The Distribution of Σ x - x^2 / σ^2 in Statistical Analysis

The standard statistical expression

Sigma; (x_i - bar{x})^2 / sigma^2 

often appears in various statistical tests and distributions. To understand which distribution this expression follows, we need to define the context in which it is used. Specifically, we are dealing with a sample of observations (x_1, x_2, ldots, x_n), which are assumed to be independent and identically distributed (i.i.d.) from a given distribution.

Assuming Normal Distribution

A common assumption is that the observations come from a normal distribution. In such a case, the sample mean (bar{x}) is an unbiased estimate of the population mean, and the sample variance is:

[ S^2 frac{sum_{i1}^{n} (x_i - bar{x})^2}{n-1} ]

When we scale this by the population variance (sigma^2), we get:

[ frac{sum_{i1}^{n} (x_i - bar{x})^2}{sigma^2} ]

Under these conditions, this statistic follows a chi-square distribution with (n-1) degrees of freedom, denoted as (chi^2_{n-1}).

Normality and Independence Assumptions

It’s important to consider the assumptions of normality and independence. If the sample represents a large portion of the population or if the underlying distribution is highly skewed, these assumptions may not hold. For example, income data is often highly skewed and not normally distributed. In such cases, the chi-square approximation may not be accurate.

Reciprocal of the Chi-Square Statistic

Another common form involves the reciprocal of the chi-square statistic:

[ V frac{1}{U} ]

where (U) is the chi-square statistic.

To find the density of (V), we use the transformation of random variables. If (f_U(u)) is the density function of (U), then the density function of (V) is given by:

[ f_V(v) f_Uleft(frac{1}{v}right) cdot left|frac{d}{dv}left(frac{1}{v}right)right| ]

Since (frac{d}{dv}left(frac{1}{v}right) -frac{1}{v^2}), the density function becomes:

[ f_V(v) -f_Uleft(frac{1}{v}right) cdot frac{1}{v^2} ]

However, the negative sign is usually ignored as the density function must be non-negative, resulting in:

[ f_V(v) f_Uleft(frac{1}{v}right) cdot frac{1}{v^2} ]

Conclusion

In summary, the statistic (frac{sum (x_i - bar{x})^2}{sigma^2}) follows a chi-square distribution with (n-1) degrees of freedom. If you need the distribution of the reciprocal (V frac{1}{U}), the density function can be derived through the transformation of random variables.

Keywords

chi-square distribution statistical inference sample variance