1
Type I and Type II Errors and the NeymanPearson
Lemma: Lecture XXIII
I.
Introduction
A.
In general there are two kinds of hypotheses: one type concerns the form
of the probability distribution (i.e. is the random variable normally
distributed) and the second concerns parameters of a distribution function
(i.e. what is the mean of a distribution).
B.
The second kind of distribution is the traditional stuff of econometrics. We
may be interested in testing whether the effect of income on consumption
is greater than one, or whether the effect of price on the level consumed is
equal to zero.
1.
The second kind of hypothesis is termed a simple hypothesis.
Under this scenario, we test the value of a parameter against a
single alternative.
2.
The first kind of hypothesis (whether the effect of income on
consumption is greater than one) is termed a composite hypothesis.
Implicit in this test is several alternative values.
C.
Hypothesis testing involves the comparison between two competing
hypothesis, or conjectures.
1.
The null hypothesis, denoted
0
H
, is sometimes referred to as the
maintained hypothesis.
2.
The competing hypothesis to be accepted if the null hypothesis is
rejected is called the alternative hypothesis.
D.
The general notion of the hypothesis test is that we collect a sample of
data
1
,
n
XX
. This sample is a multivariate random variable,
n
E
. (The
text refers to this as an element of a Euclidean space).
1.
If the multivariate random variable is contained in space
R
, we
reject the null hypothesis.
2.
Alternatively, if the random variable is in the complement of the
space
R
, we fail to reject the null hypothesis.
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 Fall '09
 CARRIKER

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