Question

Let $X$ be a uniformly distributed random variable in $[a, b]$. The values of an independently drawn sample of size five from $X$ are given by 1.3, 0.8, 9.5, 20.2, 8.2. Let $\hat{a}$ and $\hat{b}$ denote the Maximum Likelihood Estimates for the parameters $a$ and $b$, respectively. Then, 
 

• A. $\hat{a} = 0.8;\ \hat{b} = 20.2$
• B. $\hat{a} = 1.3;\ \hat{b} = 9.5$
• C. $\hat{a} = 1.3;\ \hat{b} = 8.2$
• D. $\hat{a} = 0;\ \hat{b} = 21$

Hint:

For uniform distributions, the MLEs of parameters are simply the minimum and maximum of the observed data values.

Answer 1

The Correct Option is A

Step 1: Likelihood for uniform distribution.
For $X \sim U(a,b)$, \[ L(a,b) = \frac{1}{(b - a)^n}, \quad \text{if } a \le x_i \le b\ \forall i. \] To maximize likelihood, the interval $[a,b]$ must be as narrow as possible while containing all sample points.
Step 2: Use sample extremes.
MLEs are: \[ \hat{a} = \min(x_i), \quad \hat{b} = \max(x_i). \] Hence, \[ \hat{a} = 0.8, \quad \hat{b} = 20.2. \]
Step 3: Conclusion.
The MLEs for $(a,b)$ are $\hat{a} = 0.8$ and $\hat{b} = 20.2$.