Suppose the random variable, X, follows a geometric distribution with parameter 0 (0 < 0 < 1). Let X₁, X2,, X₁ 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-1Xj.

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Suppose the random variable, X, follows a geometric distribution with parameter 0 (0 <
0 < 1). Let X₁, X2,,X₁ 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-1Xj.
Transcribed Image Text:Suppose the random variable, X, follows a geometric distribution with parameter 0 (0 < 0 < 1). Let X₁, X2,,X₁ 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-1Xj.
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(e) Show how the mle ofis related to its mme. Explain if it is always true.
(f) Verify if XX; is a sufficient statistic for or not.
(g) Justify if = is an unbiased estimator for 0.
X
Transcribed Image Text:(e) Show how the mle ofis related to its mme. Explain if it is always true. (f) Verify if XX; is a sufficient statistic for or not. (g) Justify if = is an unbiased estimator for 0. X
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Follow-up Question
(c) Find the maximum likelihood estimator (mle) of 0.
(d) Find the method of moments estimator (mme) of 0.
Transcribed Image Text:(c) Find the maximum likelihood estimator (mle) of 0. (d) Find the method of moments estimator (mme) of 0.
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