Difference between revisions of "Similarity region"
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''similar region'' | ''similar region'' | ||
A generally used abbreviation of the term "critical region similar to a sample space" as used in mathematical statistics for a [[Critical region|critical region]] with non-randomized similarity of a statistical test. | A generally used abbreviation of the term "critical region similar to a sample space" as used in mathematical statistics for a [[Critical region|critical region]] with non-randomized similarity of a statistical test. | ||
− | Let | + | Let $ X $ |
+ | be a random variable taking values in a sample space $ ( \mathfrak X , \mathfrak B , {\mathsf P} _ \theta ) $, | ||
+ | $ \theta \in \Theta $, | ||
+ | and consider testing the compound hypothesis $ H _ {0} $: | ||
+ | $ \theta \in \Theta _ {0} \subset \Theta $ | ||
+ | against the alternative $ H _ {1} $: | ||
+ | $ \theta \in \Theta _ {1} = \Theta \setminus \Theta _ {0} $. | ||
+ | Suppose that in order to test $ H _ {0} $ | ||
+ | against $ H _ {1} $, | ||
+ | a non-randomized [[Similar test|similar test]] of level $ \alpha $( | ||
+ | $ 0 < \alpha < 1 $) | ||
+ | has been constructed, with critical function $ \phi ( x) $, | ||
+ | $ x \in \mathfrak X $. | ||
+ | As this test is non-randomized, | ||
+ | |||
+ | $$ \tag{1 } | ||
+ | \phi ( x) = \ | ||
+ | \left \{ | ||
− | + | \begin{array}{ll} | |
+ | 1, &{ x \in K \subset \mathfrak X, } \\ | ||
+ | 0, &{ x \notin K, } \\ | ||
+ | \end{array} | ||
+ | \right . | ||
+ | $$ | ||
− | where | + | where $ K $ |
+ | is a certain set in $ \mathfrak X $, | ||
+ | called the critical set for the test (according to this test, the hypothesis $ H _ {0} $ | ||
+ | is rejected in favour of $ H _ {1} $ | ||
+ | if the event $ \{ X \in K \} $ | ||
+ | is observed in an experiment). Also, the constructed test is a similar test, which means that | ||
− | + | $$ \tag{2 } | |
+ | \int\limits _ { \mathfrak X } \phi ( x) d {\mathsf P} _ \theta = \alpha | ||
+ | \ \textrm{ for } \textrm{ all } \theta \in \Theta _ {0} . | ||
+ | $$ | ||
− | It follows from (1) and (2) that the critical region | + | It follows from (1) and (2) that the critical region $ K $ |
+ | of a non-randomized similar test has the property: | ||
− | + | $$ | |
+ | {\mathsf P} _ \theta \{ X \in K \} = \alpha | ||
+ | \ \textrm{ for } \textrm{ all } \theta \in \Theta _ {0} . | ||
+ | $$ | ||
− | Accordingly, J. Neyman and E.S. Pearson emphasized the latter feature of the critical set of a non-randomized similar test and called | + | Accordingly, J. Neyman and E.S. Pearson emphasized the latter feature of the critical set of a non-randomized similar test and called $ K $ |
+ | a "region similar to the sample space" $ \mathfrak X $, | ||
+ | in the sense that the two probabilities $ {\mathsf P} _ \theta \{ X \in K \} $ | ||
+ | and $ {\mathsf P} _ \theta \{ X \in \mathfrak X \} $ | ||
+ | are independent of $ \theta \in \Theta _ {0} $. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.L. van der Waerden, "Mathematische Statistik" , Springer (1957)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J. Neyman, E.S. Pearson, "On the problem of the most efficient tests of statistical hypotheses" ''Philos. Trans. Roy. Soc. London Ser. A'' , '''231''' (1933) pp. 289–337</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E.L. Lehmann, H. Scheffé, "Completeness, similar regions, and unbiased estimation I" ''Sankhyā'' , '''10''' (1950) pp. 305–340</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E.L. Lehmann, H. Scheffé, "Completeness, similar regions, and unbiased estimation II" ''Sankhyā'' , '''15''' (1955) pp. 219–236</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.L. van der Waerden, "Mathematische Statistik" , Springer (1957)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J. Neyman, E.S. Pearson, "On the problem of the most efficient tests of statistical hypotheses" ''Philos. Trans. Roy. Soc. London Ser. A'' , '''231''' (1933) pp. 289–337</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E.L. Lehmann, H. Scheffé, "Completeness, similar regions, and unbiased estimation I" ''Sankhyā'' , '''10''' (1950) pp. 305–340</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> E.L. Lehmann, H. Scheffé, "Completeness, similar regions, and unbiased estimation II" ''Sankhyā'' , '''15''' (1955) pp. 219–236</TD></TR></table> |
Latest revision as of 14:55, 7 June 2020
similar region
A generally used abbreviation of the term "critical region similar to a sample space" as used in mathematical statistics for a critical region with non-randomized similarity of a statistical test.
Let $ X $ be a random variable taking values in a sample space $ ( \mathfrak X , \mathfrak B , {\mathsf P} _ \theta ) $, $ \theta \in \Theta $, and consider testing the compound hypothesis $ H _ {0} $: $ \theta \in \Theta _ {0} \subset \Theta $ against the alternative $ H _ {1} $: $ \theta \in \Theta _ {1} = \Theta \setminus \Theta _ {0} $. Suppose that in order to test $ H _ {0} $ against $ H _ {1} $, a non-randomized similar test of level $ \alpha $( $ 0 < \alpha < 1 $) has been constructed, with critical function $ \phi ( x) $, $ x \in \mathfrak X $. As this test is non-randomized,
$$ \tag{1 } \phi ( x) = \ \left \{ \begin{array}{ll} 1, &{ x \in K \subset \mathfrak X, } \\ 0, &{ x \notin K, } \\ \end{array} \right . $$
where $ K $ is a certain set in $ \mathfrak X $, called the critical set for the test (according to this test, the hypothesis $ H _ {0} $ is rejected in favour of $ H _ {1} $ if the event $ \{ X \in K \} $ is observed in an experiment). Also, the constructed test is a similar test, which means that
$$ \tag{2 } \int\limits _ { \mathfrak X } \phi ( x) d {\mathsf P} _ \theta = \alpha \ \textrm{ for } \textrm{ all } \theta \in \Theta _ {0} . $$
It follows from (1) and (2) that the critical region $ K $ of a non-randomized similar test has the property:
$$ {\mathsf P} _ \theta \{ X \in K \} = \alpha \ \textrm{ for } \textrm{ all } \theta \in \Theta _ {0} . $$
Accordingly, J. Neyman and E.S. Pearson emphasized the latter feature of the critical set of a non-randomized similar test and called $ K $ a "region similar to the sample space" $ \mathfrak X $, in the sense that the two probabilities $ {\mathsf P} _ \theta \{ X \in K \} $ and $ {\mathsf P} _ \theta \{ X \in \mathfrak X \} $ are independent of $ \theta \in \Theta _ {0} $.
References
[1] | E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986) |
[2] | B.L. van der Waerden, "Mathematische Statistik" , Springer (1957) |
[3] | J. Neyman, E.S. Pearson, "On the problem of the most efficient tests of statistical hypotheses" Philos. Trans. Roy. Soc. London Ser. A , 231 (1933) pp. 289–337 |
[4] | E.L. Lehmann, H. Scheffé, "Completeness, similar regions, and unbiased estimation I" Sankhyā , 10 (1950) pp. 305–340 |
[5] | E.L. Lehmann, H. Scheffé, "Completeness, similar regions, and unbiased estimation II" Sankhyā , 15 (1955) pp. 219–236 |
Similarity region. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Similarity_region&oldid=49583