Reference no: EM132398828
STATISTICS 421 Assignment -
Solve the following problems:
1. Let X1, X2, X3 ∼rs b(1, p), the Bernoulli parent with unknown parameter given by the success rate p ∈ [0, 1].
a. Obtain the joint sampling PDF pX^-,S^2 (x-, s2; p) = P(X- = x-, S2 = s2) of the sample mean X- = 1/3 i=1Σ3Xi and sample variance S2 = ½ i=1Σ3(Xi - X-)2.
b. Based on Problem 1a, are X- and S2 (statistically) dependent for all p ∈ [0, 1]? Show why your answer is 'Yes'; if your answer is 'No' - that ∃p ∈ [0, 1] such that X- and S2 are independent - doesn't this contradict the well-known characterization of the normal distribution as the only distribution for which the independence of X- and S2 holds (i.e., Student's Theorem is, in fact, an "if and only if" result, see E. Lukacs's 1942 paper "A characterization of the normal distribution", Annals of Mathematical Statistics, 13(1), pp. 91-93)? Resolve this apparent contradiction in at most 2 sentences.
2. Let X1, · · · , Xn be a random sample from some parent distribution with finite mean μ = E(Xi) and positive finite variance σ2 = var(Xi), ∀i. Consider the sample variance S2 = 1/n-1 i=1Σn(Xi-X-)2, where X- = 1/n i=1Σn Xi is the sample mean.
a. Using results in §5.1-5.2 of textbook, and without assuming normality of the parent distribution, show that
√n(S2- σ2) →D N(0, μ4 - σ4),
as n →+∞, where μ4 = E{(Xi - μ)4} is the common population's 4th central moment. In other words, show that S2 ∼a N (σ2, μ4-σ4/n); that is, show that the asymptotic distribution of S2 is N (σ2, μ4-σ4/n), for large n.
Note that the asymptotic mean σ2 of S2 is also its exact mean - not surprising, since S2 is unbiased for σ2, regardless of what n is. The same, however, may not be true of its exact and asymptotic variances.
b. Let X1, · · · , Xn ∼rs N(μ, σ2), for -∞ < μ < +∞ and σ2 > 0. Use the CDF method to obtain the exact sampling distribution of S2 from Student's Theorem.
With n = 10, μ = 0, and σ2 = 1, plot the exact PDF of S2, and to see clearly how well (or poorly) the asymptotic PDF of S2 approximates its exact distribution in this case, superimpose the plot of the latter on that of the former. Please follow conventions for such figures: number the figure, include a succinct title, differentiate the plots by using different colours or line types, include a legend if necessary, etc. You may want to use the R package ggplot2 to create your plots although they can also be constructed just as nicely (but perhaps, more easily) using R's built-in plotting functions.
Comment, in at most 2 sentences, on approximating the exact distribution of S2 by its asymptotic distribution in Problem 2a when n = 10.
c. To see what happens when we increase the sample size n, re-do Problem 2b with n = 50. Comment, in at most 2 sentences, on the impact of a 5-fold increase in the sample size from n = 10 to n = 50.
3. Problem 2 involves a parent population for which the exact sampling distribution of S2 is known, and thus, its asymptotic distribution is used mainly only for computational convenience. In Problem #3, we consider a parent population for which the exact distribution of S2 is mathematically difficult (if not impossible) to obtain, which makes knowing its asymptotic distribution, when n is large, very useful.
Problems 2b and 2c show how the asymptotic distribution's accuracy in approximating the exact distribution of S2 when sampling from a normal parent is impacted by how large (or small) the sample size is. Using a gamma parent population in the following problems, you will explore how sampling from a non-normal distribution, in addition to the sample size, affects the accuracy of the asymptotic approximation.
a. Let X1, · · · , Xn ∼rs Gamma(α = 2, β = 2), for shape parameter α and scale parameter β. Use Problem 2a to obtain the asymptotic distribution of S2 in this case by evaluating the numerator μ4 - σ4 = μ4 - (σ2)2 of its asymptotic variance after expressing μ4 in terms of raw moments μ'k = E(Xik), for k = 1, · · ·, 4, and then evaluating them using the MGF of Gamma(2, 2).
b. Because obtaining the exact sampling distribution of S2 when sampling from a gamma parent population is mathematically intractable, you can generate instead its (almost) exact sampling distribution by Monte Carlo sampling using a very large number R of Monte Carlo simulation repeats.
With α = β = 2, simulate R = 10,000 random samples, each of size n = 10, from Gamma(2, 2) using the function rgamma() in R, and then calculate S2 for each random sample to obtain R s2-values along with their mean and standard deviation (SD). Comment, in one sentence, on how close the Monte Carlo mean and SD to the exact mean and asymptotic SD, respectively, of S2.
Construct a histogram of the R s2-values using the R function hist(), and superimpose on it the asymptotic PDF of S2. Follow the same conventions used for the figure in Problem 2b for this one. In at most 2 sentences, compare the accuracy of the asymptotic distribution of S2 as an approximation to its "exact" distribution in this case with that in the normal case in Problem #2b and explain why one is more accurate than the other for the same sample size n = 10.
c. To see what happens when we increase the sample size n, re-do Problem 3b with n = 50. Comment, in at most 2 sentences, on the impact on the asymptotic approximation of increasing the sample size from n = 10 to n = 50.
Textbook - Introduction to Mathematical Statistics - Seventh Edition by Robert V. Hogg, Joseph W. McKean and Allen T. Craig. ISBN 978-0-321-79543-4.
Attachment:- STATISTICS Assignment File.rar