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Sagot :
The given point estimators are unbiased and estimator b₁ is more efficient than b₂.
A statistic used to precisely estimate a population parameter is called an "unbiased estimator." An estimator's expected value must match the population parameter's value to be considered unbiased. The discrepancy between an estimator's expected value and the parameter value is known as bias. The estimator tends if this difference is not zero.
a) To show whether the given point estimators are unbiased, first, find whether the expected value of the given estimators is equal to the population parameter μ.
[tex]\begin{aligned}E(b_1) &= E(0.10X_1 + 0.20X_2 + 0.40X_3 + 0.30X_4)\\&=0.10E(X_1) + 0.20E(X_2) + 0.40E(X_3) +0.30E(X_4)\\&=0.10\mu+0.20\mu+0.40\mu+0.30\mu\\&=\mu\\E(b_2) &= E(0.25X_1 + 0.25X_2 + 0.30X_3 + 0.20X_4)\\&=0.25E(X_1) + 0.25E(X_2) + 0.30E(X_3) +0.20E(X_4)\\& = 0.25\mu + 0.25\mu + 0.30\mu + 0.20\mu\\&=\mu \end{aligned}[/tex]
Therefore, both estimators are found to be unbiased.
b) To show which estimator is more efficient, first, find the two-point estimators' variance.
[tex]\begin{aligned}V ar(b_1)&= Var(0.10X_1+0.20X_2+0.40X_3+0.30X_4)\\&= 0.10^2V ar(X_1)+0.20^2Var(X_2)+0.40^2Var(X_3)+0.30^2Var(X_4)\\&= 0.10^2\sigma^2+ 0.20^2\sigma^2+ 0.40^2\sigma^2 + 0.30^2\sigma^2\\&=0.3\sigma^2\\Var(b_2)&= Var(0.25X_1+0.25X_2+0.30X_3+0.20X_4)\\&= 0.25^2Var(X_1)+0.25^2Var(X_2)+0.30^2Var(X_3)+0.20^2Var(X_4)\\&= 0.25^2\sigma^2 + 0.25^2\sigma^2 + 0.30^2\sigma^2 + 0.20^2\sigma^2\\&= 0.255\sigma^2\end{aligned}[/tex]
From this, we can say that Var(b₁) > Var(b₂). Therefore, b₁ is more efficient.
The complete question is -
Let X₁, X₂, X₃, and X₄ be a random sample of observations from a population with mean μ and variance σ². Consider the following two-point estimators:
b₁ = 0.10X₁ + 0.20X₂ + 0.40X₃ + 0.30X₄, and
b₂ = 0.25X₁ + 0.25X₂ + 0.30X₃ + 0.20X₄
(a) Show that both estimators are unbiased.
(b) Which estimator is more efficient, b₁ or b₂? Explain in detail.
To know more about unbiased estimators:
https://brainly.com/question/29646467
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