Editing Weighted arithmetic mean (section)

==== Survey sampling perspective ====

From a ''model based'' perspective, we are interested in estimating the variance of the weighted mean when the different <math>y_i</math> are not [[Independent and identically distributed random variables|i.i.d]] random variables. An alternative perspective for this problem is that of some arbitrary [[Survey sampling|sampling design]] of the data in which units are [[Design effect#Sources for unequal selection probabilities|selected with unequal probabilities]] (with replacement).<ref name = "Cochran1977" />{{rp|306}}

In [[Survey methodology]], the population mean, of some quantity of interest ''y'', is calculated by taking an estimation of the total of ''y'' over all elements in the population (''Y'' or sometimes ''T'') and dividing it by the population size – either known (<math>N</math>) or estimated (<math>\hat N</math>). In this context, each value of ''y'' is considered constant, and the variability comes from the selection procedure. This in contrast to "model based" approaches in which the randomness is often described in the y values. The [[survey sampling]] procedure yields a series of [[Bernoulli distribution|Bernoulli]] indicator values (<math>I_i</math>) that get 1 if some observation ''i'' is in the sample and 0 if it was not selected. This can occur with fixed sample size, or varied sample size sampling (e.g.: [[Poisson sampling]]). The probability of some element to be chosen, given a sample, is denoted as <math>P(I_i=1 \mid \text{Some sample of size } n ) = \pi_i </math>, and the one-draw probability of selection is <math>P(I_i=1 | \text{one sample draw}) = p_i \approx \frac{\pi_i}{n}</math> (If N is very large and each <math>p_i</math> is very small). For the following derivation we'll assume that the probability of selecting each element is fully represented by these probabilities.<ref name="sarndal1992">{{cite book |title = Model Assisted Survey Sampling |author=Carl-Erik Sarndal |author2=Bengt Swensson |author3=Jan Wretman |isbn= 978-0-387-97528-3 |year = 1992|publisher=Springer }}</ref>{{rp|42,43,51}} I.e.: selecting some element will not influence the probability of drawing another element (this doesn't apply for things such as [[cluster sampling]] design).

Since each element (<math>y_i</math>) is fixed, and the randomness comes from it being included in the sample or not (<math>I_i</math>), we often talk about the multiplication of the two, which is a random variable. To avoid confusion in the following section, let's call this term: <math>y'_i = y_i I_i</math>. With the following expectancy: <math>E[y'_i] = y_i E[I_i] = y_i \pi_i</math>; and variance: <math>V[y'_i] = y_i^2 V[I_i] = y_i^2 \pi_i(1-\pi_i)</math>.

When each element of the sample is inflated by the inverse of its selection probability, it is termed the <math>\pi</math>-expanded ''y'' values, i.e.: <math>\check y_i = \frac{y_i}{\pi_i}</math>. A related quantity is <math>p</math>-expanded ''y'' values: <math>\frac{y_i}{p_i} = n \check y_i</math>.<ref name="sarndal1992" />{{rp|42,43,51,52}} As above, we can add a tick mark if multiplying by the indicator function. I.e.: <math>\check y'_i = I_i \check y_i = \frac{I_i y_i}{\pi_i}</math>

In this ''design based'' perspective, the weights, used in the numerator of the weighted mean, are obtained from taking the inverse of the selection probability (i.e.: the inflation factor). I.e.: <math>w_i = \frac{1}{\pi_i} \approx \frac{1}{n \times p_i}</math>.