Summary of the Statistics for EOR course given in the second year of EOR at Tilburg University. The summary covers every topic, except MLEs, etc. are not included, because this has already been learned in the introductory course.
, 1 Confidence Intervals
A confidence interval (short CI) is used to estimate a parameter in such a
manner that there is a high probability that the true value of the parameter
lies in the interval. For a so called two-sided confidence interval, we have
Pθ [l(X) < τ (θ) < r(X)] = γ and define α = 1−γ. The value for γ is normally
fixed and is often a high number like 0.9, 0.95 or 0.99. Sometimes we need
one-sided confidence intervals. If Pθ [τ (θ) > l(X)] = γ then (l(x), ∞) is
called a left-sided 100γ% CI for τ (θ). In the same way, (−∞, r(x)) is called
a right-sided CI for τ (θ) if Pθ [τ (θ) < r(X)] = γ. The lenght of a CI is given
by r(X) − l(X).
For instance, a 95% confidence interval means that if we apply the procedure
many times, in about 95% of the cases the true value will lie in the confidence
interval. So, on average, we catch the true value in 95% of the cases.
There are a few important values which are being used consistently through-
out this chapter:
1.1 Pivotal Quantities
Q = q(X, θ) is a pivotal quantity if the probability distribution of Q does
not depend on θ. Note that Q is a function of both X and θ, so when you
write down Q, you will see a θ. The pivotal quantity Q could for instance
have a normal, chi-squared or a t-distribution. There are a few statements
with respect to a pivotal quantity.
1. If θ is a one-dimensional location parameter and θ̂ is the MLE, then θ̂ − θ
is a pivotal quantity.
θ̂
2. If θ > 0 is a one-dimensional scale parameter and θ̂ is the MLE, then θ
is
a pivotal quantity.
2
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