Estimating parameters from multivariate normal

In summary: Another approach is to use a Bayesian approach, which involves specifying prior distributions for the parameters and then using the data to update these distributions and obtain posterior distributions for the parameters. In summary, to estimate the parameters p_1, p_2 and y from data that follows a multinormal distribution, you can use either a maximum likelihood estimation approach or a Bayesian approach. Both methods involve deriving a likelihood function and using an optimization algorithm, or specifying prior distributions and updating them with the data, respectively, to obtain estimates of the parameters.
  • #1
jimmy1
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I need to estimate parameters from data that follow a mutinormal distribution. The parameters that I need to estimate are contained in the expression for the mean of the marginal normal distributions. That is each marginal distribution has mean:

[tex] \frac{p_1*c_i + p_1*y}{p_1+p_2}[/tex]

where [tex]p_1[/tex], [tex]p_2[/tex] and [tex]y[/tex] are the paramters that I need to estimate and [tex]c_i[/tex] is just a known contant associated with the ith marginal. [tex]p_1[/tex] and [tex]p_2[/tex] are random parameters and [tex]y[/tex] is a fixed parameter.

I've tried using a multinormal likelihood approach, which gives an estimate for the mean as the sample mean, but how do I get estimate of the actual parameters from this sample mean? Can I even use this approach?

Also is there an added complexity to estimating parameters when there are random parameters, as is the case with [tex]p_1[/tex] and [tex]p_2[/tex]?

Any help, on how best to estimate the paramters [tex]p_1[/tex], [tex]p_2[/tex] and [tex]y[/tex] would be great?
 
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  • #2
One approach to estimating the parameters p_1, p_2 and y from data that follows a multinormal distribution is to use a maximum likelihood estimation (MLE) approach. This involves first deriving a likelihood function for the multinormal distribution and then using an optimization algorithm to find the parameters that maximize the likelihood function. This will give you an estimate of the parameters p_1, p_2 and y. The added complexity of estimating parameters when there are random parameters comes from the fact that for each sample point the parameters p_1 and p_2 will be different. This means that the likelihood function needs to be derived and maximized for each sample point in order to obtain an estimate of the parameters.
 

What is multivariate normal distribution?

Multivariate normal distribution is a probability distribution that describes the joint distribution of a set of variables that are normally distributed. It is often used in statistics and data analysis to model complex datasets where multiple variables are involved.

What are parameters in multivariate normal distribution?

Parameters in multivariate normal distribution refer to the mean vector and covariance matrix that define the shape and location of the distribution. The mean vector represents the average values of each variable and the covariance matrix describes the relationship between each variable.

How are parameters estimated from multivariate normal distribution?

Parameters can be estimated from multivariate normal distribution using various methods such as maximum likelihood estimation, Bayesian estimation, and method of moments. These methods involve using data from a sample to estimate the parameters of the population distribution.

What are the assumptions when estimating parameters from multivariate normal distribution?

The main assumptions when estimating parameters from multivariate normal distribution are that the data follows a multivariate normal distribution, the variables are continuous, and the variables are linearly related. Additionally, the sample size should be large enough and the variables should not be highly correlated.

Why is estimating parameters from multivariate normal distribution important?

Estimating parameters from multivariate normal distribution is important because it allows us to model and analyze complex datasets with multiple variables. This can provide valuable insights and help in making accurate predictions and decisions in various fields such as finance, economics, and social sciences.

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