Let X = (X1, . . . , Xn) be a sample from the N(\mu , \sigma 2). Find the Jeffreys prior for \theta = (\mu , \sigma 2).
Added by Kimberly C.
Step 1
To find the Jeffreys prior for the parameters \(\theta = (\mu, \sigma^2)\) of a normal distribution \(N(\mu, \sigma^2)\), we will follow these steps: Show more…
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Suppose x1, ..., xn is a random sample from an exponential distribution with mean 1/θ. 1. Derive the Jeffrey's prior for θ. 2. Derive the posterior distribution of θ using the Jeffrey's prior. 3. Derive the predictive distribution of a future observation z. 4. Derive a 95% credible interval for z.
Sri K.
Let $Y_{1}, Y_{2}, \ldots, Y_{n}$ denote a random sample from a normal population with known mean $\mu_{0}$ and unknown variance $1 / \nu .$ In this case, $U=\sum\left(Y_{i}-\mu_{o}\right)^{2}$ is a sufficient statistic for $\nu,$ and $W=\nu U$ has a $x^{2}$ distribution with $n$ degrees of freedom. Use the conjugate gamma $(\alpha, \beta)$ prior for $\nu$ to do the following. a. Show that the joint density of $U, \nu$ is $$ f(u, v)=\frac{u^{(n / 2)-1} v^{(n / 2)+a-1}}{\Gamma(\alpha) \Gamma(n / 2) \beta^{\alpha} 2^{(n / 2)}} \exp \left[-v /\left(\frac{2 \beta}{u \beta+2}\right)\right] $$ b. Show that the marginal density of $U$ is $$ m(u)=\frac{u^{(n / 2)-1}}{\Gamma(\alpha) \Gamma(n / 2) \beta^{\alpha} 2^{(n / 2)}}\left(\frac{2 \beta}{u \beta+2}\right)^{(n / 2)+\alpha} \Gamma\left(\frac{n}{2}+\alpha\right) $$ c. Show that the posterior density for $\nu | u$ is a gamma density with parameters $\alpha^{*}=(n / 2)+\alpha$ and $\beta^{*}=2 \beta /(u \beta+2)$ d. Show that the Bayes estimator for $\sigma^{2}=1 / \nu$ is $\partial_{B}^{2}=(U \beta+2) /[\beta(n+2 \alpha-2)]$. [Hint: Recall Exercise $4.111(e) .]$ e. The MLE for $\sigma^{2}$ is $U / n$. Show that the Bayes estimator in part. (d) can be written as a weighted average of the MLE and the prior mean of $1 / \nu . \text { [Hint: Recall Exercise } 4.111(\mathrm{e}) .]$
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Consider a random sample $X_{1}, X_{2}, \ldots, X_{n}$ from the normal distribution with mean 0 and variance $\sigma^{2}=1 / \tau$ . (The parameter $\tau=1 / \sigma^{2}$ is called the precision of the normal distribution.) Assume a gamma-distributed prior for $\tau$ and show the posterior distribution of $\tau$ is also gamma. What are its parameters?
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