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My Exercise Problem:

Let x1, x2, ..., x7 be observations of independent random variables X1, X2, ..., X7 with E(Xi) = μ and Var(Xi) = σ2, for all i = 1, 2, ..., 7.
Let σ2obs = $c(x_2^2 + x_6^2- \frac{(x_1 + x_7)^2}{2})$ be a point estimation of σ2.
Determine the constant c such that σ2obs becomes a unbiased estimator of variance.

The correct solution:
c = 1

What I need help with
How do I solve this problem? I am very new to point estimations and "unbiased estimators of variance and standard deviations". I want to gradually become better at solving these type of questions, where should I start? Perhaps it would be good for me to start with similar problems but on a easier level, any advice where I can find such problems?

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1 Answer 1

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Considering that $E(X^2)=V(X)+E^2(X)$ and setting

$$T=X_2^2+X_6^2-\frac{(X_1+X_7)^2}{2}$$

You get that

$$E(T)=2(\sigma^2+\mu^2)-\frac{1}{2}(2\sigma^2+4\mu^2)=\dots=\sigma^2$$

Thus T (your original estimator with $c=1$) is unbiased of variance, as requested

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  • $\begingroup$ How did you get $$E(T)=2(\sigma^2+\mu^2)-\frac{1}{2}(2\sigma^2+4\mu^2)$$ ? :) $\endgroup$ Commented Aug 9, 2021 at 18:20
  • $\begingroup$ @AugustJelemson : $$E(T)=E(X_2^2)+E(X_6^2)-\frac{1}{2}E[(X_1+X_7)^2]$$ and so on...by the way did you forget to accept my answer? $\endgroup$ Commented Aug 9, 2021 at 20:53
  • $\begingroup$ $E(X^2)=V(X)+E^2(X)$ is this a well known theorem/definition that I should know? :) When can I apply this? $\endgroup$ Commented Aug 10, 2021 at 8:52
  • $\begingroup$ @August Jelemson : like in this case when you have to derive second Moment given variance and mean. In this case any $X_i$ have the same mean $\mu$ and variance $\sigma^2$ $\endgroup$ Commented Aug 10, 2021 at 9:15

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