Question

A sample of \({\rm{n}}\) captured Pandemonium jet fighters results in serial numbers\({{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}{\rm{,}}{{\rm{x}}_{\rm{3}}}{\rm{, \ldots ,}}{{\rm{x}}_{\rm{n}}}\). The CIA knows that the aircraft were numbered consecutively at the factory starting with \({\rm{\alpha }}\)and ending with\({\rm{\beta }}\), so that the total number of planes manufactured is \({\rm{\beta - \alpha + 1}}\) (e.g., if \({\rm{\alpha = 17}}\) and\({\rm{\beta = 29}}\), then \({\rm{29 - 17 + 1 = 13}}\)planes having serial numbers \({\rm{17,18,19, \ldots ,28,29}}\)were manufactured). However, the CIA does not know the values of \({\rm{\alpha }}\) or\({\rm{\beta }}\). A CIA statistician suggests using the estimator \({\rm{max}}\left( {{{\rm{X}}_{\rm{i}}}} \right){\rm{ - min}}\left( {{{\rm{X}}_{\rm{i}}}} \right){\rm{ + 1}}\)to estimate the total number of planes manufactured.

a. If\({\rm{n = 5, x\_}}\left\{ {\rm{1}} \right\}{\rm{ = 237, x\_}}\left\{ {\rm{2}} \right\}{\rm{ = 375, x\_}}\left\{ {\rm{3}} \right\}{\rm{ = 202, x\_}}\left\{ {\rm{4}} \right\}{\rm{ = 525,}}\)and\({{\rm{x}}_{\rm{5}}}{\rm{ = 418}}\), what is the corresponding estimate?

b. Under what conditions on the sample will the value of the estimate be exactly equal to the true total number of planes? Will the estimate ever be larger than the true total? Do you think the estimator is unbiased for estimating\({\rm{\beta - \alpha + 1}}\)? Explain in one or two sentences.

Short answer

a) The corresponding estimate is\({\rm{324}}\).

(b) If the sample includes both the plane with the highest serial number and the plane with the lowest serial number.

The greatest value can never exceed\({\rm{\beta }}\), and the smallest value can never be less than\({\rm{\alpha }}\).

Step-by-step solution

Definition

An estimator is a rule for computing an estimate of a given quantity based on observable data: the rule (estimator), the quantity of interest (estimate), and the output (estimate) are all distinct.

Finding corresponding estimate

(a)

Consider the given information and simplify,

\(\begin{array}{l}{\rm{n = 5}}\\{{\rm{x}}_{\rm{1}}}{\rm{ = 237}}\\{{\rm{x}}_{\rm{2}}}{\rm{ = 375}}\\{{\rm{x}}_{\rm{3}}}{\rm{ = 202}}\\{{\rm{x}}_{\rm{4}}}{\rm{ = 525}}\\{{\rm{x}}_{\rm{5}}}{\rm{ = 418}}\end{array}\)

We note that the smallest data value of the five data values is\({\rm{525}}\):

\({\rm{max}}\left( {{{\rm{x}}_{\rm{i}}}} \right){\rm{ = 525}}\)

We note that the largest data value of the five data values is\({\rm{202}}\):

\({\rm{min}}\left( {{{\rm{x}}_{\rm{i}}}} \right){\rm{ = 202}}\)

The value of the estimator then becomes:

\({\rm{max}}\left( {{{\rm{x}}_{\rm{i}}}} \right){\rm{ - min}}\left( {{{\rm{x}}_{\rm{i}}}} \right){\rm{ + 1 = 525 - 202 + 1 = 324}}\)

Finding corresponding estimate

(b)

The estimator will be exactly equal to the parameter if:

\(\begin{array}{l}{\rm{\beta = max}}\left( {{{\rm{x}}_{\rm{i}}}} \right)\\{\rm{\alpha = min}}\left( {{{\rm{x}}_{\rm{i}}}} \right)\end{array}\)

If the sample comprises the aircraft with the biggest serial number and the plane with the smallest serial number, the estimator is precisely equal to the parameter.

Because the maximum can never be higher than \({\rm{\beta }}\)and the minimum can never be lower than\({\rm{\alpha }}\), the difference between the maximum and minimum can never be more than\({\rm{\beta - \alpha + 1}}\), the estimate can never be greater than the real total.

If the sample comprises the aircraft with the biggest serial number and the plane with the smallest serial number, the estimator is precisely equal to the parameter.

Because the maximum can never be higher than \({\rm{\beta }}\)and the minimum can never be lower than\({\rm{\alpha }}\), the difference between the maximum and minimum can never be more than \({\rm{\beta - \alpha + 1}}\), the estimate can never be greater than the real total.