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Math Help - Estimating Parameters

  1. #1
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    Estimating Parameters

    Hi guys,

    Parameter Estimation Question

    I tried using [IMG][/IMG] tags, but they would not work
    Any help is greatly appreciated!
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  2. #2
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    Quote Originally Posted by lindah View Post
    Hi guys,

    Parameter Estimation Question

    I tried using [IMG][/IMG] tags, but they would not work
    Any help is greatly appreciated!
    What do you get for the mean and variance of the distribution?

    (you should also tell us that f(x)=0 for x<a and that what you give applies for x>=a)

    CB
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  3. #3
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    Hi CB,

    Yes sorry I omitted the information about the bounds

    The math tags don't seem to be working so my E[X] and E[X^2] are as follows:
    Mean and Variance, Moments

    It is at this point where I get slightly confused. I have just rearranged the parameters in each of the 2 equations to get the answer I obtained above. I've worked through the examples for a normal distribution, but its much more straightforward than this one.
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  4. #4
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    Quote Originally Posted by lindah View Post
    Hi CB,

    Yes sorry I omitted the information about the bounds

    The math tags don't seem to be working so my E[X] and E[X^2] are as follows:
    Mean and Variance, Moments

    It is at this point where I get slightly confused. I have just rearranged the parameters in each of the 2 equations to get the answer I obtained above. I've worked through the examples for a normal distribution, but its much more straightforward than this one.
    Your moments are wrong, how have you calculated them?

    CB
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  5. #5
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    For m_{1}, I took the integral with respect to x of [x.f(x)]
    For m_{2}, I took the integral of [x^2 f(x)]
    The integral boundaries were [0,infinity)
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  6. #6
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    Quote Originally Posted by lindah View Post
    For m_{1}, I took the integral with respect to x of [x.f(x)]
    For m_{2}, I took the integral of [x^2 f(x)]
    The integral boundaries were [0,infinity)
    The limits are a to infinity.

    CB
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  7. #7
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    Quote Originally Posted by CaptainBlack View Post
    The limits are a to infinity.

    CB
    Hi CB,

    In hindsight that does make sense to me now to have those limits.

    I have recalculated:
    1) E[X] = a + \theta
    2) E[X^2] = a^2 + 2a\theta + 2\theta^2

    Can I confirm if these are correct before the next step?

    Thank you!
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  8. #8
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    Quote Originally Posted by lindah View Post
    Hi CB,

    In hindsight that does make sense to me now to have those limits.

    I have recalculated:
    1) E[X] = a + \theta
    2) E[X^2] = a^2 + 2a\theta + 2\theta^2

    Can I confirm if these are correct before the next step?

    Thank you!
    Yes, they are correct, but you will find the next part easier if you work with the second central moment (the variance) the effect is the same as working with the raw second moment.

    CB
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  9. #9
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    Quote Originally Posted by CaptainBlack View Post
    Yes, they are correct, but you will find the next part easier if you work with the second central moment (the variance) the effect is the same as working with the raw second moment.
    Hi CB,

    Please bear with me for this part, as its a hurdle I can't get past for these types of questions.

    Using what I've derived above:
    var[X] = E[X^2] - \left(  E[X]   \right)^2 =\theta^2

    I equate:
    m_{1} = a + \theta
    a = m_{1} - \theta

    Similarly for m_{2} and then re-written in terms of \theta:
    m_{2} = \theta^2
    \theta = \sqrt{m_{2}} = \sqrt{\sigma^2} = \sigma

    Substituting this into the expression for a the estimates are as follows?:
    \hat{a} = \bar{X} - \sigma
    \hat{\theta} = \sigma
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  10. #10
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    Quote Originally Posted by lindah View Post
    Hi CB,

    Please bear with me for this part, as its a hurdle I can't get past for these types of questions.

    Using what I've derived above:
    var[X] = E[X^2] - \left(  E[X]   \right)^2 =\theta^2

    I equate:
    m_{1} = a + \theta
    a = m_{1} - \theta

    Similarly for m_{2} and then re-written in terms of \theta:
    m_{2} = \theta^2
    \theta = \sqrt{m_{2}} = \sqrt{\sigma^2} = \sigma

    Substituting this into the expression for a the estimates are as follows?:
    \hat{a} = \bar{X} - \sigma
    \hat{\theta} = \sigma
    That looks right (except make it clear that \sigma is the sample based SD).

    CB
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  11. #11
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    Thank you for your guidance!!!

    Linda
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  12. #12
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    Quote Originally Posted by CaptainBlack View Post
    That looks right (except make it clear that \sigma is the sample based SD).

    CB
    Hi CB,

    Can I just confirm if I am allowed to equate the Variance as the second moment or would it have to be calculated as:

    var[X] = \theta^2
    var[X]=E[X^2] - (E[X])^2
    E[X^2] = var[X] +(E[X])^2

    Now equate m_{2}=E[X^2], NOT m_{2}=var[X] as I did before?

    Doing this I should get:
    m_{2}=\theta^2 + m_{1}^2
    and so rearranging will give me:
    \theta = \sqrt{m_{2} - m_{1}^2}

    Instead of
    \theta = \hat{\sigma}
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  13. #13
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    Quote Originally Posted by lindah View Post
    Hi CB,

    Can I just confirm if I am allowed to equate the Variance as the second moment or would it have to be calculated as:

    var[X] = \theta^2
    var[X]=E[X^2] - (E[X])^2
    E[X^2] = var[X] +(E[X])^2

    Now equate m_{2}=E[X^2], NOT m_{2}=var[X] as I did before?

    Doing this I should get:
    m_{2}=\theta^2 + m_{1}^2
    and so rearranging will give me:
    \theta = \sqrt{m_{2} - m_{1}^2}

    Instead of
    \theta = \hat{\sigma}
    You start with a pair of simultaneous equations:

    \overline{X}=\mu(\theta)

    \overline{X^2}=m_2(\theta)

    Then subtract the square of the first from the second to get the equations:

    \overline{X}=\mu(\theta)

    \overline{X^2}-\overline{X}^2=m_2(\theta)-(\mu(\theta))^2

    The left hand side of the second equation above is the sample variance (that is with n rather than (n-1) in it) and the right hand side is the variance of the population.

    So equating the raw first two moments is the same as equating the means and variances, but in this case you will find it easier to work with the population variance than with the second moment (the algebra comes out nicer)

    CB
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  14. #14
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    Thanks for the clarification!
    I see what you mean now and is definitely easier to deal with the population variance
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