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Estimation of the size of a finite population

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

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Estimation of the size of a finite population

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  1. 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

  2. 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 ?

  3. 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]

  4. 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.

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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……..

  11. 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……

  12. 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 ?

  13. 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?

  14. ‘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…. ?

  15. 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

  16. ‘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…. 35 Q. 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]…

  17. 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.

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