# Theorem about trimmed mean and location parameter, explanation needed.

The theorem is found on Stephen M. Stigler paper The Asymptotic Distribution of The Trimmed Mean (1973)

Let $$S_n$$ denote trimmed mean and: $$a = \sup\{x: F(x) \le \alpha\} \\ b = \inf\{x:F(x)\ge \beta\}$$ $$A = a - \inf\{x: F(x) \ge \alpha\}, \\ B = b - \sup\{x:F(x) \le \beta\}-b$$

Further define: $$\begin{matrix}G(x) = & 0 & \text{for }x Theorem: As $$n \to \infty$$, $$L(n^{1/2}(S_n - \mu)) \to L(Z)$$, with $$Z = (\beta - \mu)^{-1} \left[Y_1 + (b-\mu) Y_2 +(\alpha - \mu) Y3 + B\max(0,Y_2) - A\max(0,Y_3)\right]$$ and $$E[Z] = \left[B(\beta(1-\beta))^{1/2} - A(\alpha(1-\alpha))^{1/2}\right] / \left[(\beta-\alpha)(2\pi)^{1/2}\right]$$, where $$Y_1\sim \mathcal{N}(0, (\beta-\alpha)\sigma^2)$$ and independent of $$(Y_1, Y_3)$$, and $$(Y2,Y3) \sim \mathcal{N}(0,C)$$, where C is

$$C = \left(\begin{matrix}\beta(1-\beta) & -\alpha(1-\beta) \\ -\alpha(1-\beta) & \alpha(1-\alpha)\end{matrix}\right)$$

First, I cannot grasp what $$Z$$ is. In the introduction it is stated that with the help of this theorem the distribution of trimmed mean will be derived. But the theorem itself is about the asymptotic distribution of $$S_n - \mu$$.

Second, what is the meaning of $$Y_1, Y_2$$ and $$Y_3$$?

Third, how could the $$C$$ covariance matrix be interpreted? Does the $$\max(0,Y_2)$$ denote the maximum variance of $$Y2$$ or what?

Proof. Given $$X_1,...,X_n$$ define random variables $$V_n$$ and $$W_n$$ as follows: if $$F(a-) = \alpha$$, let $$V_n = (\#X_i < a);$$ otherwise let $$V_n$$ count the number of $$X_i = a$$ with probability $$p_1 = (\alpha - F(a-)) / (F(a) - F(a-))$$ each (independently of others) plus the number of $$X_i < a$$. If $$F(b) = \beta$$, let $$W_n = (\# X_i \le b)$$; otherwise let $$W_n$$ count the number of $$X_i = b$$ with probability $$p_2 = (\beta - F(b-)) / (F(b) - F(b-))$$ each (independently of others) plus the number $$X_i < b$$. (In the special case $$a=b$$ and $$F(a-) < \alpha<\beta < F(a)$$, then conditional on $$(\#X_i = a) = m$$, perform $$m$$ independent trinomial trials with probabilities $$(p_1, p_2 - p_1, 1-p_2)$$, observing $$(R_1,R_2,R_3)$$. Let $$V_n = (\#X_i and $$Wn = V_n + R_2$$. Then $$V_n$$ has a binomial $$(n, \alpha)$$ distribution, $$W_n$$ has a binomial $$(n, \beta)$$ distribution, and $$Cov(V_n, W_n) = n\alpha(1-\beta)$$.

What does $$V_n$$ and $$W_n$$ signify? In the special case of $$a=b$$ , why is it supposed that $$F(-a) < F(a)$$? Also, regarding the special case where $$a=b$$, it seems to me that $$\alpha$$ must be equal to $$\beta$$, because $$a=b \Rightarrow \sup\{x: F(x) \le \alpha\} = \inf\{x:F(x)\ge \beta\}$$, and if $$\alpha \neq \beta$$, then the distribution is with two blocks of data on the sides and then one point in the middle, isn't it?