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Estimate parameters with moments method
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I am trying to know how to estimate a parameter with the moments method. The wikipedia article and similar websites are too confusing and formal for me to understand. I'm looking for a more basic and school type "how-to".
For example ;
Let $Z_1,Z_2,...,Z_n$ a simple sample of an Erlang random variable whose function of probability density is given by:
$$ f(z)begincaseslambda^2ze^-lambda z & z ge 0\0 & elseendcases$$
How do I estimate $lambda$ with the moments method ? How can I generalize the method to any (maybe not extra-hard) problem of the same type ?
EDIT :
I am now trying to solve the problem and so far;
$$E[X] = n/lambda$$
$$widehatE[X] = int_0^infty lambda z^2 e^-lambda z dz = frac2lambda^2$$
And now I'm stuck, what to do with the second moment ? Or do I equal both the theorical and empirical, giving $lambda = 2/n$ ?
statistical-inference
asked Jul 21 at 14:58
Dranna
1105
add a comment |Â
up vote
0
down vote
favorite
I am trying to know how to estimate a parameter with the moments method. The wikipedia article and similar websites are too confusing and formal for me to understand. I'm looking for a more basic and school type "how-to".
For example ;
Let $Z_1,Z_2,...,Z_n$ a simple sample of an Erlang random variable whose function of probability density is given by:
$$ f(z)begincaseslambda^2ze^-lambda z & z ge 0\0 & elseendcases$$
How do I estimate $lambda$ with the moments method ? How can I generalize the method to any (maybe not extra-hard) problem of the same type ?
EDIT :
I am now trying to solve the problem and so far;
$$E[X] = n/lambda$$
$$widehatE[X] = int_0^infty lambda z^2 e^-lambda z dz = frac2lambda^2$$
And now I'm stuck, what to do with the second moment ? Or do I equal both the theorical and empirical, giving $lambda = 2/n$ ?
statistical-inference
asked Jul 21 at 14:58
Dranna
1105
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I am trying to know how to estimate a parameter with the moments method. The wikipedia article and similar websites are too confusing and formal for me to understand. I'm looking for a more basic and school type "how-to".
For example ;
Let $Z_1,Z_2,...,Z_n$ a simple sample of an Erlang random variable whose function of probability density is given by:
$$ f(z)begincaseslambda^2ze^-lambda z & z ge 0\0 & elseendcases$$
How do I estimate $lambda$ with the moments method ? How can I generalize the method to any (maybe not extra-hard) problem of the same type ?
EDIT :
I am now trying to solve the problem and so far;
$$E[X] = n/lambda$$
$$widehatE[X] = int_0^infty lambda z^2 e^-lambda z dz = frac2lambda^2$$
And now I'm stuck, what to do with the second moment ? Or do I equal both the theorical and empirical, giving $lambda = 2/n$ ?
statistical-inference
asked Jul 21 at 14:58
Dranna
1105
I am trying to know how to estimate a parameter with the moments method. The wikipedia article and similar websites are too confusing and formal for me to understand. I'm looking for a more basic and school type "how-to".
For example ;
Let $Z_1,Z_2,...,Z_n$ a simple sample of an Erlang random variable whose function of probability density is given by:
$$ f(z)begincaseslambda^2ze^-lambda z & z ge 0\0 & elseendcases$$
How do I estimate $lambda$ with the moments method ? How can I generalize the method to any (maybe not extra-hard) problem of the same type ?
EDIT :
I am now trying to solve the problem and so far;
$$E[X] = n/lambda$$
$$widehatE[X] = int_0^infty lambda z^2 e^-lambda z dz = frac2lambda^2$$
And now I'm stuck, what to do with the second moment ? Or do I equal both the theorical and empirical, giving $lambda = 2/n$ ?
statistical-inference
asked Jul 21 at 14:58
Dranna
1105
edited Jul 21 at 16:52
asked Jul 21 at 14:58
Dranna
1105
asked Jul 21 at 14:58
Dranna
1105
asked Jul 21 at 14:58
Dranna
1105
1105
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
A very simple guide:
1) Determine the amount of parameters you need to estimate.
2) Express the moments (eg. the mean and the second moment) through the parameters you want to esitmate, those can for instance be found on wikipedia.
3) Calculate the corresponding empirical moments from your sample.
4) Now you have (2) equations that can be solved for the parameters.
I hope this helps.
Edit:
The empirical first moment is simply the sample mean. This applies for any of the moments: $frac1nsum_i=1^n X_i^k$ is the $k$'th empirical moment. The expection of $Z$, given it follows the Erlang distribution, is given by $frac2lambda$. As this type of Erlang distribution has only one parameter, $hatlambda=frac2frac1nsum_i=1^n X_i$.
answered Jul 21 at 15:21
Rasmus
364
Finding the equations is the real problem for me. The first moment is n/lambda correct ? But the empirical moment is what (lambda/lambda-1)² ? And I don't know how to proceed for the second moment
– Dranna
Jul 21 at 15:39
I edited my question with what I am now trying to do to solve the problem. I don't know how to use the second moment, can you help me out ?
– Dranna
Jul 21 at 16:53
1
I edited my response to elaborate a bit.
– Rasmus
Jul 21 at 19:24
Thank you, I now understand. I accepter your post as the answer.
– Dranna
Jul 21 at 19:27
You are welcome. Note that i changed the formula for $hatlambda$.
– Rasmus
Jul 21 at 19:29
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
A very simple guide:
1) Determine the amount of parameters you need to estimate.
2) Express the moments (eg. the mean and the second moment) through the parameters you want to esitmate, those can for instance be found on wikipedia.
3) Calculate the corresponding empirical moments from your sample.
4) Now you have (2) equations that can be solved for the parameters.
I hope this helps.
Edit:
The empirical first moment is simply the sample mean. This applies for any of the moments: $frac1nsum_i=1^n X_i^k$ is the $k$'th empirical moment. The expection of $Z$, given it follows the Erlang distribution, is given by $frac2lambda$. As this type of Erlang distribution has only one parameter, $hatlambda=frac2frac1nsum_i=1^n X_i$.
answered Jul 21 at 15:21
Rasmus
364
Finding the equations is the real problem for me. The first moment is n/lambda correct ? But the empirical moment is what (lambda/lambda-1)² ? And I don't know how to proceed for the second moment
– Dranna
Jul 21 at 15:39
I edited my question with what I am now trying to do to solve the problem. I don't know how to use the second moment, can you help me out ?
– Dranna
Jul 21 at 16:53
1
I edited my response to elaborate a bit.
– Rasmus
Jul 21 at 19:24
Thank you, I now understand. I accepter your post as the answer.
– Dranna
Jul 21 at 19:27
You are welcome. Note that i changed the formula for $hatlambda$.
– Rasmus
Jul 21 at 19:29
add a comment |Â
up vote
1
down vote
accepted
A very simple guide:
1) Determine the amount of parameters you need to estimate.
2) Express the moments (eg. the mean and the second moment) through the parameters you want to esitmate, those can for instance be found on wikipedia.
3) Calculate the corresponding empirical moments from your sample.
4) Now you have (2) equations that can be solved for the parameters.
I hope this helps.
Edit:
The empirical first moment is simply the sample mean. This applies for any of the moments: $frac1nsum_i=1^n X_i^k$ is the $k$'th empirical moment. The expection of $Z$, given it follows the Erlang distribution, is given by $frac2lambda$. As this type of Erlang distribution has only one parameter, $hatlambda=frac2frac1nsum_i=1^n X_i$.
answered Jul 21 at 15:21
Rasmus
364
Finding the equations is the real problem for me. The first moment is n/lambda correct ? But the empirical moment is what (lambda/lambda-1)² ? And I don't know how to proceed for the second moment
– Dranna
Jul 21 at 15:39
I edited my question with what I am now trying to do to solve the problem. I don't know how to use the second moment, can you help me out ?
– Dranna
Jul 21 at 16:53
1
I edited my response to elaborate a bit.
– Rasmus
Jul 21 at 19:24
Thank you, I now understand. I accepter your post as the answer.
– Dranna
Jul 21 at 19:27
You are welcome. Note that i changed the formula for $hatlambda$.
– Rasmus
Jul 21 at 19:29
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
A very simple guide:
1) Determine the amount of parameters you need to estimate.
2) Express the moments (eg. the mean and the second moment) through the parameters you want to esitmate, those can for instance be found on wikipedia.
3) Calculate the corresponding empirical moments from your sample.
4) Now you have (2) equations that can be solved for the parameters.
I hope this helps.
Edit:
The empirical first moment is simply the sample mean. This applies for any of the moments: $frac1nsum_i=1^n X_i^k$ is the $k$'th empirical moment. The expection of $Z$, given it follows the Erlang distribution, is given by $frac2lambda$. As this type of Erlang distribution has only one parameter, $hatlambda=frac2frac1nsum_i=1^n X_i$.
answered Jul 21 at 15:21
Rasmus
364
A very simple guide:
1) Determine the amount of parameters you need to estimate.
2) Express the moments (eg. the mean and the second moment) through the parameters you want to esitmate, those can for instance be found on wikipedia.
3) Calculate the corresponding empirical moments from your sample.
4) Now you have (2) equations that can be solved for the parameters.
I hope this helps.
Edit:
The empirical first moment is simply the sample mean. This applies for any of the moments: $frac1nsum_i=1^n X_i^k$ is the $k$'th empirical moment. The expection of $Z$, given it follows the Erlang distribution, is given by $frac2lambda$. As this type of Erlang distribution has only one parameter, $hatlambda=frac2frac1nsum_i=1^n X_i$.
answered Jul 21 at 15:21
Rasmus
364
edited Jul 21 at 19:27
answered Jul 21 at 15:21
Rasmus
364
answered Jul 21 at 15:21
Rasmus
364
answered Jul 21 at 15:21
Rasmus
364
364
Finding the equations is the real problem for me. The first moment is n/lambda correct ? But the empirical moment is what (lambda/lambda-1)² ? And I don't know how to proceed for the second moment
– Dranna
Jul 21 at 15:39
I edited my question with what I am now trying to do to solve the problem. I don't know how to use the second moment, can you help me out ?
– Dranna
Jul 21 at 16:53
1
I edited my response to elaborate a bit.
– Rasmus
Jul 21 at 19:24
Thank you, I now understand. I accepter your post as the answer.
– Dranna
Jul 21 at 19:27
You are welcome. Note that i changed the formula for $hatlambda$.
– Rasmus
Jul 21 at 19:29
add a comment |Â
Finding the equations is the real problem for me. The first moment is n/lambda correct ? But the empirical moment is what (lambda/lambda-1)² ? And I don't know how to proceed for the second moment
– Dranna
Jul 21 at 15:39
I edited my question with what I am now trying to do to solve the problem. I don't know how to use the second moment, can you help me out ?
– Dranna
Jul 21 at 16:53
1
I edited my response to elaborate a bit.
– Rasmus
Jul 21 at 19:24
Thank you, I now understand. I accepter your post as the answer.
– Dranna
Jul 21 at 19:27
You are welcome. Note that i changed the formula for $hatlambda$.
– Rasmus
Jul 21 at 19:29
Finding the equations is the real problem for me. The first moment is n/lambda correct ? But the empirical moment is what (lambda/lambda-1)² ? And I don't know how to proceed for the second moment
– Dranna
Jul 21 at 15:39
Finding the equations is the real problem for me. The first moment is n/lambda correct ? But the empirical moment is what (lambda/lambda-1)² ? And I don't know how to proceed for the second moment
– Dranna
Jul 21 at 15:39
I edited my question with what I am now trying to do to solve the problem. I don't know how to use the second moment, can you help me out ?
– Dranna
Jul 21 at 16:53
I edited my question with what I am now trying to do to solve the problem. I don't know how to use the second moment, can you help me out ?
– Dranna
Jul 21 at 16:53
1
1
I edited my response to elaborate a bit.
– Rasmus
Jul 21 at 19:24
I edited my response to elaborate a bit.
– Rasmus
Jul 21 at 19:24
Thank you, I now understand. I accepter your post as the answer.
– Dranna
Jul 21 at 19:27
Thank you, I now understand. I accepter your post as the answer.
– Dranna
Jul 21 at 19:27
You are welcome. Note that i changed the formula for $hatlambda$.
– Rasmus
Jul 21 at 19:29
You are welcome. Note that i changed the formula for $hatlambda$.
– Rasmus
Jul 21 at 19:29
add a comment |Â
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