Estimators from Scratch: Weights, Auxiliary Data, and Error Repair

The observed tiger counts 18, 14, 20, and 19 sum to 71, but 71 is only the sample total. An estimator must connect these four observations to the ten-forest population.

No. The inclusion probabilities were designed before the survey: A 0.80, B 0.60, E 0.70, and H 0.50. Accessibility, cost, prior maps, and size can make selection chances unequal.

The design weight is w_i=1/pi_i. A unit selected with probability 0.20 represents 5 population units, while one selected with probability 0.80 represents 1.25. Weight corrects opportunity to be observed, not the tiger count itself.

Multiply observed count by inverse inclusion probability. The four contributions are 22.50, 23.33, 28.57, and 38.00, which sum to an estimated population total of 112.40 tigers.

Yes. The Horvitz–Thompson total is Y_hat_HT=sum over sampled units of y_i/pi_i. It adds each observed value divided by its known inclusion probability and never invents values for unsampled units.

Yes. This sample gives 112.40 while the known total is 144, an error of -31.60. Unbiasedness means the average estimate across repeated design-based samples equals the truth, not that every realization is exact.

Forest area, canopy, water sources, and prey indices may be available from satellites, GIS, or prior records. These auxiliary variables are known population information related to the unknown target, tiger count.

Applying the same design weights to sampled areas gives X_hat_HT=649.76 square kilometres, while the known total area is X=870. The gap reveals that this sample under-represents a variable related to tiger count.

The area correction factor is 870/649.76=1.339. Multiplying 112.40 by 1.339 gives 150.50 tigers. Because sampled area was low, ratio estimation enlarges the tiger estimate proportionally.

The ratio total estimator is Y_hat_R=Y_hat_HT times X/X_hat_HT. It keeps the sample fixed and changes the estimate's scale using one auxiliary total known for the population.

Larger sampled forests generally contain more tigers, but equal area changes do not produce equal tiger changes. The relationship supports auxiliary correction while warning that a fixed ratio is only an approximation.

The represented-area gap is 870-649.76=220.24 square kilometres. At a working rate of 0.15 tiger per square kilometre, the correction is 33.04, giving 112.40+33.04=145.44 tigers.

The difference total estimator is Y_hat_D=Y_hat_HT+B(X-X_hat_HT). B translates an auxiliary-unit gap into expected target units; unlike ratio estimation, the correction is added rather than multiplied.

From the four sampled forests, the fitted line is y_hat_i=2.23+0.151x_i. The intercept supplies a starting level and the slope supplies the estimated increase in tigers per square kilometre.

Use known area to predict every forest, then compute residuals y_i-y_hat_i in sampled forests. Inverse-probability weighting expands those observed residuals so prediction errors beyond the sample are represented.

GREG equals the sum of predictions for all population units plus the HT-weighted residuals in the sample. Predictions use auxiliary structure; weighted residuals preserve design-based correction when the model misses.

Every estimator combines observed target data, known population information, and a correction rule. The creative decision is choosing which known information should correct which weakness.

Start with design weights, add a scale correction, add a gap correction, learn a relationship, then combine prediction with weighted error repair. Each change targets one identifiable weakness in the current estimate.

On historical or simulated populations with known truth, repeatedly draw samples, apply the estimator, and measure bias, variance, and error. Only after this validation do we use the chosen estimator once on the real survey.

Weights restore representation, auxiliary totals expose sample imbalance, models add structure, and residual corrections defend against model error. A production estimator is not just a formula; it is a justified chain from sampling design to uncertainty.