the moment estimator of θ;• the MLE of θ
Question
the moment estimator of θ;• the MLE of θ
Solution
It seems like you forgot to mention the specific distribution or context for the parameter θ. However, I'll provide a general approach for both the moment estimator and the Maximum Likelihood Estimator (MLE) of θ.
- Moment Estimator of θ: The method of moments involves equating the population moments (about the origin) to the sample moments and solving the resulting equations to obtain the moment estimators.
Step 1: Identify a distribution whose shape might reasonably describe the population of interest. This distribution will have parameters that need to be estimated.
Step 2: Equate the population moments to the sample moments. The rth moment of the population is E(X^r), and the rth sample moment is (1/n) Σ (X_i)^r, where the sum is over all sample points.
Step 3: Solve the resulting equations to get the moment estimators.
- Maximum Likelihood Estimator (MLE) of θ: The MLE involves finding the parameter values that make the observed results the most probable.
Step 1: Write down the likelihood function. This is the joint probability density/mass function of your sample, viewed as a function of the parameter(s), given the sample values.
Step 2: Take the natural logarithm of the likelihood function to get the log-likelihood. This step is optional but is usually done because it simplifies the differentiation step.
Step 3: Differentiate the log-likelihood function with respect to the parameter(s).
Step 4: Set the derivative equal to zero and solve for the parameter(s). This will give the MLE of the parameter(s).
Remember, these are general steps. The actual computation can vary depending on the specific distribution of your data.
Similar Questions
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the maximum likelihood estimate is a solution of the equation d angle theta \ d theta
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Let X1, . . . , Xn denote a random sample from a N(μ, σ2) distribution, where the mean μ and the variance σ2 are both unknown so that the param- eter vector is given by θ = (μ, σ2)T . (i) Show that the joint p.d.f. of X1, . . . , Xn belongs to the two-parameter regular exponential family, stating what the canonical parameter is equal to in terms of θ. (ii) Use the result in (i) to derive the maximum likelihood (ML) estimate of θ, θˆ = (μˆ, σˆ2)T , without the need to differentiate the log likelihood function with respect to θ.
The maximum likelihood estimate is a solution of the equation
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