Parameter Estimation

• After a family of distribution has been selected such as Poisson, Normal, Geometric ..., the next step is to estimate the parameters of the distribution.

• Sample mean and sample variance can be used to estimate the parameters in a distribution.
  • Let $X_1, X_2, ..., X_n$ be the sample of size n.
  • The sample mean is
    
    $$\overline{X} = \frac{\sum_{i=1}^n X_i}{n}$$
    
    

  • The sample variance is
    
    $$S^2 = \frac{\sum_{i=1}^n {X_i}^2 - n \overline{X}^2} {n - 1}$$
    
    

  • If the data are discrete and grouped in a frequency distribution, then we can re-write the equations as
    
    $$\overline{X} = \frac{\sum_{j=1}^k f_j X_j}{n}$$
    
    
    and
    
    $$S^2 = \frac{\sum_{j=1}^k f_j {X_j}^2 - n \overline{X}^2} {n - 1}$$
    
    

  • Example 10.5 on page 368
  • If the data are continuous, we ``discretize'' them and estimate the mean
    
    $$\overline{X} \approx \frac{\sum_{j=1}^c f_j m_j}{n}$$
    
    
    and the variance
    
    $$S^2 \approx \frac{\sum_{j=1}^c f_j {m_j}^2 - n \overline{X}^2} {n - 1}$$
    
    
    where$f_j$ is the observed frequency in the jth class interval, $m_j$ is the midpoint of the jth interval, and c is the number of class intervals.
  • Example 10.6 on page 369

• A few well-established, suggested estimators are listed in Table 10.3 on page 370, followed by examples. They come from theory of statistics.

• The examples include Poisson Distribution, Uniform Distribution, Normal Distribution, Exponential Distribution, and Weibull Distribution.