0
$\begingroup$

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: $$\begin{matrix}G(x) = & 0 & \text{for }x<a, \\ & (F(x)-\alpha)/(\beta-\alpha) & \text{for } \alpha\le b, \\ & 1 & x\ge b\end{matrix}$$ 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 $V_n = (\#X_i<a)+R_1$ 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?

$\endgroup$

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.