Question: (continuation) The value of θ is called the Feature Completeness Index (FCI). The larger the FCI, the better the product. We want to estimate the FCI. Suppose that n raters will provide scores for the product. We construct a model for the n scores by proposing a random variable \[{{X}_{i}}\] for the score of each rater. We assume the Random Sample model, with the above probability density function (from problem 1) for each \[{{X}_{i}}\] .

1. Suppose the model proposes that each realization \[{{x}_{i}}\] "should be" equal to \[\frac{\theta }{\theta +1}\] . Given realizations \[{{x}_{1}},{{x}_{2}},...,{{x}_{n}}\] , what is the least-squares estimate of the FCI?
2. Given realizations \[{{x}_{1}},{{x}_{2}},...,{{x}_{n}}\] , what is the maximum likelihood estimate of the FCI?
3. What is \[p\lim (\bar{X})\] ?
4. Provide values for a and b so that \[a\bar{X}-b\] has an asymptotic standard normal distribution.

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