4. Suppose that X1, X2, . . . , Xn are i.i.d. random variables with density function f(x) = 0 <...

Question

Question

4. Suppose that X1, X2, . . . , Xn are i.i.d. random variables with density function f(x) = ${\theta x^{\theta-1}}$ 0 < x < 1, $\theta$ > 0

a) Find a sufficient statistic for $\theta$ . Is the statistic minimal sufficient?

b) Find the MLE for $\theta$ and verify that it is a function of the statistic in a)

c) Find IX( $\theta$ ) and hence give the CRLB for an unbiased estimator of $\theta$ .

pdf means probability distribution function

math Statistics-And-Probability

Add a comment Improve this question Transcribed image text

Answer 1

Answer #1

a)

b)

c)

Answer 2

Similar Homework Help Questions

8.60-Modified: Let X1,...,Xn be i.i.d. from an exponential distribution with the density function...

8.60-Modified: Let X1,...,Xn be i.i.d. from an exponential distribution with the density function a. Check the assumptions, and find the Fisher information I(T) b. Find CRLB c. Find sufficient statistic for τ. d. Show that t = X1 is unbiased, and use Rao-Blackwellization to construct MVUE for τ. e. Find the MLE of r. f. What is the exact sampling distribution of the MLE? g. Use the central limit theorem to find a normal approximation to the sampling distribution h....
Suppose X1, X2, . . . , Xn are i.i.d. Exp(µ) with the density f(x) =...

Suppose X1, X2, . . . , Xn are i.i.d. Exp(µ) with the density f(x) = for x>0 (a) Use method of moments to find estimators for µ and µ^2 . (b) What is the log likelihood as a function of µ after observing X1 = x1, . . . , Xn = xn? (c) Find the MLEs for µ and µ^2 . Are they the same as those you find in part (a)? (d) According to the Central Limit...
Suppose X1,··· ,Xn are i.i.d. with pdf if 0 < x < 1 and 0 otherwise....

Suppose X1,··· ,Xn are i.i.d. with pdf if 0 < x < 1 and 0 otherwise. (a) Construct the MP test for the hypothesis v.s. with α=0.05. (b) Derive the power function of the test in (a). We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image

Let X1, X2, ..., Xn be a random sample from X which has pdf depending on...

Let X1, X2, ..., Xn be a random sample from X which has pdf depending on a parameter and (i) (ii) where < x < . In both these two cases a) write down the log-likelihood function and find a 1-dimensional sufficient statistic for b) find the score function and the maximum likelihood estimator of c) find the observed information and evaluate the Fisher information at = 1. f(20) We were unable to transcribe this image((z(0 – 2) - )dxəz(47)...
Distribution with shape parameter 3 and unknown scale parameter, λ. Thus the density of Xis given...

distribution with shape parameter 3 and unknown scale parameter, λ. Thus the density of Xis given by: a) Find a sufficient statistic for λ is the statistic minimal sufficient? b) Find the MLE for and verify that it is a function of the statistic in a) c) Find it) and hence give the CRLB for an unbiased estimator of λ d) Find the distribution of the sufficient statistic in (a) distribution with shape parameter 3 and unknown scale parameter, λ....
Q3 Suppose X1, X2, ..., Xn are i.i.d. Poisson random variables with expected value ). It...

Q3 Suppose X1, X2, ..., Xn are i.i.d. Poisson random variables with expected value ). It is well-known that X is an unbiased estimator for l because I = E(X). 1. Show that X1+Xn is also an unbiased estimator for \. 2 2. Show that S2 (Xi-X) = is also an unbaised esimator for \. n-1 3. Find MSE(S2). (We will need two facts) E com/questions/2476527/variance-of-sample-variance) 2. Fact 2: For Poisson distribution, E[(X – u)4] 312 + 1. (See for...

2. Suppose X1, X2, . .., Xn are a random sample from θ>0 0, otherwise Note:...

2. Suppose X1, X2, . .., Xn are a random sample from θ>0 0, otherwise Note: If X~fx(a; 0), thenXEx(0). (a) Find the CRLB of any unbiased estimator of θ (b) Is the MLE for θ the MVUE?
Let X1,X2,...,Xn denote independent and identically distributed random variables with mean µ and variance 2. State...

Let X1,X2,...,Xn denote independent and identically distributed random variables with mean µ and variance 2. State whether each of the following statements are true or false, fully justifying your answer. (a) T =(n/n-1)X is a consistent estimator of µ. (b) T = is a consistent estimator of µ (assuming n7). (c) T = is an unbiased estimator of µ. (d) T = X1X2 is an unbiased estimator of µ^2. We were unable to transcribe this imageWe were unable to transcribe...
Suppose X1, X2, . . . , Xn (n ≥ 5) are i.i.d. Exp(µ) with the...

Suppose X1, X2, . . . , Xn (n ≥ 5) are i.i.d. Exp(µ) with the density f(x) = 1 µ e −x/µ for x > 0. (a) Let ˆµ1 = X. Show ˆµ1 is a minimum variance unbiased estimator. (b) Let ˆµ2 = (X1 +X2)/2. Show ˆµ2 is unbiased. Calculate V ar(ˆµ2). Confirm V ar(ˆµ1) < V ar(ˆµ2). Calculate the efficiency of ˆµ2 relative to ˆµ1. (c) Show X is consistent and sufficient. (d) Show ˆµ2 is not consistent...

Let X1, . . . , Xn be a random sample from a population with density...

Let X1, . . . , Xn be a random sample from a population with density 8. Let Xi,... ,Xn be a random sample from a population with density 17 J 2.rg2 , if 0<、〈릉 0 , if otherwise ( a) Find the maximum likelihood estimator (MLE) of θ . (b) Find a sufficient statistic for θ (c) Is the above MLE a minimal sufficient statistic? Explain fully.

4. Suppose that X1, X2, . . . , Xn are i.i.d. random variables with density function f(x) = 0 <...

Homework Answers

Add Answer to:
4. Suppose that X1, X2, . . . , Xn are i.i.d. random variables with density function f(x) = 0 <...

Post as a guest

Earn Coins

8.60-Modified: Let X1,...,Xn be i.i.d. from an exponential distribution with the density function...

Suppose X1, X2, . . . , Xn are i.i.d. Exp(µ) with the density f(x) =...

Suppose X1,··· ,Xn are i.i.d. with pdf if 0 < x < 1 and 0 otherwise....

Let X1, X2, ..., Xn be a random sample from X which has pdf depending on...

Distribution with shape parameter 3 and unknown scale parameter, λ. Thus the density of Xis given...

Q3 Suppose X1, X2, ..., Xn are i.i.d. Poisson random variables with expected value ). It...

2. Suppose X1, X2, . .., Xn are a random sample from θ>0 0, otherwise Note:...

Let X1,X2,...,Xn denote independent and identically distributed random variables with mean µ and variance 2. State...

Suppose X1, X2, . . . , Xn (n ≥ 5) are i.i.d. Exp(µ) with the...

Let X1, . . . , Xn be a random sample from a population with density...

4. Suppose that X1, X2, . . . , Xn are i.i.d. random variables with density function f(x) = 0 <...

Homework Answers

Add Answer to: 4. Suppose that X1, X2, . . . , Xn are i.i.d. random variables with density function f(x) = 0 <...

Post as a guest

Earn Coins

Add Answer to:
4. Suppose that X1, X2, . . . , Xn are i.i.d. random variables with density function f(x) = 0 <...