Question

In: Statistics and Probability

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.

Solutions

Expert Solution

please give a thumbs up if you like the solution


Related Solutions

2. Let X1, . . . , Xn be a random sample from the distribution with...
2. Let X1, . . . , Xn be a random sample from the distribution with pdf given by fX(x;β) = β 1(x ≥ 1). xβ+1 (a) Show that T = ni=1 log Xi is a sufficient statistic for β. Hint: Use n1n1n=exp log=exp −logxi .i=1 xi i=1 xi i=1 (b) Find the pdf of Y = logX, where X ∼ fX(x;β). (c) Find the distribution of T . Hint: Identify the distribution of Y and use mgfs. (d) Find...
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...
Let X1, . . . , Xn be a random sample from a uniform distribution on...
Let X1, . . . , Xn be a random sample from a uniform distribution on the interval [a, b] (i) Find the moments estimators of a and b. (ii) Find the MLEs of a and b.
2. Let X1, ..., Xn be a random sample from a uniform distribution on the interval...
2. Let X1, ..., Xn be a random sample from a uniform distribution on the interval (0, θ) where θ > 0 is a parameter. The prior distribution of the parameter has the pdf f(t) = βαβ/t^(β−1) for α < t < ∞ and zero elsewhere, where α > 0, β > 0. Find the Bayes estimator for θ. Describe the usefulness and the importance of Bayesian estimation. We are assuming that theta = t, but we are unsure if...
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 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)
: Let X1, X2, . . . , Xn be a random sample from the normal...
: Let X1, X2, . . . , Xn be a random sample from the normal distribution N(µ, 25). To test the hypothesis H0 : µ = 40 against H1 : µne40, let us define the three critical regions: C1 = {x¯ : ¯x ≥ c1}, C2 = {x¯ : ¯x ≤ c2}, and C3 = {x¯ : |x¯ − 40| ≥ c3}. (a) If n = 12, find the values of c1, c2, c3 such that the size of...
. Let X1, X2, . . . , Xn be a random sample from a normal...
. Let X1, X2, . . . , Xn be a random sample from a normal population with mean zero but unknown variance σ 2 . (a) Find a minimum-variance unbiased estimator (MVUE) of σ 2 . Explain why this is a MVUE. (b) Find the distribution and the variance of the MVUE of σ 2 and prove the consistency of this estimator. (c) Give a formula of a 100(1 − α)% confidence interval for σ 2 constructed using the...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT