Question

Suppose that x_(1),dots,x_(n) are i.i.d. from Pareto( heta ,4), where heta >0 is unknown. Therefore, the pdf of a generic x is f_(x)(x; heta )=(4 heta ^(4))/(x^(5)),x>= heta ; heta >0 Reference the common distributions for cdf, mean, variance of a Pareto. a) Determine the method of moments estimator of heta . b) The method of moments estimator is unbiased. Determine its variance. Var[hat( heta )_(n,MaM)]= c) Consider all estimators of the form where c_(n) is a constant that can change with sample size. The MoM estimator is the sole unbiased member of this collection. Find the MSE for each collection member and determine which member has the smallest MSE when estimating heta . What is the value of c_(n) which minimizes the MSE? c_(n)= h) For what values of n is MSE_( heta )[Y_(n)]<=MSE_( heta )[hat( heta )_(n,MaM)] ? i) Since the first parameter in a Pareto distribution is a scale parameter, then the following statement is true: If W∼Pareto( heta ,eta ), then( W)/( heta )∼Pareto(1,eta ). Since hat( heta )_(n,MLE)∼ Pareto( heta , enter value), then (hat( heta )_(n,MLE))/( heta )∼ Pareto(1, enter value). Determine two constants a_(n) and b_(n) such that the following probability statement is true: Y_(n) heta Y_(n) heta Y_(n)f_(x_(n))(y; heta )=◻,y>= heta ; heta >0Y_(n) heta gW∼hat( heta )_(n,MLE)∼90%CI heta ( heta ,4), heta 2.5006,2.5912,2.6173,2.6480 2.7256,2.7309,3.2930,4.413690%jP(a_(n)<(hat( heta )_(n,MLE))/( heta ) Hint. useful to derive or look up the cdf of a pareto. j) Therefore, a generic formula for a 90%CI for heta is to kP(a_(n)<(hat( heta )_(n,MLE))/( heta ) Hint: useful to derive or look up the cdf of a pareto. f) Determine the bias, variance, and MSE of Y_(n), the MLE for heta . g Suppose that X,... X are i.i.d. from Pareto(0,4), where e > O is unknown. Therefore, the pdf of a generic X is x0; 0>0 Reference the common distributions for cdf, mean, variance of a Pareto. 4pts d) Find the Likelihood function and express it in simplified form. int adicaor fction Since the firt parameter in a Pareto distrbution is a scale parameter then the following statement is troe: IfW Part(e,, thenW/~Paret(1. 0|xxx- Since Par,cir valm, then Determine two constants a, and b, such that the following probability statement is trve: e It can be seen from the likelihood above that the data minimum is the maimmlikeoodetimator ofe.Define I,- mi{..,} Find the pdf of the random variable Y. 3pts a Determine the method of moments estimator of e. fy;0)- y8; 6>0 4pts b The method of momenis estimator is unbiased. Determine its variance. You should recopniz distribution Ulin: usefal to derive odf of a paret ] D Deterns nce,and MSE of Y,the MLE for e 3pts j) Therefore, a generic formula for a 90% CI for is pis c Consider all estimators of the form c,, where c, is a constant that can change with sample size, The MoM estimator is the sole unbiased member of this collection. Find the MSE for each collection member and determine which member has the smallest MSE when estimating O. Varian '[v.]- ME.IY. to 3gs oaug s jo dues saadauo oddn x unknown, presented in ascending order: (c,X,) - Variance VarcX- MSE,(c,) = g) Using MSE & Bias, show the MLE Ya is a consistent estimator of 6 2:7205 2:3912 3.6173 2:139 Use this data to evalate the90%condence ineal of part j to What is the value of c, which minimizes the MSE? C= hForwhat values ofn is MS,[v]MS[]

Name: suppose that x1dotsxn are iid from paretotheta 4 where theta 0 is unknown therefore the pdf of a generic x is fxxtheta 4theta 4x5xtheta theta 0 reference the common distributions for cdf 33956
Uploaded: 2023-01-13T15:00:11-08:00
Duration: 2 min 12 s
Channel: Sri K
Description: suppose that x1dotsxn are iid from paretotheta 4 where theta 0 is unknown therefore the pdf of a generic x is fxxtheta 4theta 4x5xtheta theta 0 reference the common distributions for cdf 33956

          Suppose that x_(1),dots,x_(n) are i.i.d. from Pareto(	heta ,4), where 	heta >0 is unknown.
Therefore, the pdf of a generic x is
f_(x)(x;	heta )=(4	heta ^(4))/(x^(5)),x>=	heta ;	heta >0
Reference the common distributions for cdf, mean, variance of a Pareto.
a) Determine the method of moments estimator of 	heta .
b) The method of moments estimator is unbiased. Determine its
variance.
Var[hat(	heta )_(n,MaM)]=
c) Consider all estimators of the form  where c_(n) is a constant that
can change with sample size. The MoM estimator is the sole
unbiased member of this collection. Find the MSE for each
collection member and determine which member has the smallest
MSE when estimating 	heta .
What is the value of c_(n) which minimizes the MSE?
c_(n)=
h) For what values of n is MSE_(	heta )[Y_(n)]<=MSE_(	heta )[hat(	heta )_(n,MaM)] ?
i) Since the first parameter in a Pareto distribution is a scale parameter,
then the following statement is true:
 If W∼Pareto(	heta ,eta ), then( W)/(	heta )∼Pareto(1,eta ). 
Since hat(	heta )_(n,MLE)∼ Pareto( 	heta , enter value), then (hat(	heta )_(n,MLE))/(	heta )∼ Pareto(1, enter value).
Determine two constants a_(n) and b_(n) such that the following probability
statement is true:
Y_(n)	heta Y_(n)	heta Y_(n)f_(x_(n))(y;	heta )=◻,y>=	heta ;	heta >0Y_(n)	heta  gW∼hat(	heta )_(n,MLE)∼90%CI	heta (	heta ,4),	heta 2.5006,2.5912,2.6173,2.6480
2.7256,2.7309,3.2930,4.413690%jP(a_(n)<(hat(	heta )_(n,MLE))/(	heta )
Hint. useful to derive or look up the cdf of a pareto.
j) Therefore, a generic formula for a 90%CI for 	heta  is
to
kP(a_(n)<(hat(	heta )_(n,MLE))/(	heta )
Hint: useful to derive or look up the cdf of a pareto.
f) Determine the bias, variance, and MSE of Y_(n), the MLE for 	heta .
g
Suppose that X,... X are i.i.d. from Pareto(0,4), where e > O is unknown. Therefore, the pdf of a generic X is x0; 0>0 Reference the common distributions for cdf, mean, variance of a Pareto.
4pts
d) Find the Likelihood function and express it in simplified form. int adicaor fction
Since the firt parameter in a Pareto distrbution is a scale parameter then the following statement is troe: IfW Part(e,, thenW/~Paret(1.
0|xxx-
Since  Par,cir valm, then Determine two constants a, and b, such that the following probability statement is trve:
e It can be seen from the likelihood above that the data minimum is the maimmlikeoodetimator ofe.Define I,- mi{..,} Find the pdf of the random variable Y.
3pts a Determine the method of moments estimator of e.
fy;0)-
y8; 6>0
4pts
b The method of momenis estimator is unbiased. Determine its variance.
You should recopniz distribution
Ulin: usefal to derive
 odf of a paret
]
D Deterns
nce,and MSE of Y,the MLE for e
3pts
j) Therefore, a generic formula for a 90% CI for  is
pis
c Consider all estimators of the form c,, where c, is a constant that can change with sample size, The MoM estimator is the sole unbiased member of this collection. Find the MSE for each collection member and determine which member has the smallest MSE when estimating O.
Varian
'[v.]- ME.IY.
to
3gs
oaug s jo dues saadauo  oddn x unknown, presented in ascending order:
(c,X,) - Variance VarcX- MSE,(c,) =
g) Using MSE & Bias, show the MLE Ya is a consistent estimator of 6
2:7205 2:3912 3.6173 2:139
Use this data to evalate the90%condence ineal of part j to
What is the value of c, which minimizes the MSE? C=
hForwhat values ofn is MS,[v]MS[]

suppose that x1dotsxn are iid from paretotheta 4 where theta 0 is unknown therefore the pdf of a generic x is fxxtheta 4theta 4x5xtheta theta 0 reference the common distributions for cdf 33956

Added by Lisa R.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Sri K

01/13/2023

Step 1

The population mean of a Pareto distribution with parameters \(\theta\) and 4 is given by \(E[X] = \frac{4\theta}{4-1} = \frac{4\theta}{3}\). The sample mean is the sample moment, which is the first sample moment \(\bar{X} = \frac{1}{n} \sum_{i=1}^{n} Show more…

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Suppose that x_(1),dots,x_(n) are i.i.d. from Pareto( heta ,4), where heta >0 is unknown. Therefore, the pdf of a generic x is f_(x)(x; heta )=(4 heta ^(4))/(x^(5)),x>= heta ; heta >0 Reference the common distributions for cdf, mean, variance of a Pareto. a) Determine the method of moments estimator of heta . b) The method of moments estimator is unbiased. Determine its variance. Var[hat( heta )_(n,MaM)]= c) Consider all estimators of the form where c_(n) is a constant that can change with sample size. The MoM estimator is the sole unbiased member of this collection. Find the MSE for each collection member and determine which member has the smallest MSE when estimating heta . What is the value of c_(n) which minimizes the MSE? c_(n)= h) For what values of n is MSE_( heta )[Y_(n)]<=MSE_( heta )[hat( heta )_(n,MaM)] ? i) Since the first parameter in a Pareto distribution is a scale parameter, then the following statement is true: If W∼Pareto( heta ,eta ), then( W)/( heta )∼Pareto(1,eta ). Since hat( heta )_(n,MLE)∼ Pareto( heta , enter value), then (hat( heta )_(n,MLE))/( heta )∼ Pareto(1, enter value). Determine two constants a_(n) and b_(n) such that the following probability statement is true: Y_(n) heta Y_(n) heta Y_(n)f_(x_(n))(y; heta )=◻,y>= heta ; heta >0Y_(n) heta gW∼hat( heta )_(n,MLE)∼90%CI heta ( heta ,4), heta 2.5006,2.5912,2.6173,2.6480 2.7256,2.7309,3.2930,4.413690%jP(a_(n)<(hat( heta )_(n,MLE))/( heta ) Hint. useful to derive or look up the cdf of a pareto. j) Therefore, a generic formula for a 90%CI for heta is to kP(a_(n)<(hat( heta )_(n,MLE))/( heta ) Hint: useful to derive or look up the cdf of a pareto. f) Determine the bias, variance, and MSE of Y_(n), the MLE for heta . g Suppose that X,... X are i.i.d. from Pareto(0,4), where e > O is unknown. Therefore, the pdf of a generic X is x0; 0>0 Reference the common distributions for cdf, mean, variance of a Pareto. 4pts d) Find the Likelihood function and express it in simplified form. int adicaor fction Since the firt parameter in a Pareto distrbution is a scale parameter then the following statement is troe: IfW Part(e,, thenW/~Paret(1. 0|xxx- Since Par,cir valm, then Determine two constants a, and b, such that the following probability statement is trve: e It can be seen from the likelihood above that the data minimum is the maimmlikeoodetimator ofe.Define I,- mi{..,} Find the pdf of the random variable Y. 3pts a Determine the method of moments estimator of e. fy;0)- y8; 6>0 4pts b The method of momenis estimator is unbiased. Determine its variance. You should recopniz distribution Ulin: usefal to derive odf of a paret ] D Deterns nce,and MSE of Y,the MLE for e 3pts j) Therefore, a generic formula for a 90% CI for is pis c Consider all estimators of the form c,, where c, is a constant that can change with sample size, The MoM estimator is the sole unbiased member of this collection. Find the MSE for each collection member and determine which member has the smallest MSE when estimating O. Varian '[v.]- ME.IY. to 3gs oaug s jo dues saadauo oddn x unknown, presented in ascending order: (c,X,) - Variance VarcX- MSE,(c,) = g) Using MSE & Bias, show the MLE Ya is a consistent estimator of 6 2:7205 2:3912 3.6173 2:139 Use this data to evalate the90%condence ineal of part j to What is the value of c, which minimizes the MSE? C= hForwhat values ofn is MS,[v]MS[]