# 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  
[PERSON] [25] proposed a method of estimating Gumbel distribution parameters based on order statistics, requiring the sample elements \(Z_{i}\) to be placed in ascending order. The estimators of the investigated distribution parameters take the form of a linear function with respect to positional statistics:  
\[\hat{b}_{(n)}=\frac{1}{\hat{\alpha}_{(n)}}=\sum\limits_{i=1}^{n}b_{i}Z_{i},\ \hat{U}=\sum\limits_{i=1}^{n}a_{i}Z_{i} \tag{9}\]  
where: Best Linear Unbiased Estimators (BLUE) are with minimal variance. The coefficients \(b_{i},\ a_{i}\) were collated in the paper [25] for sample size of \(n=1,2,\ldots,16\). The aforementioned work also describes a method of estimating parameters \(b,\ U\) for arbitrarily large sample with a size of \(n\) having given coefficients for an \(m\)-element sample if \(m<n\). For an \(n\)-element sample, the coefficients \(b_{i}^{\prime},\ a_{i}^{\prime}\) can be calculated using the following formulas:  
\[ \begin{array}{c}b_{i}^{\prime}=\sum\limits_{j=1}^{m}b_{j}\cdot(j/i)p(n,m,i, j),\\ i=1,2,\ldots,n\\ a_{i}^{\prime}=\sum\limits_{j=1}^{m}a_{j}\cdot(j/i)p(n,m,i,j),\end{array}  \tag{10}\]  
where:  
\[p(n,m,i,j)=\left( \begin{array}{c}i\\ j\end{array} \right)\left( \begin{array}{c}n-i\\ m-j\end{array} \right)/\left( \begin{array}{c}n\\ m\end{array} \right) \tag{11}\]  
is the hypergeometric probability function. The formulas below allow the calculation of \"good\" estimators \(\hat{b}_{(n)}^{\prime},\ \hat{U}_{(n)}^{\prime}\) for an \(n\)-element sample:  
\[\hat{b}_{(n)}^{\prime}=\frac{1}{\hat{\alpha}_{(n)}^{\prime}}=\sum\limits_{i=1} ^{n}b_{i}^{\prime}\cdot Z_{i},\ \hat{U}_{(n)}^{\prime}=\sum\limits_{i=1}^{n}a_{i}^{\prime}\cdot Z_{i} \tag{12}\]  
The samples analyzed in this work contain significantly more than 16 measurement results; therefore, a set of BLUE coefficients \(b_{j}\), \(a_{j}\) for \(m=16\) was used in the calculations. The evaluation of the results obtained with four methods proved to be more difficult than the estimation of the Gumbel distribution parameters. As we have four Gumbel distributions, we face the question which of them best approximates the empirical distribution and can be used to determine the characteristic value \(Z_{k}\) of the location of the zero isotherm in the ground [26]. The test statistics of chi-squared, Kolmogorov-Smirnov and Cramer von Mises were adopted as a measure of adjustment of the theoretical to the empirical distribution. Low values of these statistics indicate that the given theoretical distribution truly reflects the empirical distribution. Amid the known compliance tests, the chi squared test is mentioned first. It requires the decomposition of the \(Z\) axis into \(r+1\) intervals \(I_{1},\ I_{2},\ \cdots,\ I_{r+1}\) by the numbers \(-\infty=g_{0}<g_{1}<\cdots<g_{r}<g_{r+1}=\infty\). The difference in the value of the cumulative distribution function \(F\) at the ends of the interval is equal to the probability \(p\) that the random variable \(Z\) will take a value belonging to that interval:  
\[p_{j}=F\big{(}g_{j}\big{)}-F\big{(}g_{j-1}\big{)},\ j=1,2,\ldots,r+1 \tag{13}\]  
The chi squared test statistic for a measurement sequence consisting of \(n\) maximum freezing depths is given by the formula:  
\[\chi^{2}=\sum\limits_{j=1}^{r+1}\frac{\big{(}n_{j}-np_{j}\big{)}^{2}}{np_{j}} \tag{14}\]where: \(n_{j}\)--the number of measured values of \(Z\) within \(I_{j}\), while \(np_{j}\)--is the expected number of measurement results which, according to the considered distribution, should be in \(I_{j}\)[29]. An examination of distribution compliance by means of statistics (14) leaves the choice of the number of intervals \(I_{j}\) into which the axis \(Z\) is divided and the points \(g_{j}\) constituting the limits of these intervals. In this work because of the ambiguity mentioned chi-squared test was implemented in two variants.