Reference no: EM132265619

Assignment - Principles Of Statistical Inference (PSI) Practical Exercises

These practical exercises make use of a number of our previous examples, and require the justification and calculation of MLEs, standard errors and confidence intervals for the various situations.

Exercise 1 -

 In the Module 1 notes we considered a series of examples in which we discussed likelihood functions for a disease prevalence study, based on the binomial distribution.

a. Review the exercises and write down the likelihood function and log-likelihood function for the population prevalence θ, in terms of the number of individuals in the sample, n, and the number of individuals in the sample who have the disease, y. Hint: this is just the likelihood of a binomial sample.

b. Using differentiation, write down the likelihood equation.

c. Based on the likelihood equation determine θ^{^}, the MLE of θ. Hence justify that the sample prevalence is the best estimate of the population prevalence.

d. Determine the expected information, in terms of θ and n.

e. Using part d, evaluate standard errors associated with the MLE, for the two samples discussed in Module 1. Hence calculate confidence intervals for the population prevalence based on each of these samples.

Exercise 2 -

Read the section on reparameterisation in the Extended Example, and review the log-likelihood function for the case of a common event rate λ. Let θ = log(λ). We will now consider the likelihood and MLE based on this transformed parameter.

Give general answers unless you are asked to substitute the data.

a. Using the log-likelihood for λ, write down the log-likelihood function as a function of θ, by substituting in θ = log(λ).

b. Using the log-likelihood in part a, write down the likelihood equation for θ, and hence determine θ^{^} the MLE for θ. Evaluate θ^{^} using the data in the Extended Example.

c. Verify that θ^{^} = log(λ^{^}), where λ^{^} is the MLE that was derived in the Extended Example. Show this algebraically not just for this particular data. This is an example of a general property of MLEs - what property is this?

d. After differentiating the left hand side of the likelihood equation, obtain the expected information. For the data in Extended Example, use the expected information to calculate a standard error based on θ^{^}, and hence a confidence interval for θ. In your answer include:

• The formula for the expected information.
• The formula for the variance.
• Substitute the data and evaluate the standard error.
• The 95% confidence interval.

e. Using part d, calculate a confidence interval for λ. Evaluate this confidence interval. Compare this confidence interval to the confidence interval for λ calculated in the Extended Example.

f. Consider the two approaches to calculating confidence intervals in part e. Would it be possible (in other samples) for either of these two approaches to yield confidence intervals which include values that are not in the parameter space i.e negative values? Which approach is more desirable from this point of view?