Question

In: Statistics and Probability

Let X1, X2, … , Xn be a random sample from a normal population with zero...

Let X1, X2, … , Xn be a random sample from a normal population with zero mean and unknown variance σ^2 . Find the maximum likelihood estimator of σ^2

Solutions

Expert Solution


Related Solutions

: 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...
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 be a random sample of size n from...
Let X1, X2, . . . , Xn be a random sample of size n from a Poisson distribution with unknown mean µ. It is desired to test the following hypotheses H0 : µ = µ0         versus     H1 : µ not equal to µ0 where µ0 > 0 is a given constant. Derive the likelihood ratio test statistic
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 Exp(?). Find the MVUE of the...
Let X1, X2, ..., Xn be a random sample from Exp(?). Find the MVUE of the median of this exponential distribution.
Let x1, x2,x3,and x4 be a random sample from population with normal distribution with mean ?...
Let x1, x2,x3,and x4 be a random sample from population with normal distribution with mean ? and variance ?2 . Find the efficiency of T = 17 (X1+3X2+2X3 +X4) relative to x= x/4 , Which is relatively more efficient? Why?
Let X1,X2,...,Xn be a random sample from a uniform distribution on the interval (0,a). Recall that...
Let X1,X2,...,Xn be a random sample from a uniform distribution on the interval (0,a). Recall that the maximum likelihood estimator (MLE) of a is ˆ a = max(Xi). a) Let Y = max(Xi). Use the fact that Y ≤ y if and only if each Xi ≤ y to derive the cumulative distribution function of Y. b) Find the probability density function of Y from cdf. c) Use the obtained pdf to show that MLE for a (ˆ a =...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT