Estimation of the size of a finite population

BIKAS K SINHA Faculty [1979-2011] INDIAN STATISTICAL INSTITUTE KOLKATA & Ex-Member [2006-2009] National Statistical Commission GoI

How many bacteria in the jar ?

• Capture – Recapture Technique :

Innovative Statistical Method for ‘ascer-taining’ the size [N] of a finite population

Demonstration with Marbles…..same size and shape…..almost same color…..no distinguishing features as such....

Q. How many are there ?

Capture-Recapture [CR] Method

• ‘Capture’ a few items (k) and ‘Mark’ them and ‘Release’ in the population.

• 2. Next Recatch AT RANDOM a few (n) Items & Count the Number (X) ‘Recaptured’.

• N = Population Size [unknown]• k = Initial Catch Size [for Marking] • n = Random Catch Size [Pre-Fixed]

Capture-Recapture [CR] Method

• X = No. ‘Marked” items in the chosen sample• Population Proportion of “Marked” = k/N• Sample Proportion of “Marked” = X / n

• “Estimating Equation” : k / N = X / n• Implies : N^ = kn / X• Q. What if “X = 0 ‘” ? ….N^ = Infinity !!! • Compromise : N^ = k(n+a)/a with a >0.

Estimation of N….

• k \ n

• 10 20

• 5 X = 2, N^ = 25 X = 3, N^ = 34

• 10 X = 2, N^ = 50 X = 3, N^ = 67

Ascertaining the Size of a Finite Population : CMR Method

• 1. ‘Capture’ a few items (k) : ‘Mark’ & ‘Release’ in the population.

• 2. Recatch one-by-one & Inspect & Release UNTIL Initially Marked Items are Recaptured ‘m’ times

• N = Population Size [unknown]• k = Initial Catch Size [for Marking] • n = Second Catch Size UNTIL Marked Items are

Recaptured ‘m’ times [m being prespecified] • N^ = kn/m

Estimation of Size of a Finite Population….

• k \ m 2 3 5 n = 15 n = 25 N^ = 2.5n = 38 N^ = 1.67n = 42

2 3 10 n = 8 n = 13 N^ = 5n = 40 N^ = 3.3n= 44

CMR Method : Modified….

• Recapture one-by-one & Inspect & Release …..BUT….Keep Aside the Marked Items ….STOP as soon as ‘m’ Marked Items are found in the process of sampling.

• N^ = {(k+1)n/m} – 1

Estimation of N….

• k \ m 2 3 5 n = 9 n = 17 N^ = 3n - 1 N^ = 2n - 1 = 26 = 33 10 n = 7 n = 13 N^ = 5.5n - 1 N^ = 3.67n - 1 = 38 = 47

Size of a Finite Population…

• Sequential Search…..CMRR Method

• Capture One Item -Mark & Release

• Recapture & Inspect :

Stop [if Already Marked]

Mark & Release [if NOT Already Marked]

Continue until one Marked is Discovered

s = No. of attempts made after First Entry

N^ = s(s+1) / 2……..

Size of a Finite Population…

• s : 1 2 3 4 5 ……10• N^ : 1 3 6 10 15 …….55 Q. What if more items [k] are marked

initially ? N^ = (s+k+1)_c_2 – k_c_2 k = 5 s = 1 2 3 4 5……. N^ = 11 18 26 35 45……

New Game…..

• Estimation of Total Number of Units produced….in a production process….

• Units Serially Numbered as 1, 2, …

• No Omission of numbers …..

• No Duplication of numbers…..

• How far does it go ?

Marbles : Serially Numbered ?

Natural Numbering : 1, 2,…,N? How many ?

Pick one marble : Holds the number ’19’

“Best” Judgment for N ? …..

Next marble : ’11’….. BAD NEWS ?

Next…..’5’ …..Ooooopppsssss!!!!!!!

Next….’26’ …………Great !!!

Next……’9’……what’s this…..most erratic !

Next…..’30’…Better ‘stop’ Random Choice?

‘Best’ Guess for ‘N’ ?

• Guessed Value of N

• 19….. ?

• 19 ...11…. ?

• 19…11…5… ?

• 19..11..5.. 26… ?

• 19…11…5…26…9… ?

• 19…11..5…26…9….30…. ?

Thought Process….• Concept of Partitioning of Popl. Units..• Median : 50 % cut-off value• Upper 50 % : ………..X..……….[1/2]• Upper 67 % : ………x……..X………[2/3]• Upper 75 % : ……x…..x…..X…..[3/4]• Upper 80 % : …..x…..x…..x…..X….[4/5]• Upper 83 % : …x…x…x…x…X…[5/6]• And so on…….[n/(n+1)] at the n-th stage• N. n/(n+1) = X = Max. Value….N^ =X(n+1)/n

‘Best’ Guess for ‘N’ …. Guessed Value of N• 19….. 38• 19 …11…. 29• 19…11…5… 26• 19..11..5.. 26… 32 / 33 • 19…11…5…26…9… 31• 19…11..5…26…9….30…. 35Q. Why ‘waste [?]’ all other information…Q. Is there any ‘extra’ information in the rest,

beyond what is captured by the largest number ? …..Decisively NOT…..except for how many are there …the sample size [n]…

References….

Maximum Likelihood Estimation of a

Finite Population Size

[Co-authors : Md. Mesbahul Alam &

A. H. Rahmatullah Imon]

J. Stat. Theory Appl. 5 (2006), 306—315.