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Unformatted text preview: § 3 Likelihood and Sufficiency § 3.1 Introduction 3.1.1 Sample data: X = ( X 1 ,...,X n ) Postulated parametric family of probability functions (i.e. either mass functions or pdf’s) : { p ( ·  θ ) : θ ∈ Θ } e.g. — • X 1 ,...,X n are iid Poisson( θ ): p ( x 1 ,...,x n  θ ) = n Y i =1 P ( X i = x i  θ ) = n Y i =1 e θ θ x i x i ! , θ ∈ (0 , ∞ ) . • X 1 ,...,X n are iid N ( μ,σ 2 ): p ( x 1 ,...,x n  θ ) = n Y i =1 • 1 √ 2 πσ 2 exp ‰ ( x i μ ) 2 2 σ 2 ‚ θ = ( μ,σ ) ∈ (∞ , ∞ ) × (0 , ∞ ) . If we observe X = x = ( x 1 ,...,x n ), what can we learn from x about the true value of θ ? We wish to answer this question by means of statistical inference . 3.1.2 Raw sample data, i.e. x , contain information relevant to our inference about θ . We want to extract such information completely yet “economically”. This amounts to finding an efficient way to “summarize” data. Answer: Likelihood function — a mathematical device summarizing all information available in x which is relevant to θ . It measures the plausibility of each θ ∈ Θ being the true θ that gives rise to x . 3.1.3 Suppose the random vector X has a probability function belonging to the parametric family { p ( ·  θ ) : θ ∈ Θ } . Definition. Given that X is observed (realised) to be x , the likelihood function of θ is defined to be ‘ x ( θ ) = p ( x  θ ) , i.e. the probability function of X , evaluated at X = x , but considered as a function of θ . 17 The loglikelihood function is S x ( θ ) = ln ‘ x ( θ ) , which gives equivalent information but is more convenient to work with than ‘ x ( θ ). 3.1.4 Common special case — X iid: X is a random sample, i.e. X = ( X 1 ,...,X n ) are iid with each X i ∼ probability function f ( x  θ ). Then the (joint) probability function of X is p ( x  θ ) = n Y i =1 f ( x i  θ ) and so ‘ x ( θ ) = p ( x  θ ) = n Y i =1 f ( x i  θ ) , where x = ( x 1 ,...,x n ) is the realisation of X . 3.1.5 When examining a likelihood function ‘ x ( θ ), we are mainly interested in the relative likelihoods of different values of θ . The actual magnitude of the likelihood function itself is unimportant. Without loss of information we may ignore multiplicative factors in the likelihood function that do not depend on θ . For example, if ( X 1 ,...,X n ) iid ∼ Poisson( θ ), we may take the likelihood function to be ‘ x ( θ ) = e nθ θ ∑ i x i , omitting the factor ( Q i x i !) 1 . 3.1.6 Example § 3.1.1 There are two decks of playing cards: • deck A — conventional, has 52 cards; • deck B — has 13 cards of the same suit ( ♥ ). Your friend draws two cards randomly without replacement from the same deck and gets an ace and a king of hearts. Without asking your friend, which deck do you think your friend is more likely to draw the cards from?...
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This note was uploaded on 05/04/2011 for the course STAT 1302 taught by Professor Smslee during the Spring '10 term at HKU.
 Spring '10
 SMSLee
 Statistics, Probability

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