Consider the independent observations x1, x2, . . . , xn from the gamma distribution with...

Consider the independent observations x1, x2, . . . , xn from the gamma distribution with pdf f(x) = (1/ Γ(α)β^α)x^(α−1)e ^(−x/β), x > 0 and 0 otherwise.

a. Write out the likelihood function

b. Write out a set of equations that give the maximum likelihood estimators of α and β.

c. Assuming α is known, find the likelihood estimator Bˆ of β.

d. Find the expected value and variance of Bˆ

Solutions

Expert Solution

orchestra answered 1 year ago

Next > < Previous

Related Solutions

Let X1,X2, . . . , Xn be a random sample from the uniform distribution with...

Let X1,X2, . . . , Xn be a random sample from the uniform distribution with pdf f(x; θ1, θ2) = 1/(2θ2), θ1 − θ2 < x < θ1 + θ2, where −∞ < θ1 < ∞ and θ2 > 0, and the pdf is equal to zero elsewhere. (a) Show that Y1 = min(Xi) and Yn = max(Xi), the joint sufficient statistics for θ1 and θ2, are complete. (b) Find the MVUEs of θ1 and θ2.

Let X1, X2, · · · , Xn (n ≥ 30) be i.i.d observations from N(µ1,...

Let X1, X2, · · · , Xn (n ≥ 30) be i.i.d observations from N(µ1, σ12 ) and Y1, Y2, · · · , Yn be i.i.d observations from N(µ2, σ22 ). Also assume that X's and Y's are independent. Suppose that µ1, µ2, σ12 , σ22 are unknown. Find an approximate 95% confidence interval for µ1µ2.

Let X1, … , Xn. be a random sample from gamma (2, theta) distribution. a) Show...

Let X1, … , Xn. be a random sample from gamma (2, theta) distribution. a) Show that it is the regular case of the exponential class of distributions. b) Find a complete, sufficient statistic for theta. c) Find the unique MVUE of theta. Justify each step.

Let X1, X2, . . . , Xn represent a random sample from a Rayleigh distribution...

Let X1, X2, . . . , Xn represent a random sample from a Rayleigh distribution with pdf f(x; θ) = (x/θ) * e^x^2/(2θ) for x > 0 (a) It can be shown that E(X2 P ) = 2θ. Use this fact to construct an unbiased estimator of θ based on n i=1 X2 i . (b) Estimate θ from the following n = 10 observations on vibratory stress of a turbine blade under specified conditions: 16.88 10.23 4.59 6.66...

Let X1, X2, ..., Xn be a random sample of size from a distribution with probability...

Let X1, X2, ..., Xn be a random sample of size from a distribution with probability density function f(x) = λxλ−1 , 0 < x < 1, λ > 0 a) Get the method of moments estimator of λ. Calculate the estimate when x1 = 0.1, x2 = 0.2, x3 = 0.3. b) Get the maximum likelihood estimator of λ. Calculate the estimate when x1 = 0.1, x2 = 0.2, x3 = 0.3.

. Let X1, X2, ..., Xn be a random sample of size 75 from a distribution...

. Let X1, X2, ..., Xn be a random sample of size 75 from a distribution whose probability distribution function is given by f(x; θ) = ( 1 for 0 < x < 1, 0 otherwise. Use the central limit theorem to approximate P(0.45 < X < 0.55)

We have a random sample of observations on X: x1, x2, x3, x4,…,xn. Consider the following...

We have a random sample of observations on X: x1, x2, x3, x4,…,xn. Consider the following estimator of the population mean: x* = x1/2 + x2/4 + x3/4. This estimator uses only the first three observations. a) Prove that x* is an unbiased estimator. b) Derive the variance of x* c) Is x* an efficient estimator? A consistent estimator? Explain.

Let X1,...,Xn be a random sample from a gamma distribution with shape parameter α and rate...

Let X1,...,Xn be a random sample from a gamma distribution with shape parameter α and rate β (note that this may be a different gamma specification than you are used to). Then f(x | α, β) = (βα/Γ(α))*x^(α−1) * e^(−βx). where x, α, β > 0 (a) Derive the equations that yield the maximum likelihood estimators of α and β. Can they be solved explicitly? Hint: don’t forget your maximum checks, and it may help to do some internet searching...

Suppose X1, X2, ..., Xn is a random sample from a Poisson distribution with unknown parameter...

Suppose X1, X2, ..., Xn is a random sample from a Poisson distribution with unknown parameter µ. a. What is the mean and variance of this distribution? b. Is X1 + 2X6 − X8 an estimator of µ? Is it a good estimator? Why or why not? c. Find the moment estimator and MLE of µ. d. Show the estimators in (c) are unbiased. e. Find the MSE of the estimators in (c). Given the frequency table below: X 0...

Suppose we have a random sample of n observations {x1, x2, x3,…xn}. Consider the following estimator...

Suppose we have a random sample of n observations {x1, x2, x3,…xn}. Consider the following estimator of µx, the population mean. Z = 12x1 + 14x2 + 18x3 +…+ 12n-1xn−1 + 12nxn Verify that for a finite sample size, Z is a biased estimator. Recall that Bias(Z) = E(Z) − µx. Write down a formula for Bias(Z) as a function of n and µx. Is Z asymptotically unbiased? Explain. Use the fact that for 0 < r < 1, limn→∞i=1nri...

Subjects