Question

4. ?? , ? = 1, . . . , ? are i.i.d. and has p.d.f. ?(?) = { 0 ? < 0 ??−? + (1 − ?)2? −2? ? ≥ 0 , here 0 ≤ ? ≤ 1. Write down the likelihood function. (10 points) When ? = 1, write down the MLE of ?. (10 points) When ? = 1, write down the bias and variance of the MLE of ?. (10 points)

Answer 1

Answer:-

Given that:-

X_i, i = 1, ...., n are i.i.d and has p.d.f

here 0 c 1

* Write down the likelihood function.

First of all f(x) is wrong. f(x) will be,

likelihood function = L(c) = f (x₁.... x_n | c)

  

* When n = 1, write down the MLE of c.

when n = 1,

Clearly L(c) is maximum at c = 1, when  

and L(c) is maximum at c = 0, when x₁ < log_e² =  

=

=

* when n = 1, write down the bias and variance of the MLE of c.

Now,

  

  

  

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