Let 𝑋1,…,𝑋𝑛 be i.i.d. random variables with distribution (𝜃,𝜃) , for some unknown parameter 𝜃>0 .
Find an interval I𝜃 (that depends on 𝜃 ) centered about 𝑋⎯⎯⎯⎯⎯𝑛 such that
P(I𝜃 <- 𝜃 )=0.9for all 𝑛(i.e, not only for large 𝑛).
(Write barX_n for 𝑋⎯⎯⎯⎯⎯𝑛 . Use the estimate 𝑞0.05≈1.6448 for best results.)
I𝜃=[A𝜃, B𝜃] for
A𝜃=?
B𝜃=?
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Again, use the estimate 𝑞0.05≈1.6448 for best results.
Now, find a confidence interval Iplug-in with asymptotic confidence level 90% by plugging in 𝑋⎯⎯⎯⎯⎯𝑛 for all occurrences of 𝜃 in I𝜃 .
Iplug-in=[𝐴plug-in, 𝐵plug-in] for
𝐴plug-in=?
𝐵plug-in=?
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Finally, find a confidence interval Isolve for 𝜃 with nonasymptotic level 90% solving the bounds in I𝜃 for 𝜃 .
Isolve=[𝐴solve,𝐵solve] for
𝐴solve=?
𝐵solve=?
a) True
b) theta and variance = theta / n
c) barX_n - 1.6448*sqrt(theta/n)
d) barX_n - 1.6448*sqrt(barX_n/n)
clarification : Let 𝑋1,…,𝑋𝑛 be i.i.d. random variables with distribution N(𝜃,𝜃) , for some unknown parameter 𝜃>0 .
To find the interval I𝜃 centered around 𝑋⎯⎯⎯⎯⎯𝑛 such that P(I𝜃 < 𝜃) = 0.9 for all 𝑛, we need to calculate the quantile q0.05 and use it in the following formulas:
A𝜃 = 𝑋⎯⎯⎯⎯⎯𝑛 - q0.05/2
B𝜃 = 𝑋⎯⎯⎯⎯⎯𝑛 + q0.05/2
Using the estimate q0.05 ≈ 1.6448, we can substitute it into the formulas:
A𝜃 = 𝑋⎯⎯⎯⎯⎯𝑛 - 1.6448/2
B𝜃 = 𝑋⎯⎯⎯⎯⎯𝑛 + 1.6448/2
To find the interval I𝜃 centered about 𝑋̅_𝑛 such that P(I𝜃 < 𝜃) = 0.9 for all 𝑛, we can start by finding the distribution of 𝑋̅_𝑛.
Given that 𝑋1,…,𝑋𝑛 are i.i.d. random variables with a distribution 𝑓(𝜃,𝜃), we know that the sum of these variables 𝑋1 + 𝑋2 + ... + 𝑋𝑛 follows a gamma distribution with parameters 𝜃 and 𝑛.
The distribution of 𝑋̅_𝑛 is then the sum of 𝑋1 + 𝑋2 + ... + 𝑋𝑛 divided by 𝑛. Since the sum follows a gamma distribution with parameters 𝜃 and 𝑛, dividing by 𝑛 gives us a gamma distribution with parameters 𝜃 and 1/n.
To find the interval I𝜃, we need to find the values A𝜃 and B𝜃 such that P(A𝜃 < 𝑋̅_𝑛 < B𝜃) = 0.9.
Since the distribution of 𝑋̅_𝑛 is gamma(𝜃, 1/n), we can use the quantiles of the gamma distribution to find A𝜃 and B𝜃.
The quantile 𝑞(0.05) refers to the value below which 0.05 probability lies, which in this case is 1.6448.
To find A𝜃 and B𝜃, we need to find the lower and upper quantile of the gamma(𝜃, 1/n) distribution. Denoting them as 𝑎 and 𝑏 respectively, we have:
P(𝑎 < 𝑋̅_𝑛 < 𝑏) = 0.9
To find 𝑎 and 𝑏, we can solve the cumulative distribution function (CDF) of the gamma distribution:
Γ(𝜃, 1/n) = ∫[0,𝑎]𝑓(𝑥)𝑑𝑥 = 0.05
Γ(𝜃, 1/n) = ∫[0,𝑏]𝑓(𝑥)𝑑𝑥 = 0.95
Using a statistical software or calculator, we can find the values of 𝑎 and 𝑏.
The resulting interval I𝜃 = [A𝜃, B𝜃] will depend on the chosen value of 𝜃.
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To find the confidence interval Iplug-in with asymptotic confidence level 90%, we replace 𝜃 with 𝑋̅_𝑛 in the interval I𝜃.
That is, Iplug-in = [Aplug-in, Bplug-in] where:
Aplug-in = 𝑋̅_𝑛 - 𝑞(0.05) / √𝑛
Bplug-in = 𝑋̅_𝑛 + 𝑞(0.05) / √𝑛
Since 𝑋̅_𝑛 approaches 𝜃 in the limit, this interval provides an asymptotic approximation with a confidence level of 90%.
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To find the confidence interval Isolve for 𝜃 with nonasymptotic level 90%, we solve the bounds in I𝜃 for 𝜃.
Given I𝜃 = [A𝜃, B𝜃], we can solve for 𝜃 as follows:
P(𝑋̅_𝑛 > 𝜃) = 0.05
P(𝑋̅_𝑛 < 𝜃) = 0.95
Solving these equations for 𝜃 will give us the confidence interval Isolve = [Asolve, Bsolve] for 𝜃 with nonasymptotic level 90%.
Note that the values of Asolve and Bsolve may not be exact numbers, but rather functions of 𝑋̅_𝑛 and other parameters that depend on the distribution 𝑓(𝜃,𝜃).
It is recommended to use statistical software or computational tools to find the precise values of Asolve and Bsolve.