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2. Suppose Y1,...,Yn are IID discrete random variables with P(Y; = 0) = 60 P(Y; =...
Suppose Y1, ..., Yn denote a random sample of size n from an exponential distribu- tion...
Suppose Y1, ..., Yn denote a random sample of size n from an exponential distribu- tion with mean 0. a) (5 points) Find the bias and MSE of the estimator ôz = nY1). b) (3 points) Consider another estimator ôz = Y. Find the efficiency of ôı relative to 62. c) (7 points) Prove that 297 Yi is a pivotal quantity and find a 95% confidence interval for 0.
1. Suppose Yi,½, , Yn is an iid sample from a Bernoulli(p) population distribution, where 0< p<1 is unknown. The population pmf is py(ulp) otherwise 0, (a) Prove that Y is the maximum likelihoo...
1. Suppose Yi,½, , Yn is an iid sample from a Bernoulli(p) population distribution, where 0< p<1 is unknown. The population pmf is py(ulp) otherwise 0, (a) Prove that Y is the maximum likelihood estimator of p. (b) Find the maximum likelihood estimator of T(p)-loglp/(1 - p)], the log-odds of p.
1. Suppose Yi,½, , Yn is an iid sample from a Bernoulli(p) population distribution, where 0
Suppose Y1, Y2, ..., Yn is an iid sample from a Pareto population distribution described by...
Suppose Y1, Y2, ..., Yn is an iid sample from a Pareto population distribution described by the pdf fy(y|0) = 4ao y -0-1 y > 20, 2 where the parameter do is known. The unknown parameter is 0 > 0. (a) Find the MOM estimator of 0. (b) Find the MLE of 0.
Suppose Y1, Y2, ..., Yn are such that Y; ~ Bernoulli(p) and let X = 2h+Yi....
Suppose Y1, Y2, ..., Yn are such that Y; ~ Bernoulli(p) and let X = 2h+Yi. (a) [1 point] Use the distribution of X to show that the method of moments estimator of p is ÔMM = Lizzi. (Work that is unclear or that cannot be followed from step to step will not recieve full credit.) (b) [2 points] Show that the method of moments estimator PMM is a consistent estimator of p. Please show your work to support your...
Let Y1,Y2, …… Yn be a random sample from the distribution f(y)
Let Y1,Y2, …… Yn be a random sample from the distribution f(y) = θxθ-1 where 0 < x < 1 and 0 < θ < ∞. Show that the maximum likelihood estimator (MLE) for θ is
5. Consider a random sample Y1, . . . , Yn from a distribution with pdf f(y|θ) = 1 θ 2 xe−x/θ , 0 < x < ∞. Calcula...
5. Consider a random sample Y1, . . . , Yn from a distribution with pdf f(y|θ) = 1 θ 2 xe−x/θ , 0 < x < ∞. Calculate the ML estimator of θ. 6. Consider the pdf g(y|α) = c(1 + αy2 ), −1 < y < 1. (a) Show that g(y|α) is a pdf when c = 3 6 + 2α . (b) Calculate E(Y ) and E(Y 2 ). Referencing your calculations, explain why M1 can’t be...
Suppose that a discrete, random variable Y(with three possible outcomes) has the following distribution: prob(Y=1)=q, prob(Y=2)=p,...
Suppose that a discrete, random variable Y(with three possible outcomes) has the following distribution: prob(Y=1)=q, prob(Y=2)=p, and prob(Y=3)=1-p-q. A random variable of size 109 is drawn from this distribution and the random variables are denoted Y1, Y2,....Y109. (A) Derive the likelihood function for the parameters p and q (B) Derive the formulas for MLE of p and q
. Suppose the Y1, Y2, · · · , Yn denote a random sample from a...
. Suppose the Y1, Y2, · · · , Yn denote a random sample from a
population with Rayleigh distribution (Weibull distribution with
parameters 2, θ) with density function f(y|θ) = 2y θ e −y 2/θ, θ
> 0, y > 0
Consider the estimators ˆθ1 = Y(1) = min{Y1, Y2, · · · , Yn},
and ˆθ2 = 1 n Xn i=1 Y 2 i .
ii) (10 points) Determine if ˆθ1 and ˆθ2 are unbiased
estimators, and in...
The random vector Y = (Y1, ..., Yn)T is such that Y = Xβ + ε,...
The random vector Y = (Y1, ...,
Yn)T is such that Y = Xβ + ε, where X is an n
× p full-rank matrix of known constants, β is a p-length vector of
unknown parameters, and ε is an n-length vector of random
variables. A multiple linear regression model is fitted to the
data.
(a) Write down the multiple linear regression model assumptions in
matrix format.
(b) Derive the least squares estimator β^ of β.
(c) Using the data:...
8.4.12 Suppose that X, .., Y, are iid random variables having the ernoulli(p) distribution where ...
8.4.12 Suppose that X, .., Y, are iid random variables having the ernoulli(p) distribution where p e (0, 1) is the unknown parameter. With (0, l ), derive the randomized UMP level α test for l, P-Po p reassigned oE versus H p Po where p, is a number between 0 and 1
8.4.12 Suppose that X, .., Y, are iid random variables having the ernoulli(p) distribution where p e (0, 1) is the unknown parameter. With (0, l ),...
Suppose Y1, ..., Yn denote a random sample of size n from an exponential distribu- tion with mean 0. a) (5 points) Find the bias and MSE of the estimator ôz = nY1). b) (3 points) Consider another estimator ôz = Y. Find the efficiency of ôı relative to 62. c) (7 points) Prove that 297 Yi is a pivotal quantity and find a 95% confidence interval for 0.
1. Suppose Yi,½, , Yn is an iid sample from a Bernoulli(p) population distribution, where 0< p<1 is unknown. The population pmf is py(ulp) otherwise 0, (a) Prove that Y is the maximum likelihood estimator of p. (b) Find the maximum likelihood estimator of T(p)-loglp/(1 - p)], the log-odds of p.
1. Suppose Yi,½, , Yn is an iid sample from a Bernoulli(p) population distribution, where 0
Suppose Y1, Y2, ..., Yn is an iid sample from a Pareto population distribution described by the pdf fy(y|0) = 4ao y -0-1 y > 20, 2 where the parameter do is known. The unknown parameter is 0 > 0. (a) Find the MOM estimator of 0. (b) Find the MLE of 0.
Suppose Y1, Y2, ..., Yn are such that Y; ~ Bernoulli(p) and let X = 2h+Yi. (a) [1 point] Use the distribution of X to show that the method of moments estimator of p is ÔMM = Lizzi. (Work that is unclear or that cannot be followed from step to step will not recieve full credit.) (b) [2 points] Show that the method of moments estimator PMM is a consistent estimator of p. Please show your work to support your...
. Suppose the Y1, Y2, · · · , Yn denote a random sample from a
population with Rayleigh distribution (Weibull distribution with
parameters 2, θ) with density function f(y|θ) = 2y θ e −y 2/θ, θ
> 0, y > 0
Consider the estimators ˆθ1 = Y(1) = min{Y1, Y2, · · · , Yn},
and ˆθ2 = 1 n Xn i=1 Y 2 i .
ii) (10 points) Determine if ˆθ1 and ˆθ2 are unbiased
estimators, and in...
The random vector Y = (Y1, ...,
Yn)T is such that Y = Xβ + ε, where X is an n
× p full-rank matrix of known constants, β is a p-length vector of
unknown parameters, and ε is an n-length vector of random
variables. A multiple linear regression model is fitted to the
data.
(a) Write down the multiple linear regression model assumptions in
matrix format.
(b) Derive the least squares estimator β^ of β.
(c) Using the data:...
8.4.12 Suppose that X, .., Y, are iid random variables having the ernoulli(p) distribution where p e (0, 1) is the unknown parameter. With (0, l ), derive the randomized UMP level α test for l, P-Po p reassigned oE versus H p Po where p, is a number between 0 and 1
8.4.12 Suppose that X, .., Y, are iid random variables having the ernoulli(p) distribution where p e (0, 1) is the unknown parameter. With (0, l ),...
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