A. Parametric bootstrap simulation for the distribution of the t-statistic in no

Question

A. Parametric bootstrap simulation for the distribution of the t-statistic in non-normal samples. Consider the nonnormal (and rather skewed) setup: Let X1,...,Xn i.i.d. from a Gamma distribution with parameters: shape = a and scale = s (both positive and unknown). The mean and variance are denoted E(X) = mu and Var(X) = sigma^2. Let n=9. Assume that you observed a sample mean arX = 10 and a sample variance (using n-1 in the denominator) hatSigma^2 = 25. (ii) Estimate the parameters a and s using Method of Moments (MOM). Recall that for a Gamma distribution: E(X) = a*s and Var(X) = a*s^2.

Explanation / Answer

We are given that,

Let X1,...,Xn i.i.d. from a Gamma distribution with parameters: shape = a and scale = s (both positive and unknown). The mean and variance are denoted E(X) = mu and Var(X) = sigma^2. Let n=9. Assume that you observed a sample mean arX = 10 and a sample variance (using n-1 in the denominator) hatSigma^2 = 25.

mean = 10

and var = 25

And also we know that for gamma distribution,

mean = as

and var = as2

divide var by mean,

var / mean = as2 / as

var / mean = s

s = 25 / 10 = 2.5

And as = 10

a*2.5 = 10

a = 10 / 2.5 = 4

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