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Thread: Consistent Estimators and Consistency in Mean Square

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    Re: Consistent Estimators and Consistency in Mean Square




    Was referring to this part:

    \Pr\{|\hat{\theta} - \theta| > \epsilon\} = \left\{\begin{matrix} 0 & \text{if} & \epsilon \geq n \\ \frac {1} {n} & \text{if} & 0 < \epsilon < n \end{matrix}\right.

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    Re: Consistent Estimators and Consistency in Mean Square

    I know what you were referring to. It's just silly to care about that. You know if epsilon is greater than n if epsilon is greater than n. The thing is that it probably isn't - but that doesn't matter because as n gets large the bottom part goes to 0.

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    Re: Consistent Estimators and Consistency in Mean Square

    Yes that is not crucial. But I just wonder how do you calculate the CDF for a given probability mass function of a discrete random variable?

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    Re: Consistent Estimators and Consistency in Mean Square

    And to find if it's consistent in mean square, I need to show that it's the E[\hat\theta] - \theta tends to 0 as n tends to infinity but what is E[\hat\theta] - \theta in each case? i.e. How can I work out expectation if I am given probabilities?

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    Re: Consistent Estimators and Consistency in Mean Square

    \hat{\theta} is just a binary discrete random variable from definition. You should not have trouble with finding its moment if you have taken the elementary probability class.

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    Re: Consistent Estimators and Consistency in Mean Square

    So I have to use method of moments and find the first moment?

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    Re: Consistent Estimators and Consistency in Mean Square

    I do not mean "method of moment". How do you calculate the expectation of a discrete random variable? E.g. Bernoulli random variable?

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    Re: Consistent Estimators and Consistency in Mean Square

    Multiply the possible value by it's probability? so in the first case I have \theta (1-(1/n)) and second case (\theta+n)(1/n)

    Is this correct? Also if it is, do I need to add them together to find E[\hat\theta]

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    Re: Consistent Estimators and Consistency in Mean Square

    Yes you need to add them together. This is the basic formula for evaluating the expectation for a discrete random variable. I guess the class you are taking right now assume you are very familiar with this.

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    Re: Consistent Estimators and Consistency in Mean Square

    And to find the variance of the estimator?

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    Re: Consistent Estimators and Consistency in Mean Square

    Sorry it is really difficult to help you out if you do not know how to compute expectation and variance. Maybe you need to revise your year 1 course first.

    http://en.wikipedia.org/wiki/Varianc...andom_variable

    For me I will directly calculate E[(\hat{\theta} - \theta)^2] and show that it will not converge to 0.

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    Re: Consistent Estimators and Consistency in Mean Square


    Quote Originally Posted by BGM View Post
    Sorry it is really difficult to help you out if you do not know how to compute expectation and variance. Maybe you need to revise your year 1 course first.
    I second this.

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