User:Al83tito/Stats

MGTECON 603: Econometrics (Part II)

Introduction
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Type I/II Errors; Power Function
Type I error: the rejection of a true null hypothesis

Type II error: retaining a null hypothesis when it’s false.

Remark: fix the H$0$ and H$1$ first, then discuss what the type I/II errors are. i.e. the errors are relative.
 * Whether a small significant level is desired: If the consequences of a type I error are serious or expensive, then a very small significance level is appropriate; e.g. Two drugs are being compared for effectiveness in treating the same condition. Drug 1 is very affordable, but Drug 2 is extremely expensive. The null hypothesis is ”both drugs are equally effective,” and the alternate is ”Drug 2 is more effective than Drug 1.” In this situation, a Type I error would be deciding that Drug 2 is more effective, when in fact it is no better than Drug 1, but would cost the patient much more money. That would be undesirable from the patient’s perspective, so a small significance level is warranted (reference: “common statistical errors”, Martha Smith (UT Austin)).


 * The trade-off between two types of errors:
 * The general approach (in the textbook but not taught here) is in “statistical decision theory” (S.8.3.5).
 * Neyman approach: Choose a critical value s.t. P$0$(type I errors) is “small”, ∀θ ∈ Θ$1$.
 * typically “at most 5%”: truncating data - e.g. 6% is thrown away.
 * “P-hacking”: Once take a rule, we need to prolong the rule regardless the results - for example, we cannot push the p-value slowly by changing experiment parameters. (more on “prolonging the rule”: medical trials?)

Power Function: The power function of a test with a critical region C is the function β :
 * Θ → [0, 1] given by β(θ) = P$0$((X$1$, ..., X$θ$)′ ∈ C) = P$0$(reject H$θ$), θ ∈ Θ.
 * – P$1$(type 1 error) = β(θ),∀θ ∈ Θ$n$.
 * (i.e. Given H$θ$ is true the probability of rejecting it)
 * – P$0$(type 2 error)=1−β(θ),∀θ∈Θ$θ$.
 * (i.e. Given H$0$ is true the probability of accepting H$0$)

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Test Statistics
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Hypothesis Testing
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