Question

In: Statistics and Probability

Let X1, X2, … , X8 be a random sample of size 8 from a distribution with probability density function

where x ≥ 1. What is the likelihood ratio critical region for testing the null hypothesis ??: ? = 1 against ?1: ? ≠ 1? If ? = 0.1, determine the best likelihood ratio critical region?

Expert Solution

Answer:

Given that,

Let X1, X2, … , X8 be a random sample of size 8 from a distribution with probability density function,

where ? ≥ 1.

What is the likelihood ratio critical region for testing the null hypothesis ??: ? = 1 against ?1: ? ≠ 1? If ? = 0.1, determine the best likelihood ratio critical region:

For testing hypothesis:

Vs

=0.1

Now,

The likelihood function of sample is,

Now, n=8

Now,

We know that,

Therefore,

i.e,

We reject H_0 for smaller values of .

We reject H_0 for

Now,

The test function is,

orchestra answered 2 years ago

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 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 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 n form a uniform distribution on the interval...

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

Let X1 and X2 be a random sample from a population having probability mass function f(x=0)...

Let X1 and X2 be a random sample from a population having probability mass function f(x=0) = 1/3 and f(x=1) = 2/3; the support is x=0,1. a) Find the probability mass function of the sample mean. Note that this is also called the sampling distribution of the mean. b) Find the probability mass function of the sample median. Note that this is also called the sampling distribution of the median. c) Find the probability mass function of the sample geometric...

Let X1,X2,...,Xn be a random sample from any distribution with mean μ and moment generating function...

Let X1,X2,...,Xn be a random sample from any distribution with mean μ and moment generating function M(t). Assume that M(t) is finite for some t > 0. Let c>μ be any constant. Let Yn = X1+X2+···+Xn. Show that P(Yn ≥ cn) ≤ exp[−n a(c)] where P(Yn ≥ cn) ≤ exp[−n a(c)] a(c) = sup[ct − ln M (t)]. t > 0

: 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...

Question

Let X1, X2, … , X8 be a random sample of size 8 from a distribution with probability density function

Solutions

Expert Solution

Related Solutions

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 75 from a distribution...

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

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

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

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

Let X1 and X2 be a random sample from a population having probability mass function f(x=0)...

Let X1,X2,...,Xn be a random sample from any distribution with mean μ and moment generating function...

: Let X1, X2, . . . , Xn be a random sample from the normal...

. Let X1, X2, . . . , Xn be a random sample from a normal...