Bayen, in Python Programming and Numerical Methods, 2021 14.1.2 Vectors Assume then that Var Y = d A 2 σ 2, where d A 2is a known constant inversely related to area. It is also clear that the assumption that the variance is constant with respect to the area is unrealistic. It is clearly easier to begin with an assumption that the average number of barrels per square mile is log–normal. Unfortunately, if the V i's were assumed log–normal then the sample mean would not be log–normal. The average number of barrels per square mile is like a sample mean and the variance of the sample mean should be σ 2 / A, where σ 2 is the common variance of the V i's. In addition to the reasons presented there we can see that if V were to represent the total hydrocarbon in a region it could be written as a sum V 1 + … + V A, with each V i being the hydrocarbons in a particular one-square-mile region and A being the area of the region. In the discussion of the Initial Model, Y = ln V and XB = ln( f(w)), and as noted there the assumption of constant variance for Y(e.g., u) is not a realistic one. If ( Y 1, X 1),…, ( Y n, X n) is a random sample then the least squares and maximum likelihood estimators, for B andσ 2, are the same. Where X is a row vector, B is a column vector, and u is normally distributed with mean zero and variance σ 2.
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