## 2 A Genetic Algorithm Markov Chain and Related Preliminaries
The weights in this weighted average are assigned by generating a random number \(\varphi\)\(\in\)(0, 1) and averaging respective genes from the parents using the following formula:  
\[\theta_{ci}=\theta_{ai}\times\varphi+[(1-\varphi)\times\theta_{bi}]. \tag{11}\]  
In Equation (11), the higher-fitness base parent is assumed to be the parent \"\(a\)\" with genes \(P_{a}=[\theta_{a1}\)..., \(\theta_{az}]\). There are some advantages of arithmetic crossover. First, arithmetic crossover produces gene values in the child that are weighted averages of the corresponding gene values of the parents. Second, these values are not random but are generated within the upper bounds of the respective gene values of both parents. Additionally, there are no biases in the arithmetic crossover, as all genes have an equal probability of crossover, and the weighted average improves the precision of parameters when parents have somewhat similar genes. As an example of bias, the first gene of parent \(a\) and the last gene of parent \(b\) will always be present in child \(c\) when a single-point crossover is used because the crossover point will always include the first gene and retain the last gene in the child. This situation is eliminated in arithmetic crossover, since the decision vector may contain one in the first and last gene locations. Let \(Q^{*}\) be the population at a generation where steady-state distribution is achieved with a desired level of accuracy. The steady-state population \(Q^{*}\) is a sample generated from an unknown non-parametric distribution whose likelihood function is maximized using the GA's minimizing least-square error fitness function. Using \(Q^{*}\) as a sample for posterior distribution, the sample with regression parameters shown in Equation (4) can be obtained by multiplying matrix \(Q^{*}\) by a scalar \(\lambda\). Once the resulting matrix is obtained, non-parametric distribution confidence intervals for regression parameters can be established. Earlier, it was assumed that the value of scalar \(\lambda\) was known and was assumed to be a constant. There are two ways to determine the value of \(\lambda\). In the first sequential search  
Figure 1: Two crossover operations. approach, an initial value of \(\lambda=10\) is selected, GA experiments are run, and the population member with the best fitness is found. Using the genes of this population member, the value of \(\lambda\), and Equation (5), the regression function in Equation (4) is estimated, and its root-mean-square (RMS) error on the dataset is computed and stored for comparison. Next, the value of \(\lambda\) is incremented by 1 to the next value, and the procedure is repeated for a new value of \(\lambda\) and a new value of RMS is computed. If the RMS for the current value (i.e., \(\lambda=11\)) is better than the previously stored RMS value (i.e., \(\lambda=10\)), then the current value of RMS is stored for comparison and \(\lambda\) is incremented by \(1\) to the next value (i.e., \(\lambda=12\)), and a similar procedure is repeated. Otherwise, the previous value of \(\lambda\) is selected as the final value of \(\lambda\). The second approach is a direct approach that benchmarks the value of \(\lambda\) by using the results from traditional parametric statistical regression function parameters. Let these parameters from the traditional parametric statistical regression function be represented as a vector \(\left[\beta_{1}^{{}^{\prime}},\ldots,\beta_{n+1}^{{}^{\prime}}\right]^{\text {T}}\); then, the value of \(\lambda\) is computed using the following expression:  
\[\lambda=max\Big{(}\Big{|}\beta_{1}^{{}^{\prime}}\Big{|},\ldots,\Big{|}\beta_{ n+1}^{{}^{\prime}}\Big{|}\Big{)}+1. \tag{12}\]  
The second approach has one additional benefit that helps speed up the convergence of the GA procedure.