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Mathematical Statistics for Economics and Business
A primary goal of scientific research often concerns the verification or refutation of assertions, conjectures, currently accepted laws, or descriptions relating to a given economic, sociological, psychological, physical, or biological process or population. Statistical hypothesis testing concerns the use of probability samples of observations from processes or populations of interest, together with probability and mathematical statistics principles, to judge the validity of stated assertions, conjectures, laws, or descriptions in such a way that the probability of falsely rejecting a correct hypothesis can be controlled, while the probability of rejecting false hypotheses is made as large as possible. The precise nature of the types of errors that can be made, how the probabilities of such errors can be controlled, and how one designs a test so that the probability of rejecting false hypotheses is as large as possible is the subject of this chapter.
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Compare this set to the
range of
X
over Ω introduced in our discussion of minimal sufficient statistics,
Section 7.4.
The terminology “acceptable region” is sometimes replaced by “
acceptance region” in the literature. We will see later that while the behavior of sample outcomes may be “acceptable” to
H given the characteristics of the probability space implied by
H, there are statistical reasons why one might not want to literally conclude
acceptance of
H on the basis of this “acceptable” behavior. We will clarify this subtle but important distinction with additional rigorous rationale later, which will motivate further why we choose to use the terminology “acceptable”.
For convenience, we have chosen to “connect the dots” and display the graph as a continuous curve. We will continue with this practice wherever it is convenient and useful.
If the magnitudes of the costs or losses incurred when errors are omitted can be expressed in terms of a loss function, then a formal analysis of expected losses can lead to a choice of type I and type II error probabilities. For an introduction to the ideas involved, see Mood, A., F. Graybill, and D. Boes, (1974).
Introduction to the Theory of Statistics, 3rd Ed., New York: McGrawHill, pp. 414–418.
In parametric models, sets of fully specified probability distributions will be identified, whereas in semiparametric models, only a subset of moments or other characteristics of the underlying probability distributions are generally identified by the hypotheses.
The properties we will examine do not exhaust the possibilities. See E. Lehmann, (1986),
Testing Statistical Hypotheses, John Wiley, NY.
Recall that
\( {{\sup}_{{{\mathbf{\Theta}} \in H}}}\{\pi \left( {\mathbf{\Theta}} \right)\} \) denotes the smallest upper bound to the values of
π(
Θ) for
Θ ∈
H (i.e., the supremum). If the maximum of
π(
Θ) for
Θ ∈
H exists, then sup is the same as max.
Recall the previous footnote, and the fact that
\( {{\inf}_{{\Theta \in \bar{H}}}} \) {
π(
Θ)} denotes the largest lower bound to the values of
π(
Θ) for
Θ
∈
\( \bar{H} \) (i.e., the infimum). The sup and inf of
π(
Θ) are equivalent to max and min, respectively, when the maximum and/or minimum exists.
In the case of continuous
X, the choice of size is generally a continuous interval contained in [0,1]. If
X is discrete, the set of choices for size is generally finite, as previous examples have illustrated.
The maximum is achievable in this case, and equals .05 when
μ = 15.
Note that
\( \mathop{{\min }}\limits_{{\mu > 15}} \left\{ {{{\pi}_{\rm{n}}}(\mu )} \right\} \)does not exist in this case. The largest possible lower bound (i.e., the infimum) is .05, which is < π
_{n}(μ), ∀ μ > 15.
Neyman, J. and E.S. Pearson, “On the Problem of the Most Efficient Tests of Statistical Hypotheses,”
Phil. Trans., A, vol. 231, 1933, p. 289.
Recall that the support of a density function is the set of
xvalues for which
f(
x;
Θ
_{0}) > 0, i.e., {
x:
f(
x;
Θ
_{0}) > 0} is the support of the density
f(
x;
Θ
_{0}).
This limitation can be overcome, in principle, by utilizing what are known as
randomized tests. Essentially, the test rule is made to depend not only on the outcomes of
X but also on auxiliary random variables that are independent of
X, so as to allow any level of test size to be achieved. However, the fact that the test outcome can depend on random variables that are independent of the experiment under investigation has discouraged its use in practice. For an introduction to the ideas involved, see Kendall, M. and A. Stuart, (1979)
The Advanced Theory of Statistics, Vol. 2, 4th Edition, New York: MacMillan, 1979, p. 180–181. Also, see problem 9.8.
This can be shown via the MGF approach, since the MGF of
\( {\sum\nolimits_{{i = 1}}^{{100}} {{{X}_i}} = \prod\nolimits_{{i = 1}}^{{100}} {{{M}_{{{{X}_i}}}}} (t) = \prod\nolimits_{{i = 1}}^{{100}} {{{\left( {1  \theta t} \right)}}^{{  1}}} = {{{\left( {1  \theta t} \right)}}^{{  100}}} \ \rm{for} \ {\it t} < {{\theta}^{{  1}}}} \), which is of the gamma form with
β =
θ, α = 100.
As we noted previously, it is possible to use a
randomized test to achieve a size of .05 exactly, but the test can depend on the outcome of a random variable that has nothing to do with the experiment being analyzed. See problem 9.8 for an example of this approach. Randomized tests are not often used in practice.
To this point, we have established UMP tests in the class of
all tests of a certain level
α. The reader should note that in all cases examined heretofore, we have shown that UMP level α tests were also unbiased. This is clearly different than examining
only unbiased tests of a certain level
α, and within this restricted set of tests, attempting to find one that is UMP.
Results are available for a more general class of densities referred to as Polya distributions, which subsumes the exponential class densities as a special case. However, the mathematics involved in analyzing the more general distributions is beyond the scope of our study. Interested readers can consult the work of S. Karlin, (1957), “Polya Type Distributions II,”
Ann. Math. Stat., 28, pp. 281–308.
We are assuming that
C
_{ r } is such that a power function is defined, i.e.,
C
_{ r } can be assigned probability by
f(
x;Θ), Θ
∈Ω.
One can show using the monotone likelihood ratio approach that a UMP level
α test of
H
_{0} versus
H
_{ a } does
not exist. Recall Theorem 9.9.
The algorithm actually used was the NLSYS procedure in the GAUSS Matrix language.
The algorithm actually used was the NLSYS procedure in the GAUSS matrix language.
We are assuming that
C
_{ r } is such that a power function is defined, i.e.,
C
_{ r } can be assigned probability by
f(
x;
Θ),
Θ ∈ Ω.
Note that
c = (
c
_{1},…,
c
_{ k }) could be viewed as an alternative parameterization of the exponential class of densities, where
with
\( {{d}_{*}}\left( {c} \right) = \ln {{\left( {\int_{{  \infty }}^{\infty } \cdots \int_{{  \infty }}^{\infty } {\exp } \left( {\sum\nolimits_{{i = 1}}^k {{{c}:g:}} \left( {\bf x} \right) + z\left( {\bf x} \right)} \right){{I}_A}\left( {\bf x} \right)d{\bf x}} \right)}^{{  1}}} \)(use summation in the discrete case). This parameterization is referred to as the
natural parameterization of the exponential class of densities. Note that the definition of
d
_{*}(
c) is a direct result of the fact that the density must integrate (or sum) to 1.
$$ {{f}_{*}}\left( {{\bf x};{\bf c}} \right) = \exp \left( {\sum\limits_{{i = 1}}^k {{{c}_i}} {{g}_i}\left( {\bf x} \right) + {{d}_{*}}\left( {\bf c} \right) + z\left( {\bf x} \right)} \right){{I}_A}\left( {\bf x} \right),{c}\in {{\Omega}_c}, $$
Except, perhaps, on a set having probability zero.
The surface area of an ndimensional hypersphere is given by
\( {A = \left( {n{{r}^{{n  1}}}{{\pi}^{{n/2}}}} \right)/\Gamma \left( {\left( {n/2} \right) + 1} \right),} \) where
r is the radius of the hypersphere. See R.G. Bartle, (1976),
The Elements of Real Analysis, 2nd Ed., John Wiley, pp. 454–455, and note that the surface area can be defined by differentiating the volume of the hypersphere with respect to
r. For
n = 2, the bracketed expression becomes simply 2
π(
s*
_{2})
^{1/2}, which is the familiar 2
πr.
Expanding
\( {\sum\nolimits_{{i = 1}}^n {{{{\left[ {\left( {{{x}_i}  \bar{x}} \right) + \left( {\bar{x}  {{\mu}_0}} \right)} \right]}}^2}} } \) leads to the result.
If the likelihood ratio is strictly increasing in
t(
x), then c can be chosen to satisfy
g(
c) =
k
^{−1}.
 Title
 Hypothesis Testing Theory
 DOI
 https://doi.org/10.1007/9781461450221_9
 Author:

Ron C. Mittelhammer
 Publisher
 Springer New York
 Sequence number
 9
 Chapter number
 9