=p\). The same probability of all intervals is ensured by the next boundary points \(g_{j}\) being defined as:  
\[g_{j}=U-\frac{1}{\alpha}\ln\bigl{[}-\ln\bigl{(}p+F\bigl{(}g_{j-1}\bigr{)} \bigr{)}\bigr{]},\ldots j=1,2,\cdots,r \tag{15}\]  
except the last: \(g_{r+1}=\infty\). Other tests refer to hypothetical cumulative distribution function and verify its compliance with empirical cumulative distribution function. Kolmogorov-Smirnov statistics \(D_{n1}\) verifies the quality of the adjustment on the basis of the largest absolute value of the hypothetical \(F(Z)\) and empirical \(F_{n}(Z)\) cumulative distribution function difference:  
\[D_{n1}=\sup_{z}\lvert F_{n}(Z)-F(Z)\rvert \tag{16}\]  
In Formula (16), unlike in [30], the \(\sqrt{n}\) factor is omitted, because only equally numerous sets of values \(F_{n}(Z)\) and \(F(Z)\) are compared with each other and without any reference to samples of a different size \(n\). Another variation of Kolmogorov-Smirnov statistics found in the literature was also used, namely:  
\[D_{n}^{+}=\max_{1\leq i\leq n}\left\lvert\frac{i}{n}-F(Z_{i:n})\right\rvert,\ D_{n}^{-}=\max_{1\leq i\leq n}\left\lvert F (Z_{i:n})-\frac{i-1}{n}\right\rvert \tag{17}\]  
from where:  
\[D_{n2}=\max\bigl{\{}D_{n}^{+},\ D_{n}^{-}\bigr{\}} \tag{18}\]  
The above Formulas (16)-(18) define the maximum discrepancy between the postulated Gumbel distribution and the measured depths of the zero isotherm location. So, the idea arose to do a similar check for medium differences of hypothetical and geometrical cumulative distribution function. The subsequent tests are equivalent to statistics (16)-(18) and check not the maximum, but the average distance of the hypothetical and empirical cumulative distribution function. According to this reasoning, statistics (16) representing the average spacing of the cumulative distribution function is given by the formula:  
\[D_{n1s}=\sum_{i=1}^{n}\frac{\lvert F_{n}(Z_{i})-F(Z_{i})\rvert}{n} \tag{19}\]  
Whereas the equivalent of statistics (18) in relation to the average value is expressed by the formulas:  
\[D_{n}^{+}=\frac{1}{n}\sum_{i=1}^{n}\left\lvert\frac{i}{n}-F(Z_{i:n})\right\rvert,\ D_{n}^{-}=\frac{1}{n}\sum_{i=1}^{n}\left\lvert F(Z_{i:n})-\frac{i-1}{n}\right\rvert \tag{20}\]  
and next:  
\[D_{n2s}=\frac{1}{2}\bigl{(}D_{n}^{+}+D_{n}^{-}\bigr{)} \tag{21}\]Another test used in this work, comparing the cumulative distribution function with order statistics of measurement results, is [PERSON]'s statistics [31] in the form:  
\[p_{j}=F\big{(}g_{j}\big{)}-F\big{(}g_{j-1}\big{)},\ j=1,2,\ldots,r+1 \tag{13}\]0  
The best estimation method was considered to be the one indicated by most of the mentioned tests. For example, if the chi squared statistic is the smallest for the Gumbel distribution parameters \(\alpha,U\) determined by the maximum likelihood method, it means that the chi squared test indicates the maximum likelihood method as the best of four applied methods.