The Treaty of Versailles
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Determining an estimate for p
The estimate for p will be the value of [Equation goes here ; ; download the original to see it.] That makes this function a maximum. That means, we have to differentiate it and set the derivative to zero, and hence find the maximum. We set this problem up as follows [Equation goes here ; ; download the original to see it.] (We will assume for the present that this value will be a maximum; to prove that it is a maximum we would strictly have to differentiate a second time and use the usual criterion that the second derivative is negative for a maximum. We will not bother with this tedious extra work here!) This sets up an apparently difficult additional problem, to which there is, fortunately a simple answer ; ; how does one go about differentiating this? [Equation goes here ; ; download the original to see it.] However, this problem is simply solved by using logarithmic differentiation instead. Note that we are not really interested in the derivative of [Equation goes here ; ; download the original to see it.] but rather in just when it takes a maximum. But since this function is always positive it always has a logarithm; and the function [Equation goes here ; ; download the original to see it.] is an always increasing function. This means that if has a maximum at [Equation goes here ; ; download the original to see it.]then [Equation goes here ; ; download the original to see it.] also has a maximum at [Equation goes here ; ; download the original to see it.]. Differentiating the logarithm of [Equation goes here ; ; download the original to see it.] is much easier than differentiating [Equation goes here ; ; download the original to see it.] itself, as we shall show.Note also the introduction of the symbol ; ; the p with the hat on it. The hat is the standard symbol used to denote an estimate of a population parameter derived from a sample; it is here the value of p that makes a maximum ; ; which is why it is called the ; ;maximum likelihood estimator ; ; Now to differentiate [Equation goes here ; ; download the original to see it.]; to do this we use the properties of logarithms that [Equation goes here ; ; download the original to see it.]So firstly,[Equation goes here ; ; download the original to see it.]Now to differentiate it we must remember the chain rule, and the rule that [Equation goes here ; ; download the original to see it.]Remember the derivative of a constant is also zero; hence[Equation goes here ; ; download the original to see it.]For turning points, when we have[Equation goes here ; ; download the original to see it.]Determining the specific estimateWe have shown that[Equation goes here ; ; download the original to see it.]So given that [Equation goes here ; ; download the original to see it.][Equation goes here ; ; download the original to see it.]
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Contents of
The Treaty of Versailles
1 The Treaty of Versailles, constraints on the peace-makers
2 Maximum Likelihood Estimators
3 Progress of the Paris Peace Conference and the Treaty of Versailles
4 Finding the maximum likelihood estimator
5 Contemporary reactions to the Treaty of Versailles
6 Determining an estimate for p
7 Assessement of the Treaty of Versailles
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