PDF of Order Statistics for RVs: \gamma_1,\,\ldots,\,\gamma_M

[EngWiPy]

Hello,

Suppose that we have the following set of independent and identically distributed RVs: $$\gamma_1,\,\gamma_2,\,\ldots,\,\gamma_M$$. Arranging them in descending order as: $$\gamma_{1:M}\ge\gamma_{2:M}\ge\cdots\ge\gamma_{M:M}$$. Now suppose we select the largest $$m\leq M$$ order statistics. What is the PDF of the selected set? Mathematically:

$$f_{\gamma_{1:M},\,\ldots,\,\gamma_{m:M}}(\gamma_{1:M},\,\ldots,\,\gamma_{m:M})=??$$

Thanks in advance

 

[EnumaElish]

What would f be for m = 1? How do you get there?

 

[EngWiPy]

What would f be for m = 1? How do you get there?

$$f_{\gamma_{1:M}}(\gamma)=\frac{d}{d\,\gamma}F_{\gamma_{1:M}}(\gamma)=\frac{d}{d\,\gamma}\text{Pr}\left[\gamma_{1:M}\le\gamma\right]=\frac{d}{d\,\gamma}\text{Pr}\left[\gamma_{1}\le\gamma,\gamma_{2}\le\gamma,\ldots,\gamma_{M}\le\gamma\right]=\frac{d}{d\,\gamma}\left[F_{\gamma}(\gamma)\right]^M$$

where $$F_{\gamma}(\gamma)$$ is the CDF of the original set of RVs.

But when we pick a subset of the $$m^{\text{th}}$$ largest order statistics, how can we treat the statistics? I mean I have the final answer from books and papers, but I didn't understand how they derive it.