Suppose that 𝜃 1, ..., 𝜃 I are a random sample from N(0, 𝜎2). It is known that the sample median of the |𝜃 i| converges in probability to the population median of the |𝜃 i| as I → ∞. (a) Show...

Suppose that 𝜃

1, ..., 𝜃

I are a random sample from N(0, 𝜎2). It is known

that the sample median of the |𝜃

i| converges in probability to the population

median of the |𝜃

i| as I → ∞.

(a) Show that the median of the positive part of a N(0, 1) random variable is

0.6745.

(b) From (a), show that median |𝜃

i

|∕𝜎 converges to 0.6745 as I → ∞, which

is equivalent to (1/0.6745) median |𝜃

i

| = 1.4826 median |𝜃

i

| → 𝜎 as I →

∞. [In (4.20), the scaling factor 1.4826 is rounded off to 1.5.]