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Let X1, X2, ..., Xn denote a random sample of size n from a population whose...
Let X1, X2, ..., Xn denote a random sample of size n from a population whose density fucntion is given by 383x-4 f S x f(x) = 0 elsewhere where ß > 0 is unknown. Consider the estimator ß = min(X1, X2, ...,Xn). Derive the bias of the estimator ß.
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...
Let X1, X2, ...,Xn denote a random sample of size n from a Pareto distribution. X(1)...
Let X1, X2, ...,Xn denote a random sample of size n from a Pareto distribution. X(1) = min(X1, X2, ..., Xn) has the cumulative distribution function given by: αη 1 - ( r> B X F(x) = . x <B 0 Show that X(1) is a consistent estimator of ß.
Let X1, X2, ..., Xn be a random sample from a Gamma( a , ) distribution....
Let X1, X2, ..., Xn be a random sample from a Gamma( a , ) distribution. That is, f(x;a,0) = loga xa-le-210, 0 < x <co, a>0,0 > 0. Suppose a is known. a. Obtain a method of moments estimator of 0, 0. b. Obtain the maximum likelihood estimator of 0, 0. c. Is O an unbiased estimator for 0 ? Justify your answer. "Hint": E(X) = p. d. Find Var(ë). "Hint": Var(X) = o/n. e. Find MSE(Ô).
Let X1, X2, ..., Xn be a random sample of size n from a population that...
Let X1, X2, ..., Xn be a random sample of size n from a population that can be modeled by the following probability model: axa-1 fx(x) = 0 < x < 0, a > 0 θα a) Find the probability density function of X(n) max(X1,X2, ...,Xn). b) Is X(n) an unbiased estimator for e? If not, suggest a function of X(n) that is an unbiased estimator for e.
Let X1, X2, ...... Xn be a random sample of size n from EXP() distribution , ,...
Let X1, X2, ......
Xn be a random sample of size n from
EXP()
distribution ,
, zero , elsewhere.
Given, mean of distribution
and variances
and mgf
a) Show that the mle
for
is
. Is
a consistent estimator for
?
b)Show that Fisher information
. Is mle of
an efficiency estimator for
? why or why not? Justify your answer.
c) what is the mle estimator of
? Is the mle of
a consistent estimator for
?
d) Is...
Let X1, ..., Xn be a random sample from a population with pdf f(x 1/8,0 <...
Let X1, ..., Xn be a random sample from a population with pdf f(x 1/8,0 < x < θ, zero elsewhere. Let Yi < < Y, be the order statistics. Show that Y/Yn and Yn are independent random variables
Let X1, X2, ...,Xn be a random sample of size n from a population that can...
Let X1, X2, ...,Xn be a random sample of size n from a population that can be modeled by the following probability model: axa-1 fx(x) = 0 < x < 0, a > 0 θα a) Find the probability density function of X(n) = max(X1, X2, ...,xn). b) Is X(n) an unbiased estimator for e? If not, suggest a function of X(n) that is an unbiased estimator for 0.
Let X1, X2, ..., Xn represent a random sample from each of the distributions having the...
Let X1, X2, ..., Xn represent a
random sample from each of the distributions having the following
pdf. Please find the maximum likelihood estimator for each
case:
(c) f(x; θ)--e-x/e,0 < x < 00, 0 < θ < oo, zero elsewhere (d) f(x; θ) e- , θ x < 00,-00 < θ < 00, zero elsewhere In each case, find the mie of a (x-6)
5. Let X1, X2,. , Xn be a random sample from a distribution with pdf of...
5. Let X1, X2,. , Xn be a random sample from a distribution with pdf of f(x) (0+1)x,0< x<1 a. What is the moment estimator for 0 using the method of moments technique? b. What is the MLE for 0?
Let X1, X2, ..., Xn denote a random sample of size n from a population whose density fucntion is given by 383x-4 f S x f(x) = 0 elsewhere where ß > 0 is unknown. Consider the estimator ß = min(X1, X2, ...,Xn). Derive the bias of the estimator ß.
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...
Let X1, X2, ...,Xn denote a random sample of size n from a Pareto distribution. X(1) = min(X1, X2, ..., Xn) has the cumulative distribution function given by: αη 1 - ( r> B X F(x) = . x <B 0 Show that X(1) is a consistent estimator of ß.
Let X1, X2, ..., Xn be a random sample from a Gamma( a , ) distribution. That is, f(x;a,0) = loga xa-le-210, 0 < x <co, a>0,0 > 0. Suppose a is known. a. Obtain a method of moments estimator of 0, 0. b. Obtain the maximum likelihood estimator of 0, 0. c. Is O an unbiased estimator for 0 ? Justify your answer. "Hint": E(X) = p. d. Find Var(ë). "Hint": Var(X) = o/n. e. Find MSE(Ô).
Let X1, X2, ..., Xn be a random sample of size n from a population that can be modeled by the following probability model: axa-1 fx(x) = 0 < x < 0, a > 0 θα a) Find the probability density function of X(n) max(X1,X2, ...,Xn). b) Is X(n) an unbiased estimator for e? If not, suggest a function of X(n) that is an unbiased estimator for e.
Let X1, X2, ......
Xn be a random sample of size n from
EXP()
distribution ,
, zero , elsewhere.
Given, mean of distribution
and variances
and mgf
a) Show that the mle
for
is
. Is
a consistent estimator for
?
b)Show that Fisher information
. Is mle of
an efficiency estimator for
? why or why not? Justify your answer.
c) what is the mle estimator of
? Is the mle of
a consistent estimator for
?
d) Is...
Let X1, ..., Xn be a random sample from a population with pdf f(x 1/8,0 < x < θ, zero elsewhere. Let Yi < < Y, be the order statistics. Show that Y/Yn and Yn are independent random variables
Let X1, X2, ...,Xn be a random sample of size n from a population that can be modeled by the following probability model: axa-1 fx(x) = 0 < x < 0, a > 0 θα a) Find the probability density function of X(n) = max(X1, X2, ...,xn). b) Is X(n) an unbiased estimator for e? If not, suggest a function of X(n) that is an unbiased estimator for 0.
Let X1, X2, ..., Xn represent a
random sample from each of the distributions having the following
pdf. Please find the maximum likelihood estimator for each
case:
(c) f(x; θ)--e-x/e,0 < x < 00, 0 < θ < oo, zero elsewhere (d) f(x; θ) e- , θ x < 00,-00 < θ < 00, zero elsewhere In each case, find the mie of a (x-6)
5. Let X1, X2,. , Xn be a random sample from a distribution with pdf of f(x) (0+1)x,0< x<1 a. What is the moment estimator for 0 using the method of moments technique? b. What is the MLE for 0?
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