Question

Suppose that \({X_1},....,{X_{10}}\)form a random sample from a normal distribution for which both the mean and the variance are unknown. Construct a statistic that does not depend on any unknown parameters and has the \(F\)distribution with three and five degrees of freedom.

Short answer

The required statistic is the ratio of the sample variance of the first four observations to the sample variance of the last 6 observations.

Step-by-step solution

To find the required statistic Vfor the following

Given \({X_1}, \cdots ,{X_{10}}\mathop \~\limits^{{\rm{ iid }}} \mathcal{N}\left( {\mu ,{\sigma ^2}} \right)\),where both \(\mu \)and \({\sigma ^2}\)are unknown.

Consider

\(\begin{array}{l}{T_1} = _{i = 1}^4{\left( {{X_i} - {{\bar X}_1}} \right)^2}\;\;\\{T_2} = _{j = 5}^{10}{\left( {{X_j} - {{\bar X}_2}} \right)^2}\end{array}\)

Where\({\bar X_1}\) is the sample mean of the first four observations, and \({\bar X_2}\)is the sample mean of the last 6 observations.

Then \({T_1}\)and \({T_2}\)are independent, and

\(\begin{array}{l}\frac{{{T_1}}}{{{\sigma ^2}}}\~\chi _3^2\;\;\;\\\frac{{{T_2}}}{{{\sigma ^2}}}\~\chi _5^2\end{array}\)

Thus,

\(\begin{array}{c}V = \frac{{\frac{{{T_1}}}{{3{\sigma ^2}}}}}{{\frac{{{T_2}}}{{5{\sigma ^2}}}}}\\ = \frac{5}{3}\frac{{{T_1}}}{{{T_2}}}\~{F_{3,5}}\end{array}\)

Since \({T_1}\)and \({T_2}\)do not involve any unknown parameters, \(V\)is our required statistic.

The required statistic is the ratio of the sample variance of the first four observations to the sample variance of the last 6 observations.