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Consider a random sample of size $n$ from a distribution with CDF $F(x) = 1 − x^{−2}$ if $x > 1$, and zero otherwise. Determine whether each of the following sequences has a limiting distribution, if so, then give the limiting distribution.

(a) $X_{1:n}$

(b) $X_{n:n}$

(c) $n^{−1/2}X_{n:n} $

I have the following definition: enter image description here

Also the CDF for $Y_1 = X_{1:n}$ is $$F_{Y_{1}}(y_{1}) = 1 - [1 - F_{X} (y_{1})]^n$$ And the CDF for $Y_{n} = X_{n:n} $ is $$F_{Y_{n}}(y_{n}) = [F_{X}(y_{n})]^n$$ Where $F_{X}$ is the CDF of the random variable $X $.

But I don't know how to conclude if the sequences has a limiting distribution and deduce it. Some help please. I'm not very familiar with order statistics, but I found that problem interesting.

Curious
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1 Answers1

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(a) $F_{Y_1}(y_1)=1-(1-\frac 1 {y_1^{2}})^{n} \to 1-0=1$ for $y_1 >1$. For $y_1 <1$ we have $F_{Y_1}(y_1)=0$ so $X_{1:n}$ converges in distribution to the constant random variable $1$.

(b) $F_{Y_n}(y_n) \to 0$ for $y_1 >1$ so $Y_n$ does not converge in distribution to any rando variable. [ We cannot have $F(t)=0$ for all $t>1$ for a distribution function $F$].

(c) The distribution function of $n^{-1/2}X_{n:n}$ is $(1- 1/({ny^{2}}))^{n} \to e^{-1/y^{2}}$. So $n^{-1/2}X_{n:n}$ converges to a random variable with density function $e^{-1/y^{2}}, 0<y<\infty$.

Kavi Rama Murthy
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