classnotes1 - Second-order inclusion probability the...

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ST 310 Notes September 9, 2009 John Bunge Population: a set, each element of which has (at least) a label (identifier) and a value x (of primary interest to the study). For our purposes there are N< elements or units. Frame: a master list of the labels (IDs) of the population units. Note that there may be serious differences between the available frame (e.g., the phone book) and the target population (e.g., registered voters). Sample: a subset of the population. Random sample: a sample chosen or selected by some random mechanism. Simple random sample (SRS): a sample of fixed size n N , selected in such a way that every such sample is equally likely. Number of such (possible) samples: ± N n ² = N ! / ( n !( N n )!) . First-order inclusion probability: the probability, denoted π i , that the i th (given) unit will appear in the sample, under a given design. SRS: π i n/N, i =1 ...N .
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Unformatted text preview: Second-order inclusion probability: the probability, denoted π ij , that a given pair of units (say i and j ) will appear in the sample, under a given design. SRS: π ij ≡ n ( n − 1) / ( N ( N − 1)) , 1 ≤ i 6 = j ≤ N . • Population total: τ := ∑ N i =1 x i . Population mean: μ := 1 N ∑ N i =1 x i . Popula-tion variance: σ 2 := 1 N − 1 ∑ N i =1 ( x i − μ ) 2 . • Horvitz-Thompson estimator (HTE) of the population total: ˆ τ = ∑ i ∈ sample x i π i . HTE of mean: ˆ μ = 1 N ∑ i ∈ sample x i π i . SRS: ˆ τ = N ¯ x ; ˆ μ = ¯ x . HTE is unbiased under any design: E (ˆ τ ) = τ . 1 • SRS: Var ( N ¯ x ) = N 2 (1 − n/N ) σ 2 n . Finite population correction (fpc): 1 − n/N . E ( s 2 ) = σ 2 . SE ( N ¯ x ) = N q 1 − n/N s √ n . SE (¯ x ) = q 1 − n/N s √ n . 2...
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classnotes1 - Second-order inclusion probability the...

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