Page 289 - Probability and Statistical Inference

266 5. Concepts of Stochastic Convergence

Next, we rewrite

Hence, one obtains

using (5.4.6) and the facts that and (1 +
(1/ν) x ) = 1. Refer to the Section 1.6 as needed. Thus, we conclude:
2 –1/2

With 0 < α < 1, let us define the upper 100α% point t for the Student’s tν
ν,α
distribution through the following relationship:

that is, the area under the pdf of tν on the rhs of the point tν is α. Similarly
,α
let z stand for the upper 100α% point for the standard normal distribution.
α
These percentage points have been tabulated in the Appendix. It is known that
for any fixed 0 < α < 1, one has

But what is interesting is that this convergence is also monotone. For any
fixed 0 < α < 1, it is known that