Roo Inequality and Statistical Problems

Roo Inequality and Statistical Problems

• The Cramér-Rao Lower Bound (CRLB) is used to determine the minimum variance of an unbiased estimator. It provides a lower limit to the variance of estimators of a parameter.

Problem 1

• Given observations ( X_1, X_2, \ldots, X_n ) from an exponential distribution with parameter ( \theta ).
• To find the CRLB for the variance of an estimator ( VE ) of ( \theta ).

Problem 2

• Observations ( X_i ) follow a Bernoulli distribution with parameter ( \theta ).
• The goal is to determine the Minimum Variance Unbiased Estimator (MVUE) for ( \theta ) based on a sample size of ( n ).

Problem 3

• Observations ( x_i ) follow a Binomial distribution parameterized by ( \theta^2 ) (with ( \theta^2 ) known).
• The task is to obtain the Cramer-Rao Lower Bound (CRLB) for the variance of an estimator of ( \theta ).

Problem 4

• Consider ( X_i ) distributed as exponential with mean ( \frac{1}{\theta} ) and Bernoulli with success probability ( \theta ).

• The likelihood function ( L(p|X) ) for these observations is given as the product of ( \theta^{x_i} (1-\theta)^{1-x_i} ).

• The likelihood can be expressed as:

  • ( L = \theta^{\sum x_i} (1 - \theta)^{n - \sum x_i} ).

• The natural logarithm of the likelihood:

  • ( \ln L = \sum x_i \ln \theta + (n - \sum x_i) \ln (1 - \theta) ).

• First derivative of the log-likelihood function:

  • ( \frac{\partial \ln L}{\partial \theta} = \frac{\sum x_i}{\theta} - \frac{n - \sum x_i}{1 - \theta} ).

• Second derivative of the log-likelihood function:

  • ( \frac{\partial^2 \ln L}{\partial \theta^2} = -\frac{\sum x_i}{\theta^2} - \frac{n - \sum x_i}{(1 - \theta)^2} ).

• The information function (Inc):

  • ( Inc(\theta) = E\left(-\frac{\partial^2 \ln L}{\partial \theta^2}\right) = E\left[\frac{n - 2\sum x_i}{\theta(1 - \theta)^2}\right] ).

• The critical point of the second derivative for setting equations to zero involves determining:

  • ( \frac{2\sum x_i}{\theta} = \frac{(n - \sum x_i)(-1)}{1 - \theta} ).

• Overall, focuses on deriving estimators and evaluating their efficiency through CRLB, maintaining an unbiased estimate for provided distributions.

This quiz covers the Cramér-Rao Lower Bound (CRLB) and its applications in various statistical problems. You will tackle problems involving exponential, Bernoulli, and Binomial distributions, focusing on finding minimum variance unbiased estimators and CRLBs for different parameters. Test your understanding of estimators and statistical variance concepts.