5. X ~ Uniform(-ΞΈ/2, ΞΈ/2) λΌκ³ κ°μ ν©μλ€. μ¦, Xμ νλ₯ λ°λν¨μ (pdf)λ
Suppose X ~ Uniform(-ΞΈ/2, ΞΈ/2), i.e., X has the pdf
$f(x;\theta) = \frac{1}{\theta} \mathbb{I}\{|x| \le \frac{\theta}{2}\}$
(a) μ΄κ²μ μΌλ°μ μΈ μ μΉμ± κ°μ μ λ§μ‘±νμ§ μλ λΆν¬μ μμ
λλ€. μ΄ λΆν¬κ° λ§μ‘±νμ§ μλ μ΅μ νλμ μ μΉμ± κ°μ μ μ μνμμμ€.
This is an example of a distribution that does NOT satisfy the usual regularity assumptions. State at least one regularity assumption that this distribution fails to satisfy.
(b) (a)λ‘ μΈν΄ μμ
μμ λ°°μ΄ μ 리λ€λ‘λΆν° μ¦μ λ°λΌμ€μ§ μλ μ΅λκ°λ₯λμΆμ λ (mle)μ μμ± μ€ νλλ₯Ό μ μνμμμ€.
State at least one property of mle that does NOT immediately follow from the theorems from the class as a consequence of part (a).