Unformatted text preview: 8.3 The LRT statistic is Î» ( y ) = sup Î¸ â‰¤ Î¸ L ( Î¸  y 1 ,...,y m ) sup Î˜ L ( Î¸  y 1 ,...,y m ) . Let y = âˆ‘ m i =1 y i , and note that the MLE in the numerator is min { y/m,Î¸ } (see Exercise 7.12) while the denominator has y/m as the MLE (see Example 7.2.7). Thus Î» ( y ) = Â· 1 if y/m â‰¤ Î¸ ( Î¸ ) y (1Î¸ ) my ( y/m ) y (1y/m ) my if y/m > Î¸ , and we reject H if ( Î¸ ) y (1Î¸ ) my ( y/m ) y (1y/m ) my < c. To show that this is equivalent to rejecting if y > b , we could show Î» ( y ) is decreasing in y so that Î» ( y ) < c occurs for y > b > mÎ¸ . It is easier to work with log Î» ( y ), and we have log Î» ( y ) = y log Î¸ + ( my ) log (1Î¸ )y log Â¸ y m Â¹( my ) log Â± my m Â² ,...
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.
 Spring '12
 Dr.Hackney
 Statistics, Binomial

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