Question

Let X₁, ..., Xₙ be independent and identically distributed random variables following a Uniform distribution U([a, b]) with unknown parameters θ₁ = a and θ₂ = b.

(a) Using the Maximum Likelihood Estimation (MLE) method, find the MLEs of θ₁ and θ₂.

(b) Argue carefully using indicator functions to justify your MLEs for θ₁ and θ₂.

Answer 1

The Maximum Likelihood Estimation (MLE) method can be used to find the MLEs of unknown parameters. To find the MLE of a parameter, we look for the sample minimum or maximum depending on the condition. The MLEs for the given independent and identically distributed random variables following a Uniform distribution U([a, b]) are X(1) for θ1 and X(n) for θ2.

Step-by-step explanation:

The Maximum Likelihood Estimation (MLE) method can be used to find the MLEs of the unknown parameters θ1 and θ2 for the given independent and identically distributed random variables X1, ..., Xn following a Uniform distribution U([a, b]).

To find the MLE for θ1, which is equivalent to finding the minimum value of a, we need to look for the smallest value for which all the observed values of X1, ..., Xn are greater than or equal to. This minimum value is the sample minimum, denoted as X(1). Therefore, the MLE of θ1 is X(1).

To find the MLE for θ2, we need to look for the largest value for which all the observed values of X1, ..., Xn are less than or equal to. This maximum value is the sample maximum, denoted as X(n). Therefore, the MLE of θ2 is X(n).

Using the indicator functions, we can justify these MLEs. The indicator function I(X ≥ a), which equals 1 if X ≥ a and 0 otherwise, can be used to indicate if all the observed values are greater than or equal to a. Similarly, the indicator function I(X ≤ b), which equals 1 if X ≤ b and 0 otherwise, can be used to indicate if all the observed values are less than or equal to b. By maximizing the probabilities defined by these indicator functions, we can find the MLEs of θ1 and θ2.