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suppose $X_1,X_2,\ldots,X_n$ have joint distribution. if $X_1\sim\mathcal{N}(0,1)$ and for $j=0,1,\ldots,n-1$ conditional distribution $X_{j+1}|X_1=x_1,\ldots,X_j=x_j\sim\mathcal{N}(\rho x_j,1)$, then find $MLE$ parameter $\rho$.

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Any comment on my post showing the hypothesis is never met? –  Did May 3 '12 at 14:59

2 Answers 2

up vote 2 down vote accepted

This looks like a homework problem, so I will give you hints in a series of steps for you to carry out.

  1. Write the joint distribution function using the formula $f(X_1, \dots, X_n) = f(X_1)f(X_2|X_1)f(X_3|X_1,X_2) \cdots f(X_n|X_1,\dots,X_{n-1})$.

  2. Use the information provided to write this expression explicitly in terms of gaussian density functions.

  3. Take the natural log of the result of step 2 and write the log of the product as a sum of logs.

  4. Differentiate the result of step 3 with respect to $\rho$.

  5. Set the result of step 4 equal to zero and solve for $\rho$.

Your answer should be an expression involving $x_i x_{i+1}$ terms and $x_i^2$ terms.

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Assume the distribution of $X_1$ is $\mathcal N(0,1)$ and the conditional distribution of $X_2$ conditionally on $X_1=x$ is $\mathcal N(\varrho x,1)$. Then the distribution of $X_2$ is $\mathcal N(0,1+\varrho^2)$.

Hence, the hypotheses of the post are never met (except if $n=1$ or $\varrho=0$).

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