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NIPRL Chapter 7. Statistical Estimation and Sampling Distributions 7.1 Point Estimates 7.2 Properties of Point Estimates 7.3 Sampling Distributions 7.4 Constructing Parameter Estimates 7.5 Supplementary Problems

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NIPRL 7.1 Point Estimates Parameters Parameters –In statistical inference, the term parameter is used to denote a quantity, say, that is a property of an unknown probability distribution. –For example, the mean, variance, or a particular quantile of the probability distribution –Parameters are unknown, and one of the goals of statistical inference is to estimate them.

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NIPRL Figure 7.1 The relationship between a point estimate and an unknown parameter θ

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NIPRL Figure 7.2 Estimation of the population mean by the sample mean

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NIPRL Statistics Statistics –In statistical inference, the term statistics is used to denote a quantity that is a property of a sample. –Statistics are functions of a random sample. For example, the sample mean, sample variance, or a particular sample quantile. –Statistics are random variables whose observed values can be calculated from a set of observed data.

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NIPRL Estimation Estimation –A procedure of “ guessing ” properties of the population from which data are collected. –A point estimate of an unknown parameter is a statistic that represents a “ guess ” at the value of. Example 1 (Machine breakdowns) –How to estimate P(machine breakdown due to operator misuse) ? Example 2 (Rolling mill scrap) – How to estimate the mean and variance of the probability distribution of % scrap ( ) ?

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NIPRL 7.2 Properties of Point Estimates Unbiased Estimates (1/5) Definitions - A point estimate for a parameter is said to be unbiased if - If a point estimate is not unbiased, then its bias is defined to be

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NIPRL Unbiased Estimates (2/5) Point estimate of a success probability -

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NIPRL Unbiased Estimates(3/5) Point estimate of a population mean -

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NIPRL Unbiased Estimates(4/5) Point estimate of a population variance

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NIPRL Unbiased Estimates (5/5)

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NIPRL Minimum Variance Estimates (1/4) Which is the better of two unbiased point estimates?

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NIPRL Minimum Variance Estimates (2/4)

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NIPRL Minimum Variance Estimates (3/4) An unbiased point estimate whose variance is smaller than any other unbiased point estimate: minimum variance unbised estimate (MVUE) Relative efficiency Mean squared error (MSE) –How is it decomposed ? –Why is it useful ?

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NIPRL Minimum Variance Estimates (4/4)

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NIPRL Example: two independent measurements Point estimates of the unknown C They are both unbiased estimates since The relative efficiency of to is Let us consider a new estimate Then, this estimate is unbiased since What is the optimal value of p that results in having the smallest possible mean square error (MSE)?

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NIPRL Let the variance of be given by Differentiating with respect to p yields that The value of p that minimizes Therefore, in this example, The variance of

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NIPRL The relative efficiency of to is In general, assuming that we have n independent and unbiased estimates having variance respectively for a parameter, we can set the unbiased estimator as The variance of this estimator is

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NIPRL Mean square error (MSE): Let us consider a point estimate Then, the mean square error is defined by Moreover, notice that

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NIPRL 7.3 Sampling Distribution Sample Proportion (1/2)

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NIPRL Sample Proportion (2/2) Standard error of the sample mean

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NIPRL Sample Mean (1/3) Distribution of Sample Mean

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NIPRL Sample Mean (2/3) Standard error of the sample mean

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NIPRL Sample Mean (3/3)

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NIPRL Sample Variance (1/2) Distribution of Sample Variance

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NIPRL Theorem: if is a sample from a normal population having mean and variance, then are independent random variables, with being normal with mean and variance and being chi-square with n-1 degrees of freedom. (proof) Let Then, or equivalently, Dividing this equation by we get Cf. Let X and Y be independent chi-square random variables with m and n degrees of freedom respectively. Then, Z=X+Y is a chi-square random variable with m+n degrees of freedom.

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NIPRL In the previous equation, Therefore,

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NIPRL Sample Variance (2/2) t-statistics

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NIPRL 7.4 Constructing Parameter Estimates The Method of Moments (1/3) Method of moments point estimate for One Parameter

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NIPRL The Method of Moments (2/3) Method of moments point estimates for Two Parameters

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NIPRL The Method of Moments (3/3) Examples - –What if the distribution is exponential with the parameter ?

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NIPRL Maximum Likelihood Estimates (1/4) Maximum Likelihood Estimate for One Parameter

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NIPRL Maximum Likelihood Estimates (2/4) Example - -

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NIPRL Maximum Likelihood Estimates (3/4) Maximum Likelihood Estimate for Two Parameters

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NIPRL Maximum Likelihood Estimates (4/4) Example

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NIPRL Examples (1/6) Glass Sheet Flaws - The method of moment

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NIPRL Examples (2/6) The maximum likelihood estimate:

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NIPRL Examples (3/6) Example 26: Fish Tagging and Recapture

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NIPRL Examples (4/6)

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NIPRL Examples (5/6) Example 36: Bee Colonies

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NIPRL Examples (6/6)

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NIPRL MLE for For some distribution, the MLE may not be found by differentiation. You have to look at the curve of the likelihood function itself.

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