d R Y 1 Y 2 2 Recall that random variable following a t distribution with \u03bd

D r y 1 y 2 2 recall that random variable following a

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(d)R=Y1Y22Recall that random variable following at-distributionwithνdegrees of freedom can be constructed from arandom variableZthat follows a standard normal distribution (i.e.,ZN(0,1)), and another random variableU*that follows a chi-square distribution withνdegrees of freedom (U*∼ X2ν), byZrU*νtν,providedZandU*are independent. 2. Consider the special case of the homoskedastic Gaussian regression model with no interceptand with one explanatoryvariable, given byYi=βxi+iwithiiidN(0, σ2)fori∈ {1, . . . , n}. Assume that an observation pair(xi, yi)is available for eachi∈ {1, . . . , n}. Condition on thexivaluesthroughout. Perform all of the following steps from first principles based on PSTAT 120A/B definitions (without usingmatrix results) 2
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(a) Write down the likelihood function forβandσ2based on the observations. (b) Find the maximum likelihood estimators ofβandσ2, clearly showing your working.
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