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Topic 7 Statistical Estimation and Sampling Distributions
Statistical Inference A statistical method which involves investigation of properties (estimation) concerning the unknown population parameters based on sample statistic results. A statistical method which involves investigation of properties (estimation) concerning the unknown population parameters based on sample statistic results.
Point Estimates The parameter, which is denoted by θ, is an unknown property of a population. For example, mean, variance, proportion or particular quantile of the probability distribution. The statistic is a property of a sample. For example, sample mean, sample variance, proportion or a particular sample quantile Estimation is a procedure by which the information contained within a sample is used to investigate properties of the population from which the sample is drawn
Point Estimates of Parameters Estimate Population Parameters ( ) …. with Sample Statistics ( ) Mean ( µ ) Standard Deviation ( ) S Proportion ( p ) A point estimate of an unknown parameter θ is a statistic that represents a “best guess” at the value of θ. There may be more than one sensible point estimate of a parameter. For example,
A point estimates for a parameter is said to be unbiased if Unbiased and Biased Point Estimates Unbiasedness is a very good property for a point estimate to possess. If a point estimate is not unbiased then its bias can be defined to be
Point Estimate of a Success Probability The obvious point estimate of p is Notice that the number of successes X has a binomial distribution, X ~ B(n,p). Therefore So that indeed an unbiased point estimate of p
Point Estimate of a Population Mean Clearly it is since [ Remember, fair coin (n = 2, μ = p = ½) and fair dice (n = 6, μ = p =1/6 ] So that Then indeed an unbiased point estimate of μ
Point Estimate of a Population Variance We know that the sample variance Then
Point Estimate of a Population Variance Since then
Point Estimate of a Population Variance We notice that Remember,
Putting this all together gives Point Estimate of a Population Variance
Minimum Variance Estimates The best situation is constructing a point estimate that is unbiased and that also has the smallest possible variance An unbiased point estimate that has a smaller variance than any other point estimate is called a minimum variance unbiased estimate (MVUE). The efficiency of MVUE is shown by its relative efficiency The relative efficiency of an unbiased point estimate to an unbiased point estimate is given by
Mean Square Errors In the case that two point estimates have different expectations and different variances, we prefer the point estimate that minimizes the value of mean square error (MSE) which is defined to be
ExercisesExercises Suppose that E(X 1 ) = μ, Var(X 1 ) = 10, E(X 2 ) = μ, and Var(X 2 ) = 15, and consider the point estimates a.Calculate the bias of each point estimate. Is any one of them unbiased b.Calculate the variance of each point estimate. Which one has the smallest variance? c.Calculate the mean square error of each point estimate. Which point estimate has the smallest mean square error when μ = 8 d.What is the relative efficiency of to the point estimate of ?
Exercise Solution a.
Exercise Solution b.
Exercise Solution c. d.
Sampling Distributions Since the summary measures of one sample vary to those of another sample, we need to consider the probability distributions or sampling distributions of the sample mean, the sample variance S 2, and the sample proportion.
Sampling Means If X 1, …, X n are observations from a population with a mean μ and a variance σ 2, then the central limit theorem indicates that the sample mean has the approximate distribution The standard deviation of the sample mean is referred to as standard error (SE) Since the standard deviation σ is usually unknown, it can be replaced by S.
is a chi-square distribution with n – 1 degrees of freedom. In the case that the variance is unknown, If X 1, …. X n are normally distributed with a mean μ, then Sample Variances If X 1, …, X n are normally distributed with a mean μ and a variance σ 2, then the sample variance S 2 has the distribution t n-1 is student’s t distribution with n – 1 degrees of freedom.
The standard error of the sample proportion is Sample Proportions If X ~ B(n, p), then the sample proportion has the approximate distribution
ExercisesExercises 1)The capacitances of certain electronic components have a normal distribution with a mean μ = 174 and a standard deviation σ = 2.8. If an engineer randomly selects a sample of n = 30 components and measures their capacitances, what is the probability that the engineer’s point estimate of the mean μ will be within the interval (173, 175)? 2)A scientist reports that the proportion of defective items from a process is 12.6%. If the scientist’s estimate is based on the examination of a random sample of 360 items from the process, what is the standard error of the scientist’s estimate? 3)The pH levels of food items prepared in a certain way are normally distributed with a standard deviation of σ = An experimenter estimates the mean pH level by averaging the pH levels of a random sample of n items. What sample size n is needed to ensure that there is a probability of at least 99% that the experimenter’s estimate in within 0.5 of the true mean value?
Exercise Solutions 1)Recall look up the table! 2)
Exercise Solutions 3)Recall The estimate is within 0.5 of the true mean value
Maximum Likelihood Estimates We have considered the obvious point estimates for a success probability, a population mean and variance. However, it is often of interest to estimate parameters that require less obvious point estimates. For example, how should the parameters of the Poisson, exponential, beta or gamma distributions be estimated? Maximum likelihood estimation is one of general and more technical methods of obtaining point estimates.
Maximum Likelihood Estimate for One Parameter If a data set consists of observations x 1, x 2, …, x n from a probability distribution f (x, ) depending upon one unknown parameter, the maximum likelihood estimate of the parameter is found by maximizing the likelihood function In practice, the maximization of the likelihood function is usually performed by taking the derivative of the natural log of the likelihood function.
ExampleExample Suppose again that x 1, x 2, …, x n are a set of Bernoulli observation, with each taking the value 1 (success) with probability p and the value 0 (no success) with the probability 1 – p. We can write this as The likelihood function is therefore Where x = x 1 + x 2 +…+ x n and the maximum likelihood estimate is the value that maximize this
ExampleExample and Setting this expression equal to 0 and solving for p produce
Maximum Likelihood Estimate for Two Parameter If a data set consists of observations x 1, x 2, …, x n from a probability distribution f (x, 1, 2 ) depending upon two unknown parameter, the maximum likelihood estimate and are the values of the parameters that jointly maximize the likelihood function Again the best way to perform the joint maximization is usually to take derivatives of the log-likelihood with respect to and to set the two resulting expressions equal to 0
ExampleExample The normal distribution is an example of a distribution with two parameters, with a probability density function The likelihood of a set of normal observation is therefore
ExampleExample So that the log-likelihood is Taking derivatives with respect to the parameters values and gives
ExampleExample Setting d ln(L)/dμ = 0 gives And setting d ln(L)/dσ 2 = 0 then gives Did you see any difference from the variance estimate that we have discussed before?
ExercisesExercises Suppose that the quality inspector at the glass manufacturing company inspects 30 randomly selected sheets of the glass and records the number of flaws found in each sheet. These data values are shown as follows 0, 1, 1, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 3, 1, 2, 0, 0, 1, 0, 0 If the distribution of the number of flaws per sheet is taken to have a Poisson distribution, how should the parameter λ of the Poisson distribution be estimated? And find its value.
Exercise Solutions We should first estimate the parameter of λ. Then, the probability mass function of the data is So that the likelihood is The log-likelihood is therefore Taking its derivative w.r.t. λ and setting it to zero, we get
Exercise Solutions Therefore Since the variance of each data is λ, then The standard error of the estimate of a Poisson parameter is