Random Variables from a non-Parametric distribution know their limits

You produce a non-parametric distribution. Then you obtain, say, 10 random variables (RV) from this non-parametric distribution- much the same way as you would obtain random variables from a (parametric) normal distribution with stated mean and variance. But unlike the parametric distribution, where our RVs would occur around the mean (our parameter), RVs from a non-parametric distribution occur within the range bound by the lowest and highest mass point. This was not necessarily an intuitive concept to me, when I first stumbled across it. Which is why this mathematical proof of this range made me feel so much more comfortable:

If our estimate of the RV is a simple weighted-mean of the mass points:

\hat{\beta} = z_{1}w_{1} + ... + z_{k}w_{k}

Furthermore, since z_1 \leq \hat{\beta} \leq z_k for RV \beta:

\left[w_{1}+...+w_{k} \right]z_{1}\leq \hat{\beta}\leq \left[w_{1}+...+w_{k} \right]z_{k}

Since \sum w_i=1, we can express the inequality as:

z_1 \leq \hat{\beta} \leq z_k

On the other hand, If we know further information, like individual weights:

\hat{\beta}=z_1w_1+...+z_kw_k

Furthermore, since  for intercept \beta :

\left(w_{1}+w_{k}\right)z_1\leq \hat{\beta}\leq \left(w_{1}+w_{k}\right)z_k

Since \sum w_i=1, we can express the inequality as:

z_1\leq \hat{\beta}\leq z_k

Thus, it is proven that any estimates of an RV drawn from a non-parametric distribution will be bound by the highest and lowest mass point.

Abbas Keshvani

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