Homework Answers
a) The cumulative distribution of 
is given by
![Rightarrow F_{Y_{(1)}}(y_1)=1 - [P(Y_1 > y_1)]^n](http://img.homeworklib.com/questions/e4fadec0-4ad5-11ea-8d05-7b459ef2e4f7.png?x-oss-process=image/resize,w_560)
![Rightarrow F_{Y_{(1)}}(y_1)=1 - [1 - P(Y_1 leq y_1)]^n](http://img.homeworklib.com/questions/e554fd60-4ad5-11ea-969a-25b535a92ade.png?x-oss-process=image/resize,w_560)
![Rightarrow F_{Y_{(1)}}(y_1)=1 - [1 - (1-e^{-rac{y_1}{ heta }})]^n](http://img.homeworklib.com/questions/e58ec5e0-4ad5-11ea-80a8-f3a3283cd344.png?x-oss-process=image/resize,w_560)
b) The pdf of
is given by
c) 
So, it is an unbiased estimator,
But, 
, so 
is an unbiased estimator of 
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2. Let Yı, ..., Yn be a random sample from an Exponential distribution with density function...
5. Let Yi,Y2, , Yn be a random sample of size n from the pdf (a)...
5. Let Yi,Y2, , Yn be a random sample of size n from the pdf (a) Show that θ = y is an unbiased estimator for θ (b) Show that θ = 1Y is a minimum-variance estimator for θ.
4. Let Yi, ½, . . . , Yn be a random sample from some pdf/pmf...
4. Let Yi, ½, . . . , Yn be a random sample from some pdf/pmf f(y; θ)·Let W be a point estimator h(y, Y2, . . . , Yn) for θ. The bias of W as a point estimator for θ is defined as W Blase(W) = E(W)- The mean square error of W is defined as MSEe(W) = E(W-0)2 (a) Using properties of expected values, and the definition of variance from PSTAT 120A/B, show that MSEe(W) = Vare(W)...
Let Yı, Y2, ...,Yn be an iid sample from a population distribution described by the pdf...
Let Yı, Y2, ...,Yn be an iid sample from a population distribution described by the pdf fy(y|0) = (@+ 1) yº, o<y<1 for 0> - -1. (a) Find the MOM estimator of 0. (b) Find the maximum likelihood estimator (MLE) of 0. (c) Find the MLE of the population mean E(Y) = 0 +1 0 + 2 You do not need to prove that the above is true. Just find its MLE.
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x...
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x/0 -e fx(x) MSE(1). Hint: What is the (a) Show that distribution of Y/1)? nY1 is an unbiased estimator for 0 and find (b) Show that 02 = Yn is an unbiased estimator for 0 and find MSE(O2). (c) Find the efficiency of 01 relative to 02. Which estimate is "better" (i.e. more efficient)?
8. Let X1,...,Xn denote a random...
(1 point) Let Yı, Y2, ..., Yn be a random sample from the probability density function...
(1 point) Let Yı, Y2, ..., Yn be a random sample from the probability density function f(yla) = |aya-2/5° f(y ) 0 <y< 5 otherwise 0 for > -1. Find an estimator for a using the method of moments.
Let Yı,Y2, ..., Yn be iid from a population following the shifted exponential distribution with scale...
Let Yı,Y2, ..., Yn be iid from a population following the shifted exponential distribution with scale parameter B = 1. The pdf of the population distribution is given by fy(y\0) = y-0) = e x I(y > 0). The "shift" @ > 0 is the only unknown parameter. (a) Find L(@ly), the likelihood function of 0. (b) Find a sufficient statistic for 0 using the Factorization Theorem. (Hint: O is bounded above by y(1) min{Y1, 42, ..., .., Yn}.) (c)...
3. Consider a random sample Yı, ,Yn from a Uniform[0, θ]. In class we discussed the...
3. Consider a random sample Yı, ,Yn from a Uniform[0, θ]. In class we discussed the method of ,y,). We moment estimator θ-2Y and the maximum likelihood estimator θ-maxx,Yo, derived the Bias and MSE for both estimators. With the intent to correct the bias of the mle θ we proposed the following new estimator -Imax where the subscript u stands for "unbiased." (a) Find the MSE of (b) Compare the MSE of θυ to the MSE of θ, the original...
1. Let Y1, . . . ,Y,, be a random sample from a population with density...
1. Let Y1, . . . ,Y,, be a random sample from a population with density function 0, otherwise (a) Find the method of moments estimator of θ (b) Show that Yan.-max(Yi, . . . ,%) is sufficient for 02] (Hint: Recall the indicator function given by I(A)1 if A is true and 0 otherwise.) (c) Determine the density function of Yn) and hence find a function of Ym) that is an unbiased estimator of θ (d) Find c so...
Let Yı, ..., Yn be an independent and identically distribution sample from the distribution function f(y)...
Let Yı, ..., Yn be an independent and identically distribution sample from the distribution function f(y) = 3y?, for 0 Sy <1. (a) Show the sample mean, y converges in probability to some constant, c. Find c. (b) Find a function that converges in probability to log(C).
1. (a) Let Yi,... , Yn be a random sample from a distribution with mean θ and finite variance σ2....
1. (a) Let Yi,... , Yn be a random sample from a distribution with mean θ and finite variance σ2. Find the BLUE of θ and justify that it is, in fact, the Best Linear Unbiased Estimate. sample variance.
1. (a) Let Yi,... , Yn be a random sample from a distribution with mean θ and finite variance σ2. Find the BLUE of θ and justify that it is, in fact, the Best Linear Unbiased Estimate. sample variance.
5. Let Yi,Y2, , Yn be a random sample of size n from the pdf (a) Show that θ = y is an unbiased estimator for θ (b) Show that θ = 1Y is a minimum-variance estimator for θ.
4. Let Yi, ½, . . . , Yn be a random sample from some pdf/pmf f(y; θ)·Let W be a point estimator h(y, Y2, . . . , Yn) for θ. The bias of W as a point estimator for θ is defined as W Blase(W) = E(W)- The mean square error of W is defined as MSEe(W) = E(W-0)2 (a) Using properties of expected values, and the definition of variance from PSTAT 120A/B, show that MSEe(W) = Vare(W)...
Let Yı, Y2, ...,Yn be an iid sample from a population distribution described by the pdf fy(y|0) = (@+ 1) yº, o<y<1 for 0> - -1. (a) Find the MOM estimator of 0. (b) Find the maximum likelihood estimator (MLE) of 0. (c) Find the MLE of the population mean E(Y) = 0 +1 0 + 2 You do not need to prove that the above is true. Just find its MLE.
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x/0 -e fx(x) MSE(1). Hint: What is the (a) Show that distribution of Y/1)? nY1 is an unbiased estimator for 0 and find (b) Show that 02 = Yn is an unbiased estimator for 0 and find MSE(O2). (c) Find the efficiency of 01 relative to 02. Which estimate is "better" (i.e. more efficient)?
8. Let X1,...,Xn denote a random...
(1 point) Let Yı, Y2, ..., Yn be a random sample from the probability density function f(yla) = |aya-2/5° f(y ) 0 <y< 5 otherwise 0 for > -1. Find an estimator for a using the method of moments.
Let Yı,Y2, ..., Yn be iid from a population following the shifted exponential distribution with scale parameter B = 1. The pdf of the population distribution is given by fy(y\0) = y-0) = e x I(y > 0). The "shift" @ > 0 is the only unknown parameter. (a) Find L(@ly), the likelihood function of 0. (b) Find a sufficient statistic for 0 using the Factorization Theorem. (Hint: O is bounded above by y(1) min{Y1, 42, ..., .., Yn}.) (c)...
3. Consider a random sample Yı, ,Yn from a Uniform[0, θ]. In class we discussed the method of ,y,). We moment estimator θ-2Y and the maximum likelihood estimator θ-maxx,Yo, derived the Bias and MSE for both estimators. With the intent to correct the bias of the mle θ we proposed the following new estimator -Imax where the subscript u stands for "unbiased." (a) Find the MSE of (b) Compare the MSE of θυ to the MSE of θ, the original...
1. Let Y1, . . . ,Y,, be a random sample from a population with density function 0, otherwise (a) Find the method of moments estimator of θ (b) Show that Yan.-max(Yi, . . . ,%) is sufficient for 02] (Hint: Recall the indicator function given by I(A)1 if A is true and 0 otherwise.) (c) Determine the density function of Yn) and hence find a function of Ym) that is an unbiased estimator of θ (d) Find c so...
Let Yı, ..., Yn be an independent and identically distribution sample from the distribution function f(y) = 3y?, for 0 Sy <1. (a) Show the sample mean, y converges in probability to some constant, c. Find c. (b) Find a function that converges in probability to log(C).
1. (a) Let Yi,... , Yn be a random sample from a distribution with mean θ and finite variance σ2. Find the BLUE of θ and justify that it is, in fact, the Best Linear Unbiased Estimate. sample variance.
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