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OSU / Statistics / STAT 3470 / What is a point estimate?

# What is a point estimate? Description

##### Description: Notes for textbook chapter 6.1 and 6.2
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Chapter 6: Point EstimationWe also discuss several other topics like What is the fundamental individual rights of a free society?

6.1 General Concepts of Point EstimationDon't forget about the age old question of Which term refers to a subset of elements from the population?

Defn: a point estimate of a parameter  is a single number that can be regarded as a sensible value for . it is obtained by selecting a suitable statistic and computing its value from the given sample data. The selected statistic is called the point estimator of .

• The symbol  is used to denote both the estimator of and the point estimate resulting from a given sample thus is read as “the point estimator of  is the sample mean

• Mean square error is used to determine which estimator is better.

• Unbiased estimators

Defn: a point estimator is said to be an unbiased estimator of  if for every possible value of . If is not unbiased, the difference  is called the bias of We also discuss several other topics like In what plants is this life cycle observed?

Prop: When x is a binomial or with parameters n and p, the sample proportion  is an unbiased estimator of p.

If you want to learn more check out how is the prestige and status of a chief in a ranked society reinforced?

No matter the true value of p, the estimator of  will be centered at the true value.

If you want to learn more check out picture of kermit the frog

Principle of unbiased estimation

• When choosing among several different estimators of σ, select are that is unbiased

Prop: Let be random sample from a distribution with mean and variance , then the estimatorDon't forget about the age old question of ucr music

is unbiased for estimating

Prop: If  is a random sample from a distribution with mean , then is an unbiased estimator of . if in addition the distribution is continuous and symmetric, then and any trimmed mean are also unbiased estimators of

• Estimators with Minimum Variance
• Principle of Minimum Variance Unbiased Estimation

Among all estimators of  that are unbiased, choose the one that has minimum variance. The resulting  is called the minimum variance unbiased estimator (MVUE) of .

Let  be a random sample from a normal distribution with parameters  and . then the estimator  is the MVUE for

Some Complications

1. If the random sample comes from a normal distribution, then  is the best of the 4 estimators, since it has minimum variance among all unbiased estimators
2. If the random sample comes from a cauchy distribution, then are terrible estimators, whereas  is quite good. is bad due to sensitivity to outliers and heavy tails
3. If the underlying distribution is uniform, the best estimator is ; this estimator is greatly influenced by outliers, but lack of tails makes such observations impossible
4. The trimmed mean is never best but works well enough in all three situations

•  A trimmed mean with small trimming percentage is call a robust estimator
• Reporting a point estimate: Standard Error

Defn: The Standard error of an estimator is its standard deviation . it is the magnitude of a typical or representative deviation between an estimate and the value of θ. If the standard error itself involves unknown parameters whose values can be estimated, substitution of these estimates into  yields the estimated standard error (est. standard deviation) of the estimator.

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