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Testing hypothesis using the Neyman-Pearson Lemma

Unknown variance hypothesis testing for normal random variablesNeyman–Pearson lemma for non monotonic spacesAn Application of the Neyman-Pearson Lemma.Neyman-Pearson lemma and hypothesis testNeed little clarification about Neyman-Pearson LemmaApplications of Neyman-Pearson-LemmaWhy can't Neyman-Pearson be generalized to composite hypotheses?Understanding likelihood ratio in Neyman-Pearson LemmaUsing Neyman-Pearson Lemma to find UMP level-alpha test with composite null and alternative hypothesesNeyman-Pearson Lemma for two hypothesis pairs and three parameters

0

$begingroup$

This is the information I have, I want to test the hypotheses $H_0$ : λ = λ0 against the alternative hypotheses $H_0$ : λ = λ1,
where λ1 > λ0.

I want to do this by using the Neyman–Pearson Lemma, and that rejecting $H_0$ if M < t is a most powerful test, where M =∑ $X_k$. ( if M < t we reject, not accept).

So far I've calculated the likelihood ratio as:

$fraclambda_1lambda_0∑^n_k=1x^n(d-1)e^-∑^n_k=1x(lambda_1-lambda_0)$

How do I go from here with regards to the most powerful test?

statistics hypothesis-testing parameter-estimation

share|cite|improve this question

edited Mar 29 at 6:11

StubbornAtom

6,29831440

asked Mar 29 at 1:30

kingking

425

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Where is a sample $X_1,ldots,X_n$ in your likelihood ratio?
  $endgroup$
  – NCh
  Mar 29 at 1:34
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  My apologies @NCh I forgot to add this in, I've edited them in
  $endgroup$
  – king
  Mar 29 at 1:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is $lambda_2$? What is $x$? There is no $x$ in your task. Try to write likelihood functions separately.
  $endgroup$
  – NCh
  Mar 29 at 1:37
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sorry @NCh $lambda_2$ was a typo, it was supposed to be $lambda_0$, I guess I've been doing maths a bit too long. So if I write out the likelihood functions separately as $fraclambda_1lambda_0$ would that make it easier for me to do?
  $endgroup$
  – king
  Mar 29 at 1:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Likelihood function $L_lambda(X_1,ldots,X_n)=P_lambda(X_1)cdot P_lambda(X_2) cdotldots cdot P_lambda(X_n)$ and nothing else.
  $endgroup$
  – NCh
  Mar 29 at 1:47
  

  
  
  
  

 | 
show 4 more comments

0

$begingroup$

This is the information I have, I want to test the hypotheses $H_0$ : λ = λ0 against the alternative hypotheses $H_0$ : λ = λ1,
where λ1 > λ0.

I want to do this by using the Neyman–Pearson Lemma, and that rejecting $H_0$ if M < t is a most powerful test, where M =∑ $X_k$. ( if M < t we reject, not accept).

So far I've calculated the likelihood ratio as:

$fraclambda_1lambda_0∑^n_k=1x^n(d-1)e^-∑^n_k=1x(lambda_1-lambda_0)$

How do I go from here with regards to the most powerful test?

statistics hypothesis-testing parameter-estimation

share|cite|improve this question

edited Mar 29 at 6:11

StubbornAtom

6,29831440

asked Mar 29 at 1:30

kingking

425

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Where is a sample $X_1,ldots,X_n$ in your likelihood ratio?
  $endgroup$
  – NCh
  Mar 29 at 1:34
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  My apologies @NCh I forgot to add this in, I've edited them in
  $endgroup$
  – king
  Mar 29 at 1:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What is $lambda_2$? What is $x$? There is no $x$ in your task. Try to write likelihood functions separately.
  $endgroup$
  – NCh
  Mar 29 at 1:37
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sorry @NCh $lambda_2$ was a typo, it was supposed to be $lambda_0$, I guess I've been doing maths a bit too long. So if I write out the likelihood functions separately as $fraclambda_1lambda_0$ would that make it easier for me to do?
  $endgroup$
  – king
  Mar 29 at 1:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Likelihood function $L_lambda(X_1,ldots,X_n)=P_lambda(X_1)cdot P_lambda(X_2) cdotldots cdot P_lambda(X_n)$ and nothing else.
  $endgroup$
  – NCh
  Mar 29 at 1:47
  

  
  
  
  

 | 
show 4 more comments

$begingroup$

This is the information I have, I want to test the hypotheses $H_0$ : λ = λ0 against the alternative hypotheses $H_0$ : λ = λ1,
where λ1 > λ0.

I want to do this by using the Neyman–Pearson Lemma, and that rejecting $H_0$ if M < t is a most powerful test, where M =∑ $X_k$. ( if M < t we reject, not accept).

So far I've calculated the likelihood ratio as:

$fraclambda_1lambda_0∑^n_k=1x^n(d-1)e^-∑^n_k=1x(lambda_1-lambda_0)$

How do I go from here with regards to the most powerful test?

statistics hypothesis-testing parameter-estimation

share|cite|improve this question

edited Mar 29 at 6:11

StubbornAtom

6,29831440

asked Mar 29 at 1:30

kingking

425

$endgroup$

share|cite|improve this question

edited Mar 29 at 6:11

StubbornAtom

6,29831440

asked Mar 29 at 1:30

kingking

425

share|cite|improve this question

edited Mar 29 at 6:11

StubbornAtom

6,29831440

asked Mar 29 at 1:30

kingking

425

edited Mar 29 at 6:11

StubbornAtom

6,29831440

edited Mar 29 at 6:11

StubbornAtom

6,29831440

asked Mar 29 at 1:30

kingking

425

asked Mar 29 at 1:30

kingking

425