Generalized likelihood ratio statistic for two binomial distributions
$begingroup$
This 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:
H0 : p1 = p2
HA : p1 < p2 or p1>p2
(a) Find the generalized likelihood ratio statistic Λ for testing H0 vs. HA based on the data X and Y .
I am slightly unsure of the distribution of X-Y. I think that X-Y~Binomial(n-m,p) under the null, but then under the alternative, what would the distribution be? And are there any suggestions on calculating the MLE of X-Y in order to calculate the GLRT
statistics
asked Dec 14 '15 at 0:04
Jacob RodgersJacob Rodgers
18210
$endgroup$
add a comment |
$begingroup$
This 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:
H0 : p1 = p2
HA : p1 < p2 or p1>p2
(a) Find the generalized likelihood ratio statistic Λ for testing H0 vs. HA based on the data X and Y .
I am slightly unsure of the distribution of X-Y. I think that X-Y~Binomial(n-m,p) under the null, but then under the alternative, what would the distribution be? And are there any suggestions on calculating the MLE of X-Y in order to calculate the GLRT
statistics
asked Dec 14 '15 at 0:04
Jacob RodgersJacob Rodgers
18210
$endgroup$
add a comment |
$begingroup$
This 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:
H0 : p1 = p2
HA : p1 < p2 or p1>p2
(a) Find the generalized likelihood ratio statistic Λ for testing H0 vs. HA based on the data X and Y .
I am slightly unsure of the distribution of X-Y. I think that X-Y~Binomial(n-m,p) under the null, but then under the alternative, what would the distribution be? And are there any suggestions on calculating the MLE of X-Y in order to calculate the GLRT
statistics
asked Dec 14 '15 at 0:04
Jacob RodgersJacob Rodgers
18210
$endgroup$
This 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:
H0 : p1 = p2
HA : p1 < p2 or p1>p2
(a) Find the generalized likelihood ratio statistic Λ for testing H0 vs. HA based on the data X and Y .
I am slightly unsure of the distribution of X-Y. I think that X-Y~Binomial(n-m,p) under the null, but then under the alternative, what would the distribution be? And are there any suggestions on calculating the MLE of X-Y in order to calculate the GLRT
statistics
statistics
asked Dec 14 '15 at 0:04
Jacob RodgersJacob Rodgers
18210
asked Dec 14 '15 at 0:04
Jacob RodgersJacob Rodgers
18210
asked Dec 14 '15 at 0:04
Jacob RodgersJacob Rodgers
18210
asked Dec 14 '15 at 0:04
Jacob RodgersJacob Rodgers
18210
asked Dec 14 '15 at 0:04
Jacob RodgersJacob Rodgers
18210
18210
add a comment |
add a comment |
1 Answer
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The binomial distributions are only additive when $n = m ;$ also, subtraction does not yield a binomial distribution. A likelihood estimator of $p_{1}$ is $frac{X}{n}$ and a likelihood estimator of $p_{2}$ is $frac{Y}{m} .$ To test the hypothesis $p_{1}-p_{2} = 0$ versus the alternative hypothesis $p_{1}-p_{2}neq0$ you can use the test statistic $$frac{X}{n}-frac{Y}{m} .$$ If $n$ and $m$ are sufficiently large and $p_{1}$ and $p_{2}$ are not too small (so that $min(np_{1},n(1-p_{1}))>5$ and $min(mp_{2},m(1-p_{2}))>5$), then the Central Limit Theorem may be used in the computation of the p-value of a given data-set with which you investigate the null hypothesis.
answered Dec 14 '15 at 0:38
Anders MusztaAnders Muszta
63435
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1 Answer
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1 Answer
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$begingroup$
The binomial distributions are only additive when $n = m ;$ also, subtraction does not yield a binomial distribution. A likelihood estimator of $p_{1}$ is $frac{X}{n}$ and a likelihood estimator of $p_{2}$ is $frac{Y}{m} .$ To test the hypothesis $p_{1}-p_{2} = 0$ versus the alternative hypothesis $p_{1}-p_{2}neq0$ you can use the test statistic $$frac{X}{n}-frac{Y}{m} .$$ If $n$ and $m$ are sufficiently large and $p_{1}$ and $p_{2}$ are not too small (so that $min(np_{1},n(1-p_{1}))>5$ and $min(mp_{2},m(1-p_{2}))>5$), then the Central Limit Theorem may be used in the computation of the p-value of a given data-set with which you investigate the null hypothesis.
answered Dec 14 '15 at 0:38
Anders MusztaAnders Muszta
63435
$endgroup$
add a comment |
$begingroup$
The binomial distributions are only additive when $n = m ;$ also, subtraction does not yield a binomial distribution. A likelihood estimator of $p_{1}$ is $frac{X}{n}$ and a likelihood estimator of $p_{2}$ is $frac{Y}{m} .$ To test the hypothesis $p_{1}-p_{2} = 0$ versus the alternative hypothesis $p_{1}-p_{2}neq0$ you can use the test statistic $$frac{X}{n}-frac{Y}{m} .$$ If $n$ and $m$ are sufficiently large and $p_{1}$ and $p_{2}$ are not too small (so that $min(np_{1},n(1-p_{1}))>5$ and $min(mp_{2},m(1-p_{2}))>5$), then the Central Limit Theorem may be used in the computation of the p-value of a given data-set with which you investigate the null hypothesis.
answered Dec 14 '15 at 0:38
Anders MusztaAnders Muszta
63435
$endgroup$
add a comment |
$begingroup$
The binomial distributions are only additive when $n = m ;$ also, subtraction does not yield a binomial distribution. A likelihood estimator of $p_{1}$ is $frac{X}{n}$ and a likelihood estimator of $p_{2}$ is $frac{Y}{m} .$ To test the hypothesis $p_{1}-p_{2} = 0$ versus the alternative hypothesis $p_{1}-p_{2}neq0$ you can use the test statistic $$frac{X}{n}-frac{Y}{m} .$$ If $n$ and $m$ are sufficiently large and $p_{1}$ and $p_{2}$ are not too small (so that $min(np_{1},n(1-p_{1}))>5$ and $min(mp_{2},m(1-p_{2}))>5$), then the Central Limit Theorem may be used in the computation of the p-value of a given data-set with which you investigate the null hypothesis.
answered Dec 14 '15 at 0:38
Anders MusztaAnders Muszta
63435
$endgroup$
The binomial distributions are only additive when $n = m ;$ also, subtraction does not yield a binomial distribution. A likelihood estimator of $p_{1}$ is $frac{X}{n}$ and a likelihood estimator of $p_{2}$ is $frac{Y}{m} .$ To test the hypothesis $p_{1}-p_{2} = 0$ versus the alternative hypothesis $p_{1}-p_{2}neq0$ you can use the test statistic $$frac{X}{n}-frac{Y}{m} .$$ If $n$ and $m$ are sufficiently large and $p_{1}$ and $p_{2}$ are not too small (so that $min(np_{1},n(1-p_{1}))>5$ and $min(mp_{2},m(1-p_{2}))>5$), then the Central Limit Theorem may be used in the computation of the p-value of a given data-set with which you investigate the null hypothesis.
answered Dec 14 '15 at 0:38
Anders MusztaAnders Muszta
63435
answered Dec 14 '15 at 0:38
Anders MusztaAnders Muszta
63435
answered Dec 14 '15 at 0:38
Anders MusztaAnders Muszta
63435
answered Dec 14 '15 at 0:38
Anders MusztaAnders Muszta
63435
63435
add a comment |
add a comment |
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