Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\rho$.

share|improve this question

closed as unclear what you're asking by Did, Ahaan S. Rungta, Tim Raczkowski, Davide Giraudo, Daniel Robert-Nicoud Jul 23 at 20:00

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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.

share|improve this answer

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 $\varrho=0$).

share|improve this answer

Not the answer you're looking for? Browse other questions tagged or ask your own question.