Using this data find the MLE $\\hat{\\mu}$ of $\\mu$ .
\n \n \n \n$\\hat{\\mu}=\\;\\;$[[0]]
\n \n \n \nEnter to 2 decimal places.
\n \n \n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "showPrecisionHint": false, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "mlephi+tol", "minValue": "mlephi-tol", "showCorrectAnswer": true, "marks": 1}], "type": "gapfill", "prompt": "Let \\[\\phi = \\simplify[std]{{t}*ln(mu+{b})+{u}*({a}/(mu+{d}))+{v}e^(mu/{c})+{w}*(mu^2+{g})}\\]
\nUsing the invariance property of maximum likelihood estimators, what is the MLE $\\hat{\\phi}$ of $\\phi$?
\n$\\hat{\\phi}=\\;\\;$[[0]] (make sure that you use an estimator for $\\mu$ which is accurate to 4 decimal places).
\nEnter to 2 decimal places.
", "showCorrectAnswer": true, "marks": 0}], "statement": "\nIn an experiment we take $\\var{n}$ observations $x_i$ from a Normal distribution with unknown mean $\\mu$ and variance $\\var{var}$.
\nFrom this sample we find:
\n\\[\\sum_{i=1}^{\\var{n}}x_i = \\var{m}\\]
\n ", "tags": ["checked2015", "cr1", "distributions", "functions", "invariance property of maximum likelihood estimators", "known variance", "MAS2302", "maximum likelihood estimator", "mean", "mean ", "mle", "MLE", "Normal distribution", "normal distribution", "query", "sample", "statistics", "sum of sample", "tested1", "unknown mean", "variance"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "14/07/2012:
\nAdded tags.
\nNew tolerance variable t=0 for the numeric inputs.
\nChecked calculations.
\n1/08/2012:
\nAdded tags.
\nQuestion appears to be working correctly.
\n21/12/2012:
\n
Checked calculations for all functions appearing in this exercise. Added tested1 tag.
Rounding OK, added cr1 tag.
\nIn order to calculate the MLE for $\\phi$ need to use a more accurate value than that found for $\\mu$ to 2 decimal places. This is stated in the question now.
\nThere is no need to have the variance given. Added query tag for that.
\nAlso query the accuracy needed for the estimator of $\\mu$.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Given sum of sample from a Normal distribution with unknown mean $\\mu$ and known variance $\\sigma^2$. Find MLE of $\\mu$ and one of four functions of $\\mu$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a)
The MLE $\\hat{\\mu}$ of $\\mu$ is given by dividing the sum of the sample by the number of samples i.e.
\\[\\hat{\\mu}=\\frac{\\var{m}}{\\var{n}}=\\var{mlemu}\\] to 2 decimal places.
\nb)
In order to find the MLE $\\hat{\\phi}$ of \\[\\phi=\\simplify[std]{{t}*ln(mu+{b})+{u}*({a}/(mu+{d}))+{v}e^(mu/{c})+{w}*(mu^2+{g})}\\]
we note that $\\phi$ is 1-1 and so we can directly substitute $\\mu=\\hat{\\mu}$ into this to obtain:
\n\\[\\hat{\\phi} = \\simplify[std]{{t} * Ln({m} / {n} + {b}) + {u} * ({a} / ({m} / {n} + {d})) + {v} * e ^ (({m} / {n}) / {c}) + {w} * (({m} / {n}) ^ 2 + {g})} = \\var{mlephi}\\]
\nto 2 decimal places.
", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}