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• Point Estimation

  Point Estimation

  If I have a known distribution, but I don't know what the parameter values (e.g., , , etc.) are, how can I estimate these population parameters using a sample of size n (sample statistics)? Given a population parameter of interest, e.g., the population mean or the population proportion , we want to use a sample to compute a number that represents a ``good'' guess for the true value of the parameter. This number is called a point estimate. We will use the Greek letter (``theta'') to represent any of the parameters we will study such as , , , etc.

   

  • Estimate A point estimate/ of a parameter is a single number based on the sample data that we can consider to be the most plausible value of .
  • Estimator The statistic (formula) used to obtain the point estimate.

  To emphasize:

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    • Point Estimator = Statistic (formula)
    • Point Estimate = Number (calculated value)

   

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  • Point Estimators for Different Situations

  • Point Estimators for Different Situations

    As we have seen before, we have several natural point estimates for the various situations we will consider:

    1. For one sample from a continuous population, we use to estimate and to estimate .
    2. For independent samples from continuous populations, we use to estimate the difference and to estimate . If we know that the variances of the two populations are the same, that is, , say, then we use the pooled estimate of variance,

       

       to estimate .

    3. For paired data, we use the mean, , and variance, of the n differences in the pairs to estimate the mean and variance of the population of differences.
    4. For one sample from a 0-1 population we use the sample proportion p to estimate the population proportion , while for two independent samples we use to estimate .
    5. In the correlation setting, we use the sample correlation coefficient r to estimate the population correlation coefficient and the slope and the intercept of the least squares line to estimate the slope and intercept of the true population line.
    
    

     

     

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  • Properties of Point Estimators

  • Properties of Point Estimators

    Consider three marksmen,

     

     

     

     

    Here we have three different situations.

    • Target 1 has all its shots clustered tightly together, but none of them hit the bullseye.
    • Target 2 has a large spread, but on average the bullseye is hit.
    • Target 3 has a tight cluster around the bullseye.
    In statistical terminology, we say that
    • Target 1 is biased/ with a small variance.
    • Target 2 is unbiased/ with a large variance.
    • Target 3 is unbiased/ with a small variance.
    If you were hiring for the police department, which shooter would you want? In general in statistics, we want both unbiased and small variance--an estimator that almost always is ``on target.''

     

     

     

     
  • Minimum Variance Unbiased Estimation

  • Minimum Variance Unbiased Estimation

    It is not always obvious what the best way to estimate a parameter is. For example, the sample mean and sample median are both natural estimators of the mean of a normal population. We want our estimator to be like target 3: unbiased with the smallest possible variance. Thus, we only look at unbiased estimators and choose the one with the smallest variance. This we call the minimum variance unbiased estimator/ (MVUE) or the best unbiased estimator/ (BUE), because it is the ``best'' estimator we can get using only unbiased estimators.

    NOTE:\ For the normal distribution, the mean, median, and mode all occur at the same location. Why do we usually use the mean, , instead of the median, ? It turns out that both and are unbiased, but has a smaller variance than . In fact, is the MVUE. See the ``Minimum variance estimation'' concept lab for some examples of this.

    
    

     

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