Let X11 X12, Xini and X21, X22, X2n2 be two independent random samples ofsize n₁ and n₂ from two normal populations N(₁, 2) and N(2, 2) respectively.(a) Derive the maximum likelihood estimators (mle's) of all the parameters in the firstpopulation (X₁). Using analogy, state the mle's of the parameters of the secondpopulation.(b) Find the pooled estimator of the common variance when it is assumed that of == ². Suggest an unbiased estimator of o². |

a) Let's start by finding the maximum likelihood estimators (MLEs) of the parameters in the first…

Answered: Let X₁1 X12, Xin, and X21, X22, X2₂ be…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Suppose the random variable, X, follows a geometric distribution with parameter 0 (0 < 0 < 1). Let X₁, X₂, X, be a random sample of size n from the population of X. (a) Write down the likelihood function of the parameter. (b) Show that the log likelihood function of depends on the sample only through Σj=1 Xj. (c) Find the maximum likelihood estimator (mle) of 0.
The number of times that a person contracts a cold in a given year is a Poisson random variable with parameter λ=5. Suppose that a new wonder drug (based on large quantities of vitamin C) has just been marketed that reduces the Poisson parameter to λ=3 for 70 percent of the population. For the other 30 percent of the population, the drug has no appreciable effect on colds. If Peter tried the drug for a year and has 3 colds in that time, what is the probability that the drug is beneficial for him?
Suppose that X1,..., Xn are iid random variables with P(X; = 0) = 1 P(X; = 1) = 2 P(X; = 2) = (1 – 0) 3 1 P(X; = 3) = -(1 – 0), %3D | where 0 < 0 < 1 is the parameter of interest. a) Find the maximum likelihood estimator. Show that its asymptotic variance is 0(1 – 0) n b) Compare this variance to the variance of the method of moments estimator. (c) Give the asymptotic distribution of the maximum likelihood estimator. (d) Using the maximum likelihood estimator, what is the form of an approximate 95% confidence interval for 0?
Consider the one-way analysis of variance model Xij = µ + a; + Eij, i= 1,.., m, j= 1,..., ni, where ɛij ~ N(0,0²) are independent. Let n= n1 + · ·+ nm, ... ni 1 ni 1 X;. >Xii for i = 1,..., m, and X. = - ΣΣΧ. ni j=1 i=1 j=1 (a) Show that SS(TO) = SS(T) + SS(E), where SS(TO) = E E i=12j=1 (Xij – X..)², m m ni SS(T) Σι (X.-Χ.) and SS(E) -ΣΣ (Χ- X.) . i=1 i=1 j=1
Let X₁, X2, X3, Xn be a random sample with unknown mean EX; = µ, and unknown variance Var(X₂) = o². Suppose that we would like to estimate 0 = μ². We define the estimator as 2 • - (™)² - [ 2x]* Xk to estimate 0. Is an unbiased estimator of ? Why?
Suppose the random variable, X, follows a geometric distribution with parameter 0 (0 < 0 < 1). Let X₁, X2,..., X be a random sample of size n from the population of X. (a) Write down the likelihood function of the parameter. (b) Show that the log likelihood function of depends on the sample only through Σj=1 Xj. (c) Find the maximum likelihood estimator (mle) of 0. (d) Find the method of moments estimator (mme) of 0.
Using the natural breaks method how many students will score an “A” using the following set of scores? 90, 76, 91, 82, 86, 93, 67, 75, 80, 82, 91, 89, 85, 73, 81, 69, 82, 85, 68, 73 Group of answer choices 4 5 6 8
The information associated with the maximum likelihood estimator of a parameter 0 is 4n, where n is the number of observations. Calculate the asymptotic variance of the maximum likelihood estimator of 20 . (A) (В) (C) (D) 8n (E) 16n
Suppose we collect n samplesX1,X2, . . . ,Xn from a uniform distribution U[θ,1]. We know the maximum likelihood estimator of θ is θ = min(Xi). (1) What is the CDF of θ? (2) What is the pdf of θ? (3) Find the bias of θ. (4) Find the variance of θ. (5) Find the MSE of θ.
Let Y < Y, < Y3 be the order statistic of a random sample of size 3 from the uniform distribution having pdff (x;0) = 1/0,0
Consider an MA(2) model: STEPS -Write the equation of the M(2) model-Determine the model based on the delay operator.- Find the expected value of the model- Find the variance of the model.- Find the covariances associated with 1,2, and s steps.- Find the associated correlation indices of 1,2, and s steps.- Assume that information is available up to time $h$ and the function that contains the accumulated information f(h, h-1,……), determine the forecasts and the error associated with 1,2, and s steps.

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