Generalized likelihood ratio statistic for two binomial distributionsThis question develops hypothesis tests for the difference between two population proportions.Let X ∼ Binomial(n, p1) and Y ∼ Binomial(m, p2) and suppose X and Y are independent. The hypotheses to be tested are: H 0 : p 1 = p 2 , H A : p 1 &lt; p 2 o r p 1 &gt; p 2 (a) Find the generalized likelihood ratio statistic Λ for testing H0 vs. HA based on the data X and Y.

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Generalized likelihood ratio statistic for two binomial distributionsThis question develops hypothesis tests for the difference between two population proportions.Let X ∼ Binomial(n, p1) and Y ∼ Binomial(m, p2) and suppose X and Y are independent. The hypotheses to be tested are:       H    0    :      p    1    =      p    2    ,      H    A    :      p    1    &amp;lt;      p    2       o  r         p    1    &amp;gt;      p    2  (a) Find the generalized likelihood ratio statistic Λ for testing H0 vs. HA based on the data X and Y.

Stella Andrade · Accepted Answer

Instead of            L              −      1            (                  s        2                    (                  s          2                +                  a          2                          )          2                      )    (  t  )  =      ∫    0    t    sin  ⁡  t  ⋅  cos  ⁡  (  a  t  −  a  τ  )    d  τyou should have            L              −      1            (                  s        2                    (                  s          2                +                  a          2                          )          2                      )    (  t  )  =      1    a        ∫    0    t    sin  ⁡  (  a  τ  )  ⋅  cos  ⁡  (  a  t  −  a  τ  )    d  τbecause the convolution of functions f,g is defined as  (  f  ∗  g  )  (  t  )  =      ∫                  R              g  (  τ  )  ⋅  f  (  t  −  τ  )    d  τwhere   f  (  t  )  :=  cos  ⁡  (  a  ⋅  t  )  ⋅      1          [      0      ,      ∞      )        (  t  ),   g  (  t  )  :=      1    a    ⋅  sin  ⁡  (  a  ⋅  t  )  ⋅      1          [      0      ,      ∞      )        (  t  )

Let X ∼ Binomial(n, p1) and Y ∼ Binomial(m, p2) and suppose X and Y are independent. The hypotheses to be tested are: H0 : p1 = p2 HA : p1 < p2 or p1>p2

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