The Treaty of Versailles
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Maximum Likelihood Estimators
We use statistics drawn from samples to estimate population parameters. We have learnt principally that we can use the sample mean to estimate the population mean, and the unbiased sample variance to estimate the population variance. However, this is just one approach to estimation, and others do exist. One such alternative approach goes by the name of ; ;Maximum Likelihood Estimators ; ;. We would use this approach when a single experiment produces two or more sample statistics, each of which could be used as an estimate for a population parameter. The central idea is to base our estimate of the population parameter on the value of the population parameter that would have made the experimental outcome most likely to have occurred. The most typical situation in which this type of estimator is used is when one is estimating a probability. For example, the likelihood of a success occurring in a trial. We use the sample mean and unbiased sample variance to estimate the population mean and variance, but not a probability of an event occurring, so the context in which a maximum likelihood estimator is used is different, and an alternative approach is required. Example In genetic theory the colour of a flower may be determined by a single gene which may exist in two forms (called alleles). For a particular plant, both forms are co ; ;dominant. One allele codes for red flowers and another for white flowers. The allele for red flowers is represented by R and the allele for white flowers by W. If the flower has one R and one W allele, then the flower will be pink. The probability of having an allele coding for a red flower is p; hence the probability of having an allele code for a white flower is W. Form a probability tree to determine in terms of the population parameter p the probabilities of a flower having red, pink or white flowers. Determine the maximum likelihood estimator for p. Given that in an experiment a flowers were found to be red, b to be pink and c to be white, find an expression for the estimate of p in terms of a, b and c. Find the value of this estimate given that in fact [Equation goes here ; ; download the original to see it.] .We talk you through the solution Solution The probability tree and the model Firstly, we have to draw the probability tree. [Diagram goes here ; ; download the original to see it.] From the tree we can see that the probabilities of the three possible outcomes is [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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