Alternatively, the estimates provided by the method of moments can instead be used as initial values for an iterative solution of the maximum likelihood coupled equations in terms of the digamma functions.
42.
Also one can express the joint log likelihood per " N " iid observations in terms of the digamma function expressions for the logarithms of the sample geometric means as follows:
43.
The ligature for ??, now called number sign for " 6 ", having been visually conflated with the cursive form of the ancient letter digamma, which had this numeral function.
44.
Now the double letters are already included in the twenty-four, but the Digamma stands outside the alphabet, and therefore its number might properly be added to that of the letters.
45.
In ancient papyri, the cursive C-shaped form of numeric digamma is often indistinguishable from the C-shaped ( " lunate " ) form that was then the common form of sigma.
46.
S . Kotz suggest the logarithmic approximation to the digamma function ? ( ? ) H " ln ( ?-1 / 2 ) which results in the following approximation to the geometric mean:
47.
Where the "'trigamma function "', denoted ? 1 ( ? ), is the second of the polygamma functions, and is defined as the derivative of the digamma function:
48.
Therefore, they can be expressed as "'trigamma functions "', denoted ? 1 ( ? ), the second of the polygamma functions, defined as the derivative of the digamma function:
49.
Along with the other special numeric symbols koppa and sampi, numeric digamma / stigma normally has no distinction between uppercase and lowercase forms, ( while other alphabetic letters can be used as numerals in both cases ).
50.
Note, however, that the same symbol is used to denote the unrelated letter waw ( ) in Pamphylia ( the " Pamphylian digamma " ) and was also the form of beta ( ) used in Melos.
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