Let X1,X2,...Xn be a random sample of size n form a uniform distribution on the interval...

Let X₁,X₂,...X_n be a random sample of size n form a uniform distribution on the interval [θ₁,θ₂]. Let Y = min (X₁,X₂,...,X_n).

(a) Find the density function for Y. (Hint: find the cdf and then differentiate.)

(b) Compute the expectation of Y.

(c) Suppose θ₁= 0. Use part (b) to give an unbiased estimator for θ₂.

Expert Solution

TOPIC:Distribution of order statistic and Unbiased estimator.

milcah answered 6 months ago

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 =...

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

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...

Suppose that X1,X2,...,Xn form a random sample from a uniform distribution on the interval [θ1,θ2], where...

Suppose that X1,X2,...,Xn form a random sample from a uniform distribution on the interval [θ1,θ2], where (−∞ < θ1 < θ2 < ∞). Find MME for both θ1 and θ2.

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, . . . , 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.

Suppose that X1, ..., Xn form a random sample from a uniform distribution for on the...

Suppose that X1, ..., Xn form a random sample from a uniform distribution for on the interval [0, θ]. Show that T = max(X1, ..., Xn) is a sufficient statistic for θ.

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...

Question