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Taylor Series and Random Variables

  1. Sep 14, 2014 #1
    1. The problem statement, all variables and given/known data

    A standard procedure for finding an approximate mean and variance of a function of a variable is to use a Taylor Expansion for the function about the mean of the variable. Suppose the variable is y, and that its mean and standard deviation are "u" and "o".

    f(y) = f(u) + f'(u)(y-u) + f''(u)(((y-u)^2)/2!)) + f'''(u)((y-u)^3)/3!)) + ...

    Consider the case of f(.) as e^(.). By taking the expectation of both sides of this equation, explain why the bias correction factor given in Equation A is an overcorrection if the residual series has a negative skewness, where skewness p of a random variable y is defined by

    p = E((y-u)^3)/(o^3)

    Equation A = x^hat_t = e^(m_t + s_t)*e^((1/2)(o^2))

    where x_t is observed series, m_t is the trend, s_t is seasonal effect

    2. Relevant equations


    3. The attempt at a solution

    Im not even really sure where to start. If someone could point me in the right direction, it would be greatly appreciated
     
  2. jcsd
  3. Sep 14, 2014 #2

    statdad

    User Avatar
    Homework Helper

    Try what is suggested. If you begin with expected values like this:
    [tex]
    E(f(y)) = f(\mu) + f'(\mu)E(y-\mu) + \frac{f'(\mu)}{2!} E((y-\mu)^2) + \frac{f'''(\mu)}{3!} E((y-\mu)^3)
    [/tex]

    What do the assumptions tell you about the final term (term with the expectation of the cubic)?
     
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