Chapter 8: Sampling distributions of estimators Sections

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1 Chapter 8: Sampling distributions of estimators Sections 8.1 Sampling distribution of a statistic 8.2 The Chi-square distributions 8.3 Joint Distribution of the sample mean and sample variance Skip: p The t distributions Skip: derivation of the pdf, p Confidence intervals 8.7 Unbiased Estimators 8.8 Fisher Information STA 611 (Lecture 16) Sampling Distributions Nov 1, / 13

2 Bayesian alternative to confidence intervals Bayesian inference is the posterior distribution. Reporting a whole distribution may not be what you (or your client) want For point estimates we use Bayesian estimators Minimize the expected loss If we want an interval to go with the point estimator we simply use quantiles of the posterior distribution For example: We can find constants c 1 and c 2 so that P(c 1 < θ < c 1 X = x) γ The interval (c 1, c 2 ) is called a 100γ% Credible interval for θ Note: The interpretation is very different from interpretation of confidence intervals STA 611 (Lecture 16) Sampling Distributions Nov 1, / 13

3 Bayesian Analysis for the normal distribution Let X 1,..., X n be a random sample for N(µ, σ 2 ) In Chapter 7.3 we saw: If σ 2 is known, the normal distribution is a conjugate prior for µ Theorem 7.3.3: If the prior is µ N(µ 0, ν0 2 ) the posterior of µ is also normal with mean and variance µ 1 = σ2 µ 0 + nν 2 0 x n σ 2 + nν 2 0 and ν 2 1 = σ2 ν 2 0 σ 2 + nν 2 0 We can obtain credible intervals for µ from this N(µ 1, ν 2 1 ) posterior distribution STA 611 (Lecture 16) Sampling Distributions Nov 1, / 13

4 Bayesian Analysis for the normal distribution Let X 1,..., X n be a random sample for N(µ, σ 2 ) In Chapter 7.3 we saw: If µ is known, the Inverse-Gamma distribution is a conjugate prior for σ 2 Example : If the prior is σ 2 IG(α 0, β 0 ) the posterior of σ 2 is also Inverse-Gamma with parameters α 1 = α 0 + n 2 and β 1 = β n (x i µ) 2 i=1 We can obtain credible intervals for σ 2 from this IG(α 1, β 1 ) posterior distribution What if both µ and σ 2 are unknown? We need the joint posterior distribution of µ and σ 2 STA 611 (Lecture 16) Sampling Distributions Nov 1, / 13

5 Bayesian Analysis for the normal distribution When both µ and σ 2 are unknown Def: Precision The precision of a normal distribution is the reciprocal of the variance: τ = 1 σ 2 It is somewhat simpler to work with the precision than the variance The pdf of the normal distribution is then written as f (x µ, τ) = τ 1/2 exp ( 12 ) τ(x µ)2 2π STA 611 (Lecture 16) Sampling Distributions Nov 1, / 13

6 Normal-Gamma distribution Def: Normal-Gamma distribution Let µ and τ be random variables where µ τ has the normal distribution with mean µ and precision λ 0 τ and τ has the Gamma distribution: µ τ N(µ 0, 1/λ 0 τ) and τ Gamma(α 0, β 0 ) Then the joint distribution of µ and τ is called the Normal-Gamma distribution with parameters µ 0, λ 0, α 0 and β 0. The pdf for this distribution is f (µ, τ µ 0, λ 0, α 0, β 0 ) = (λ 0τ) 1/2 2π exp ( 12 ) λ 0τ(µ µ 0 ) 2 βα 0 0 Γ(α 0 ) τ α0 1 exp ( β 0 τ) STA 611 (Lecture 16) Sampling Distributions Nov 1, / 13

7 z Chapter 8 continued Pdf of the Normal-Gamma distribution µ 0 = 0, λ 0 = 1, α 0 = 0.5 and β 0 = 0.5 mu tau STA 611 (Lecture 16) Sampling Distributions Nov 1, / 13

8 Bayesian Analysis for the normal distribution When both µ and σ 2 are unknown Theorem 8.6.1: Conjugate prior for µ and τ Let X 1,..., X n be a random sample from N(µ, 1/τ). The Normal-Gamma distribution with parameters µ 0, λ 0, α 0 and β 0 is a conjugate prior distribution for (µ, τ) and the posterior distribution has (hyper)parameters µ 1 = λ 0µ 0 + nx n, λ 1 = λ 0 + n λ 0 + n α 1 = α and β 1 = β s2 n + nλ 0(x n µ 0 ) 2 2(λ 0 + n) where x n = 1 n n x i and sn 2 = n (x i x n ) 2 i=1 i=1 STA 611 (Lecture 16) Sampling Distributions Nov 1, / 13

9 Bayesian Analysis for the normal distribution To give credible intervals for µ and σ 2 individually we need the marginal posterior distributions From the structure of the Normal-Gamma distribution we immediately get the marginal for τ: τ X = x Gamma(α 1, β 1 ) This distribution can be used to obtain credible intervals for τ, or any function of τ. STA 611 (Lecture 16) Sampling Distributions Nov 1, / 13

10 Marginal distribution of the mean Theorem 8.6.2: The marginal distribution of µ Let the joint distribution of µ and τ be the Normal-Gamma distribution with parameters µ 0, λ 0, α 0 and β 0. Then ( ) λ0 α 1/2 0 U = (µ µ 0 ) t 2α0 It is then easy to show that (Theorem 8.6.3): β 0 E(µ) = µ 0 (if α 0 > 1/2) and Var(µ) = β 0 λ 0 (α 0 1) (if α 0 > 1) STA 611 (Lecture 16) Sampling Distributions Nov 1, / 13

11 Marginal distribution of the mean Say we have done a Bayesian Analysis and end up with the Normal-Gamma posterior with parameters µ 1, λ 1, α 1 and β 1. We can then calculate the marginal posterior means of µ and τ E(µ x) = µ 1 (if α 1 > 1/2) and E(τ x) = α 1 β 1 Can also obtain credible intervals for µ Find quantiles c 1 and c 2 such that P(c 1 < U < c 2 ) = γ Then ( ( ) 1/2 ( ) ) 1/2 β1 β1 P µ 1 + c 1 µ µ 1 + c 2 λ 1 α 1 λ 1 α 1 x = γ E.g: for a 0.95% symmetric credible interval we set c 1 = T 1 2α 1 (0.975) and c 2 = T 1 2α 1 (0.975) STA 611 (Lecture 16) Sampling Distributions Nov 1, / 13

12 Example Hotdogs (Exercise in the book) Data on calorie content in 20 different beef hot dogs from Consumer Reports (June 1986 issue): 186, 181, 176, 149, 184, 190, 158, 139, 175, 148, 152, 111, 141, 153, 190, 157, 131, 149, 135, 132 Assume that these numbers are observed values from a random sample of twenty independent N(µ, σ 2 ) random variables, where µ and σ 2 are unknown. n x n = and sn 2 = (x i x n ) 2 = Consider the Normal-Gamma prior for µ and τ with parameters µ 0 = 100, λ 0 = 3, α 0 = 2 and β 0 = Construct the apriori symmetric 95% credible interval for µ i=1 Find the posterior symmetric 95% credible interval for µ STA 611 (Lecture 16) Sampling Distributions Nov 1, / 13

13 Other priors The usual improper prior for (µ, τ) is p(µ, τ) = 1 τ < µ, τ > 0 The posterior in this case is the Normal-Gamma distribution with parameters µ 1 = x, λ 1 = n, α 1 = (n 1)/2, β 1 = s 2 n/2 Credible intervals turn out to be the same as confidence intervals (common for improper priors) Other common options: µ and τ independent (a priori) with µ N(µ 0, ν 2 0 ) and τ Gamma(α 0, β 0 ). µ and τ will be dependent in the posterior. τ Gamma(α 0, β 0 ) but improper for µ: p(µ) = 1 STA 611 (Lecture 16) Sampling Distributions Nov 1, / 13

Bayesian statistics, simulation and software

Bayesian statistics, simulation and software Module 4: Normal model, improper and conjugate priors Department of Mathematical Sciences Aalborg University 1/25 Another example: normal sample with known precision Heights of some Copenhageners in 1995: