STAT271 - Parameter Space - Rayleigh - Statistics Assignment Help

STAT271:  Parameter Space Rayleigh Statistics Assignment Help

Assignment Task:

Question 1
Let X1, X2, . . . , X7 be the 7 rvs corresponding to 7 independent coin toss experiments. Let Xi = 1 if the ith toss results in ahead and 0 if a tail. Let the probability of a head be θ. The null hypothesis to be tested is that the coin is unbiased.

(a) What is the parameter space Θ ?

(b) How many points are there in the sample space S? Give two examples of possible points.

(c) We could partition the sample space based on the number of heads in the sample. If we do this, how many basic partitions of S are there?

(d) What is the type I error rate if we set as the critical region C1, those sample points which give rise to 0 or 1 head? What would be the appropriate alternative hypothesis for this C1?

(e) If the alternative hypothesis were HA: θ 6=12, specify two possible sensible critical regions C2 (with a Type I error rate of 0.0156) and C3
(with a Type I error rate of 0.125).

(f) If in fact θ = 1/3 evaluate the power associated with critical regions C1 and C2.

Question 2
Let X1, X2, ..., Xn be i.i.d. from the following density, where θ is a positive parameter (i.e. θ > 0): fX(x) = (32x2θ−1exp n−x2θo0 < x < +∞0 otherwise

(a) Show that Y = X3/θ has an exponential distribution with mean = 2.

(b) Using standard results about the distribution of the sum of independent exponential random variables, determine a pivotal quantity for θ involving all n observations. Clearly identify the distribution this pivotal quantity follows.

(c) Derive the form of a most powerful critical region (CR) with type I error probability α for H0: θ = 1 versus H1: θ = θ1 where (θ1 > 1). Determine the constant k of this CR using a critical value from the χ2distribution.

(d) When sample size n = 8, determine precisely the critical region if α = 0.05.

(e) What is the power of the test in (d) when θ1 = 4.52 (if you use tables, give your answer as accurately as tables will allow).

Question 3

A random variable X is said to follow the Rayleigh distribution with parameter θ (denoted as X ∼ R(θ)) if the pdf of X isf(x; θ) = (xθ2 exp n−x22θ2ox > 00 otherwise

(a) Assume X1, . . . , Xn are i.i.d., all follow the R(θ) distribution. Find the method of moment estimator of θ by involving the first moments.

(b) Find the MLE of θ2 from these X1, . . . , Xn.

(c) Now suppose we have two sets of independent random variables: X1, . . . , Xn ∼ R(θ1) and Y1, . . . , Ym ∼ R(θ2) and the elements of these two sets are also independent. We wish to test the hypotheses: H0 : θ 2 1 = θ 2 2 vs HA : θ 2 1 6= θ 2 2
.

Derive a likelihood ratio test for these hypotheses. Your answers need to include:
(i) An expression for the likelihood ratio test statistic; and
(ii) the decision rule for the test.

(d) Manipulate the likelihood ratio statistic so that it involves Pn i=1 X2i / Pm j=1 Y 2 j and Pm j=1 Y2 j/Pni=1 X2i
. Demonstrate that the distributions of these two quantities under the null hypothesis are related to F-distributions.

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