Question

5. Horvitz-Thompson (HT) estimator (a) (2 marks) Show that the HT estimator es tu/su is unbiased for the population total. Clearly define any notation used. (b) (1 mark) The variance of the HT estimator is Var(ār(s)) = () Give the HT estimate of the variance based on the sample, S. (c) (1 mark) Suppose we sample with replacement where probability of selecting unit u is pu. Derive the inclusion probability for unit. (d) A sample (n = 6) was randomly selected from a population (N = 124). The inclusion probabilities and the observed y values from the sample are yi- 2-101 2 3 T4/50 4/50 4/50 4/50 1/73 1 i. (1 mark) Give the HT estimate for the proportion of units less than or equal to 2. ii. (1 mark) Give the HT estimate of the population median.

Answer 1

1 Page No Date Ansi- Horwitz - Thompson (HT) estimator :- The Horwitz - Thompson estimator is simpler ordered estimatoes . than s.be V celue that Notation. the population size. Yu-(U=1,2, ..s) be of the character under stiedy, Tu Probability unit zu included in Sample. Tuv Probability that unit Zu of Zy inclieding in sample A sample size n is drawn without replacement using arbitory probability of probability of selection of each dorewn. let the initial probability Pu=Xu of selecting unit where x = { Xu. serection o valing assume that Zu is х u=1 Then, the Horvitz Thompson estimator for population to teel is defined as, THI こ E UES yu Tu. a) The Horvitz-Thomspon estimator is unbiased estimator Foe population total. HT CS Scanned with CamScanner

Proot i consider. e Yu HT UES Tu. u e Tu E Yu HT UES Tu UES Tu where, 1 it uth au = unit included in Sample. 0.00 0 → E (WHr) { Yu E (au) UES Tu Elan) = 1. Plūnit au including in Sample) 1. Писли → E (altr) = 8 h e Tu Mua e Yu = ET-0. UES UES Tu -Y. Hence prove. b) The Vaεiance of the HT estimatoe, on the sample Considee, v(an) = E(Aut) ² - [[[fnt)]² E 2 Now, E (HT) E م ) E UEP Yu Tu a) CS Scanned with CamScanner

Page No Date 2 E Yv E uts Yu Tu + UES utved Tu • E audy E UES Yu Tu ai tee Yu Yy utval Tu Tv s E uzv=1 Yu ² + Tu. Yu Yv ir Es Tur Tu Tv 2 byl înt) S Yu² te Tu uzvil I UES Yu Yu-y2 Tuy Tu Ty {Yu + Yu Yu E UES Tuv Пu Пу Tu uzv=1 Yu2 t Yu Yu -E ies E E utv=1 Tuy TTu Пу { Yu²+ {{ Yury utval UES Tu Tuv Yu Yy. =E Yu2+ E u &v=1 Tu Tv UES Tu ( 2 ) - You EE izj=1 Tuva Tutty Yu Yy Tu Ty This is estimato e varian cens. of Horvitz thomspon based on Simple CS Scanned with CamScanner

4 -) Define a random variable a di (d = 1,2...N) if Yu is included in a sample s'of du 1) Let u=1, 2N zu = nyu NE(ru) assuming E(Lie ) >0 for cell i where E(au) = 1. P (rues) to. P (Yu ES) the probability in the sample and probability including the unit u is ccelled as inclusion d) Given Sample =n=6. Total population = N=124 uso ako also ulo 73 for the proportion of units i) HT estimator less than a CS Scanned with CamScanner

Page No. Date : Horvitz - Thompson estimatwe is, T E il Yi Tli is distinct number of units is sample. estimate For Novo, HT units is, the proportion of equel less than or -- 2. 수 ㅠ 4150 -10o 4. It> - 25 Exyen, the, HT lestymark of 'poplection Median. CS Scanned with CamScanner