Point Estimation of Parameters and Sampling Distributions

1
Point Estimation of Parameters and Sampling
Distributions

• Outlines
• Sampling Distributions and the central limit
  theorem
• Point estimation
• Methods of point estimation
• Moments
• Maximum Likelihood

2
Sampling Distributions and the central limit
theorem

• Random Sample
• Sampling distribution the probability
  distribution of a statistic.
• Ex. The probability distribution of is called
  distribution of the mean.

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Sampling Distributions of sample mean

• Consider the sampling distribution of the sample
  mean.
• Xi is a normal and independent probability, then
  

4
Central limit theorem

• n gt 30, sampling from an unknown population gt
  the sampling distribution of will be
  approximated as normal with mean µ and ?2/n.

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Central limit theorem
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Central limit theorem

• Ex. Suppose that a random variable X has a
  continuous uniform distribution
• Find the distribution of the sample mean of a
  random sample of size n40
• Method 1.Calculate the value of mean and
  variance of x
• 2.

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Sampling Distribution of a Difference

• Two independent populations.
• Suppose that both populations are normally
  distributed.
• Then, the sampling distribution of is
  normal with

µ1 ,?12
µ2 ,?22
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Sampling Distribution of a Difference

• Definition

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Sampling Distribution of a Difference

• Ex. The effective life of a jet-turbine aircraft
  engine is a random variable with mean 5000 hr.
  and sd. 40 hr. The distribution of effective life
  is fairly close to a normal distribution. The
  engine manufacturer introduces an improvement
  into the manufacturing process for the engine
  that increases the mean life to 5050 hr. and
  decrease sd. to 30 hr.
• 16 components are sampling from the old process.
• 25 components are sampling from the improve
  process.
• What is the probability that the difference in
  the two sample means is at least 25 hr?

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Point estimation

• Parameter Estimation calculation of a reasonable
  number that can explains the characteristic of
  population.
• Ex. X is normally distributed with unknown mean
  µ.
• The Sample mean( ) is a point estimator of
  population mean (µ) gt
• After selecting the sample, is the point
  estimate of µ.

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Point estimation

• Unbiased Estimators

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Point estimation

• Ex. Suppose that X is a random variable with mean
  µ and variance s2 . Let X1,X2 ,..., Xn be a
  random sample of size n from the population. Show
  that the sample mean and sample variance S2
  are unbiased estimators of µ and s2 ,
  respectively.
• proof,
• proof,

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Point estimation
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Point estimation

• Sometimes, there are several unbiased estimators
  of the sample population parameter.
• Ex. Suppose we take a random sample of size n
  from a normal population and obtain the data x1
  12.8, x2 9.4, x3 8.7, x4 11.6, x5 13.1,
  x6 9.8, x7 14.1,x8 8.5, x9 12.1, x10
  10.3.

all of them are unbiased estimator of µ
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Point estimation

• Minimum Variance Unbiased Estimator (MVUE)

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Point estimation

• MVUE for µ

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Method of Point Estimation

• Method of Moment
• Method of Maximum Likelihood
• Bayesian Estimation of Parameter

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Method of Moments

• The general idea of the method of moments is to
  equate the population moments to the
  corresponding sample moments.
• The first population moment is E(X)µ........(1)
• The first sample moment is
  ........(2)
• Equating (1) and (2),
• The sample mean is the moment estimator of the
  population mean

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Method of Moments

• Moment Estimators
• Ex. Suppose that X1,X2 ,..., Xn be a random
  sample from an exponential distribution with
  parameter ?. Find the moment estimator of ?
• There is one parameter to estimate, so we must
  equate first population moment to first sample
  moment.
• first population moment E(X)1/?, first sample
  moment

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Method of Moments

• Ex. Suppose that X1,X2 ,..., Xn be a random
  sample from a normal distribution with parameter
  µ and s2. Find the moment estimators of µ and s2.
• For µ k1 The first population moment is E(X)µ
  ........(1)
• The first sample moment is
  ........(2)
• Equating (1) and (2),
• For s2 k2 The second population moment is
  E(X2)µ2s2 ......(3)
• The second sample moment is
  ......(4)
• Equating (3) and (4),

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Method of Maximum Likelihood

• Concept the estimator will be the value of the
  parameter that maximizes the likelihood function.

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Method of Maximum Likelihood

• Ex. Let X be a Bernoulli random variable. The
  probability mass function is

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Method of Maximum Likelihood

• Ex. Let X be normally distributed with unknown µ
  and known s2. Find the maximum likelihood
  estimator of µ

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Method of Maximum Likelihood

• Ex. Let X be exponentially distributed with
  parameter ?. Find the maximum likelihood
  estimator of ?.

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Method of Maximum Likelihood

• Ex. Let X be normally distributed with unknown µ
  and unknown s2. Find the maximum likelihood
  estimator of µ, and s2.

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Method of Maximum Likelihood

• The method of maximum likelihood is often the
  estimation method that mathematical statisticians
  prefer, because it is usually easy to use and
  produces estimators with good statistical
  properties.