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Suppose that X1,..., Xn is a random sample from a gamma distribu- tion, The gamma distribution has parameters r and λ,...
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).
3. Let Xi, , Xn be a random sample from a Poisson distribution with p.m.f Assume...
3. Let Xi, , Xn be a random sample from a Poisson distribution with p.m.f Assume the prior distribution of Of λ is is an exponential with mean 1, i.e. the prior pdi g(A) e-λ, λ > 0 Note that the exponential distribution is a special gamma distribution; and a general gamma distribution with parameters α > 0 and β > 0 has the pd.f. h(A; α, β)-16(. otherwise Also the mean of a gamma random variable with the pd.f.h(Χα,...
Suppose that x1, . . . , xn are a random sample from a B(α, β)...
Suppose that x1, . . . , xn are a random sample from a B(α, β) distribution: f(x; α, β) = x^(α-1) (1-x)^(β-1) Here E[X] = α/(α + β) and E[X^2 ] = ((α + 1)α)/{(α + β + 1)(α + β)}. (a) Show that the method of moments, using the first two moments, gives the equations 0 = α(1 − m1 ) − βm1 m1 − m2 = α(m2 − m1 ) + βm2 (b) Determine the method of moments...
6.9 Find the method of moments estimators of the parameters, and e, in the gamma bution...
6.9 Find the method of moments estimators of the parameters, and e, in the gamma bution with the probability density function: 6.10 f(x) = – forro T(0) based on a random sample X. X... X. (Hint: Equate the mean and variance of the gamma distribution, the formulas for which are given in Section 2.8.3. to the correspondine sample quantities and i12 - A, respectively, and solve.) Find the method of moments estimators of the parameters, and in the beta distribution...
Let X1, X2, ..., Xn be a random sample from the distribution with probability density function...
Let X1, X2, ..., Xn be a random sample from the distribution with probability density function (0+1) A_1 fx(x) = fx(x; 0) = 20+1-xº(8 ?–1(8 - x), 0 < x < 8, 0> 0. a. Obtain the method of moments estimator of 8, 7. Enter a formula below. Use * for multiplication, / for divison, ^ for power. Use mi for the sample mean X and m2 for the second moment. That is, m1 = 7 = + Xi, m2...
1. Let X1, ..., Xn be a random sample from a distribution with the pdf le-x/0,...
1. Let X1, ..., Xn be a random sample from a distribution with the pdf le-x/0, x > 0, N = (0,00). (a) Find the maximum likelihood estimator of 0. (b) Find the method of moments estimator of 0. (c) Are the estimators in a) and b) unbiased? (d) What is the variance of the estimators in a) and b)? (e) Suppose the observed sample is 2.26, 0.31, 3.75, 6.92, 9.10, 7.57, 4.79, 1.41, 2.49, 0.59. Find the maximum likelihood...
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X...
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X with Bernoulli distribution has a probability mass function (pmf) of with E(X) = p and Var(X) = p(1-p). (a) Find the method of moments (MOM) estimator of p. (b) Find a sufficient statistic for p. (Hint: Be careful when you write the joint pmf. Don't forget to sum the whole power of each term, that is, for the second term you will have (1...
1. Let X1, ..., Xn be a random sample from a distribution with cumulative dist: 10,...
1. Let X1, ..., Xn be a random sample from a distribution with cumulative dist: 10, <<0 F(x) = (/), 0<x<B | 1, >B > (a) For this part, assume that is known and B is unknown. Find the method of moments estimator Boom of B. (b) For this part, assume that both 6 and B are unknown. Find the maximum likelihood estimators of 8 and B.
2. (a) Suppose that x1,... , Vn are a random sample from a gamma distribution with...
2. (a) Suppose that x1,... , Vn are a random sample from a gamma distribution with shape parameter α and rate parameter λ, Here α > 0 and λ > 0. Let θ-(α, β). Determine the log-likelihood, 00), and a 2-dimensional sufficient statistic for the data (b) Suppose that xi, ,Xn are a random sample from a U(-9,0) distribution. f(x; 8) otherwise Here θ > 0, Determine the likelihood, L(0), and a one-dimensional sufficient statistic. Note that the likelihood should...
Suppose that x = (x1,…..,xn) is a sample from the Gamma Distribution. Determine the minimal sufficient...
Suppose that x = (x1,…..,xn) is a sample from the Gamma Distribution. Determine the minimal sufficient statistics for this model.
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).
3. Let Xi, , Xn be a random sample from a Poisson distribution with p.m.f Assume the prior distribution of Of λ is is an exponential with mean 1, i.e. the prior pdi g(A) e-λ, λ > 0 Note that the exponential distribution is a special gamma distribution; and a general gamma distribution with parameters α > 0 and β > 0 has the pd.f. h(A; α, β)-16(. otherwise Also the mean of a gamma random variable with the pd.f.h(Χα,...
6.9 Find the method of moments estimators of the parameters, and e, in the gamma bution with the probability density function: 6.10 f(x) = – forro T(0) based on a random sample X. X... X. (Hint: Equate the mean and variance of the gamma distribution, the formulas for which are given in Section 2.8.3. to the correspondine sample quantities and i12 - A, respectively, and solve.) Find the method of moments estimators of the parameters, and in the beta distribution...
Let X1, X2, ..., Xn be a random sample from the distribution with probability density function (0+1) A_1 fx(x) = fx(x; 0) = 20+1-xº(8 ?–1(8 - x), 0 < x < 8, 0> 0. a. Obtain the method of moments estimator of 8, 7. Enter a formula below. Use * for multiplication, / for divison, ^ for power. Use mi for the sample mean X and m2 for the second moment. That is, m1 = 7 = + Xi, m2...
1. Let X1, ..., Xn be a random sample from a distribution with the pdf le-x/0, x > 0, N = (0,00). (a) Find the maximum likelihood estimator of 0. (b) Find the method of moments estimator of 0. (c) Are the estimators in a) and b) unbiased? (d) What is the variance of the estimators in a) and b)? (e) Suppose the observed sample is 2.26, 0.31, 3.75, 6.92, 9.10, 7.57, 4.79, 1.41, 2.49, 0.59. Find the maximum likelihood...
7. Let X1,....Xn random sample from a Bernoulli distribution with parameter p. A random variable X with Bernoulli distribution has a probability mass function (pmf) of with E(X) = p and Var(X) = p(1-p). (a) Find the method of moments (MOM) estimator of p. (b) Find a sufficient statistic for p. (Hint: Be careful when you write the joint pmf. Don't forget to sum the whole power of each term, that is, for the second term you will have (1...
1. Let X1, ..., Xn be a random sample from a distribution with cumulative dist: 10, <<0 F(x) = (/), 0<x<B | 1, >B > (a) For this part, assume that is known and B is unknown. Find the method of moments estimator Boom of B. (b) For this part, assume that both 6 and B are unknown. Find the maximum likelihood estimators of 8 and B.
2. (a) Suppose that x1,... , Vn are a random sample from a gamma distribution with shape parameter α and rate parameter λ, Here α > 0 and λ > 0. Let θ-(α, β). Determine the log-likelihood, 00), and a 2-dimensional sufficient statistic for the data (b) Suppose that xi, ,Xn are a random sample from a U(-9,0) distribution. f(x; 8) otherwise Here θ > 0, Determine the likelihood, L(0), and a one-dimensional sufficient statistic. Note that the likelihood should...
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