class: center, middle, inverse, title-slide .title[ # Sampling ] .author[ ### Mahendra Mariadassou, INRAE
.small[from original slides by Tristan Mary-Huard] ] .date[ ### Shandong University, Weihai (CN)
Summer School 2022 ] --- --- class: middle, center, inverse # Technical results ### Mean and variance of a sum --- ## Refresher Recall the following definitions/properties: `$$\begin{align} E[Y] &= \sum_{k=1}^K y_k P(Y = y_k) \\ V[Y] &= E\left[ \left(Y-E[Y]\right)^2\right]\\ &= E[Y^2]-E[Y]^2\\ Cov(X,Y) &= E\left[ (X-E[X])(Y-E[Y]) \right]\\ &= E[XY]-E[X]E[Y]\\ \end{align}$$` -- What about: - `\(f(Y)\)` for a general numeric function `\(f: \mathbb{R} \to \mathbb{R}\)` - `\(aY + b\)` (when `\(a, b \in \mathbb{R}\)`) - `\(f(X, Y)\)` for a general numeric function `\(f: \mathbb{R}^2 \to \mathbb{R}\)` - `\(X + Y\)`, `\(XY\)` --- ## Expectation of `\(f(Y)\)` Falling back to the definition, we have $$ E[f(Y)] = \sum_{k=1}^K f(y_k) P(Y = y_k) $$ In particular, the expectation of `\(aY + b\)` can be expressed simply as -- $$ E[aY + b] = aE[Y] + b $$ .blue[Proof:] -- --- ## Expectation of `\(f(Y_1, Y_2)\)` Falling back to the definition, we have `$$\begin{align} E[f(Y_1, Y_2)] & = \sum_{k_1=1}^{K_1} \sum_{k_2=1}^{K_2} f(y_{k_1}, y_{k_2}) P(Y_1 = y_{k_1}, Y_2 = y_{k_2}) \end{align}$$` In particular, the expectation of `\(X + Y\)` can be expressed simply: -- $$ E[Y_1 + Y_2] = E[Y_1] + E[Y_2] $$ -- Proof: --- ## Another formula for the variance Using the previous result, we have an alternative form for the variance: $$ V[X] = E\left[ (X - E[X])^2 \right] = E[X^2] - E[X]^2 $$ -- Proof: `$$\begin{align} V[X] & = E\left[ (X - E[X])^2 \right] \\ & = E\left[ X^2 - 2XE[X] + E[X]^2 \right] \\ & = E[X^2] - 2E\left[XE[X]\right] + E[X]^2 \\ & = E[X^2] - 2E[X]E\left[X\right] + E[X]^2 \\ & = E[X^2] - 2E[X]^2+ E[X]^2 \\ & = E[X^2] - E[X]^2 \end{align}$$` --- ### Expectation of `\(Y_1 Y_2\)` - The .blue[expectation of a sum] of random variables is the .blue[sum of the expectations] - The .blue[expectation of a product] is -- - no simple formula in general (but important exceptions) - it depends on the covariance between `\(Y_1\)` and `\(Y_2\)` -- <img 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" style="display: block; margin: auto;" /> In each case, `\(E[Y_1] = 0\)` and `\(E[Y_2] = 0\)` but we have .question[quizz1] .pull-left[ - .blue[A:] `\(E[Y_1 Y_2] = \dots\)` - .blue[B:] `\(E[Y_1 Y_2] = \dots\)` - .blue[C:] `\(E[Y_1 Y_2] = \dots\)` ] -- .pull-right[ - `\(E[Y_1 Y_2] = -0.48\)` - `\(E[Y_1 Y_2] = -0.01\)` - `\(E[Y_1 Y_2] = 0.47\)` ] --- ## Special case of independent variables If `\(Y_1\)` and `\(Y_2\)` are .alert[independant], noted `\(Y_1 \perp Y_2\)`, the result is simpler... -- `$$\begin{align} E[Y_1 Y_2] &= \sum_{k_1=1}^{K_1} \sum_{k_2=1}^{K_2} y_{k_1} \times y_{k_2} \underbrace{P(Y_1 = y_{k_1}, Y_2 = y_{k_2})}_{= P(Y_1 = y_{k_1}) \times P(Y_2 = y_{k_2})} \\ & = \sum_{k_1=1}^{K_1} \sum_{k_2=1}^{K_2} y_{k_1} \times y_{k_2} P(Y_1 = y_{k_1}) \times P(Y_2 = y_{k_2}) \\ & = \sum_{k_1=1}^{K_1} \sum_{k_2=1}^{K_2} y_{k_1}P(Y_1 = y_{k_1}) \times y_{k_2}P(Y_2 = y_{k_2}) \\ & = \left( \sum_{k_1=1}^{K_1} y_{k_1}P(Y_1 = y_{k_1}) \right) \times \left( \sum_{k_2=1}^{K_2} y_{k_2}P(Y_2 = y_{k_2}) \right) \\ & = E[Y_1] \times E[Y_2] \end{align}$$` -- - In particular `\(Cov(Y_1, Y_2) = E[Y_1 Y_2] - E[Y_1]E[Y_2] = 0\)` - More generally, `\(E[g_1(Y_1) \times g_2(Y_2)] = E[g_1(Y_1)] \times E[g_2(Y_2)]\)` --- ## Variance of a sum Using the previous result, we have $$ V(Y_1 + Y_2) = \dots $$ -- `$$\begin{align} V(Y_1 + Y_2) & = E\left[ (Y_1 + Y_2)^2 \right] - E\left[ Y_1 + Y_2 \right]^2 \\ & = E[Y_1^2] + 2E[Y_1 Y_2] + E[Y_2^2] - \left( E[Y_1]^2 + 2E[Y_1]E[Y_2] + E[Y_2]^2 \right) \\ & = \underbrace{E[Y_1^2] - E[Y_1]^2}_{=V(Y_1)} + 2 \underbrace{E[Y_1 Y_2] - E[Y_1]E[Y_2]}_{=Cov(Y_1, Y_2)} + \underbrace{E[Y_2^2] - E[Y_2]^2}_{=V(Y_2)} \\ & = V(Y_1) + 2Cov(Y_1, Y_2) + V(Y_2) \end{align}$$` -- In particular, if `\(Y_1 \perp Y_2\)`, then `\(V(Y_1 + Y_2) = V(Y_1) + V(Y_2)\)`. --- ## Sum of independent random variables If `\(Y_1, \dots, Y_n\)` are .blue[i.i.d.], the .blue[sum] `\(S_n = Y_1 + \dots + Y_n\)` has very simple mean and variance: -- .blue[Mean] $$ E[S_n] = E[Y_1 + \dots + Y_n] = E[Y_1] + \dots + E[Y_n] = nE[Y_1] $$ -- .remark[Remark:] The first equality is .alert[always] true and the second requires identical means. -- .blue[Variance] $$ V(S_n) = V[Y_1 + \dots + Y_n] = V[Y_1] + \dots + V[Y_n] = nV[Y_1] $$ -- .remark[Remark:] The equality requires .blue[independence] and the second requires identical variances. --- ## Application to the Binomial distribution Remember that a binomial variable `\(Y \sim \mathcal{B}(K, p)\)` is the sum of `\(K\)` independent Bernoulli trials: $$ X = Y_1 + \dots + Y_K \quad \text{where the } Y_i \text{ are i.i.d. with } Y_i \sim \mathcal{B}(p) $$ Using the previous results, it results that `$$\begin{align} E[X] & = K E[Y_1] = Kp \\ V[X] & = K V[Y_1] = Kp(1 -p) \end{align}$$` --- ## Application to the Poisson distribution We mentioned before that, if `\(X \sim \P(\lambda)\)`, then `\(V[X] = \lambda\)`. Remember that `\(E[X] = \lambda\)`. `$$\begin{align} V(X) & = E[X^2] - E[X]^2 \\ &= E[X(X-1) + E[X] - E[X]^2 \\ & = E[X(X-1)] + E[X] - E[X]^2 \\ & = \lambda^2 + \lambda - \lambda^2 \\ & = \lambda \end{align}$$` --- class: middle, inverse, center # A first try at estimation ## Finite population --- ### The Circle dataset .pull-left[ - Population of `\(n = 100\)` circles - Indexed by `\(i = 1, \dots, n\)` - Each characterized by its diameter `\(D_i\)` ] .pull-right[ <img src="data:image/png;base64,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" style="display: block; margin: auto;" /> ] --- ## List of diameters
.blue[Goal]: Provide an **estimate** of the mean circle diameter `\(E[D]=\mu_D\)`. --- ## Obtaining an estimate Requires three steps: .pull-left[ - .blue[Step 1] Collect some data, i.e. a **sample** - .blue[Step 2] Apply some **estimator** to the sample - .blue[Step 3] Get the estimate: ] -- .pull-right[ - `\(D_1,...,D_n\)` collected *at random* - `\(\overline{D}= \frac{1}{n} \sum_{i=1}^{n}D_i\)` - `\(\widehat{\mu}_D= \frac{1}{n} \sum_{i=1}^{n}d_i\)` ] -- .remark[The estimator is a random variable, the estimate is a numeric value] --- ## Remarks and questions .blue[One remark...] - .alert[ALWAYS] distinguish between the true expectation `\(\mu_D\)` and its estimate `\(\widehat{\mu}_D\)`. .blue[... and three questions] - Any (implicit) **assumption** about the way data were **collected** ? - We defined an intuitive estimator: is it a good one ? How can we **assess** its **performance** ? - If intuition fails to provide an estimator (or a good one), can we think of a **systematic strategy** to obtain a *good* estimator ? --- ## Several ways to "sample at random" In the Circle example, one can - Select the circles "by eye", - Use the droppin' pen technics, - Select identifiers at random, - Any other meaningful strategy .def[Definition] A **S**imple **R**andom **S**ampling is such that all samples of size `\(n\)` have the same probability `\({n \choose N}^{-1}\)` to be drawn. -- .def[Property]: In a SRS, each individual has the same probability `\(n/N\)` to be selected. --- ## Applying SRS to the Circle dataset Here are the estimates obtained from 32 SRS trials, with `\(n=10\)`: `$$\displaystyle{\frac{1}{10}\sum_{i=1}^{10}d_i}=$$` ``` Estimate 01: 5.3076 Estimate 02: 5.8976 Estimate 03: 3.8780 Estimate 04: 4.1243 Estimate 05: 5.0101 Estimate 06: 4.5968 Estimate 07: 5.5176 Estimate 08: 4.2012 Estimate 09: 6.0778 Estimate 10: 6.3875 Estimate 11: 6.4796 Estimate 12: 6.4881 Estimate 13: 4.3832 Estimate 14: 5.9892 Estimate 15: 5.3621 Estimate 16: 5.3960 Estimate 17: 5.3632 Estimate 18: 4.2552 Estimate 19: 6.8789 Estimate 20: 4.7823 Estimate 21: 5.6886 Estimate 22: 5.2215 Estimate 23: 6.9647 Estimate 24: 6.7541 Estimate 25: 5.4322 Estimate 26: 4.9458 Estimate 27: 4.6955 Estimate 28: 5.4238 Estimate 29: 6.0931 Estimate 30: 3.8522 Estimate 31: 4.9817 Estimate 32: 5.2376 ``` -- .blue[*True mean:* 5.427] -- .remark[Remark:] One cannot evaluate the performance of an estimator through an estimate since the estimate depends both on the estimator **and** the sample. .center[ Performance has to be evaluated **on average** over **all** possible samples ] --- ## Performance evaluation Estimator `\(\overline{D}\)` is a random variable (which depends on the sample). One can define: - the bias `\(B(\overline{D}) = E[\overline{D}]-\mu_D\)` - the variance `\(V(\overline{D})\)`. .blue[Mean Square Error] One defines `$$MSE(\overline{D}) = E[(\overline{D}-\mu_D)^2].$$` .remark[Remark] How is `\(MSE(\overline{D})\)` related to bias and variance ? -- `$$MSE(\overline{D}) = B^2(\overline{D}) + V(\overline{D})$$` -- .blue[Why is all of this useful at all ?] .question[quizz2] --- class: middle, inverse, center # SRS in finite population --- ## General framework Consider a population `\(\mathcal{P}\)` of size `\(N\)` and note `\(y_i\)` the value of variable `\(Y\)` measured on individual `\(i\)`. Define `$$\begin{eqnarray*} E[Y] &=& \frac{1}{N}\sum_{i=1}^{N}y_i =\mu\\ \text{and } V[Y] &=& \frac{1}{N}\sum_{i=1}^{N}(y_i-\mu)^2 = \sigma^2 \end{eqnarray*}$$` -- Assume `\(\mu\)` is estimated by applying the empirical mean to sample `\(S\)`: `$$\begin{equation} \overline{Y} = \frac{1}{n}\sum_{i\in S}Y_i = \frac{1}{n}\sum_{i=1}^{N}\varepsilon_i y_i \quad \text{where} \quad \varepsilon_i = \begin{cases} 1 & \text{if } i\in S \\ 0 & \text{otherwise} \end{cases} \end{equation}$$` -- .remark[Remark] `\(n\)` (sample) is generally much smaller than `\(N\)` (population) !! --- ## Theoretical properties of `\(\overline{Y}\)` Consider a SRS, i.e. `\(P(S)={n \choose N}^{-1}\)`, then `$$\begin{eqnarray*} E[\overline{Y}] &=& \mu \\ V[\overline{Y}] &=& \frac{1}{n}\left(1-\frac{n}{N}\right)\frac{N}{N-1}\sigma^2 =\frac{1}{n}\left(1-\frac{n}{N}\right)(\sigma^*)^2 \end{eqnarray*}$$` where `\((\sigma^*)^2= \frac{N}{N-1}\sigma^2\)` is the corrected population variance (more on that later) .question[quizz3] .blue[Proof:] Next slide -- Application to the circle dataset: `\(N=100\)`, `\(n=10\)`, `\(\sigma^2\)`=7.779 `\(\Rightarrow V[\overline{Y}] =\frac{1}{10}\times\frac{100}{99}\times\left(1-\frac{10}{100}\right)\times\)` 7.779 = 0.707 --- ## Proof --- ## Stratified sampling Assume that population `\(\mathcal{P}\)` has `\(H\)` strata `\(\mathcal{P}_1,...,\mathcal{P}_H\)`. .pull-left[ Define - `\(N_h\)`: size of stratum `\(\mathcal{P}_h\)`, `\(h=1,...,H\)`, - `\(N=\sum_h N_h\)`: population size, - `\(\mu_h=\frac{1}{N_h}\sum_{i\in\mathcal{P}_h}y_i\)`: mean of stratum `\(h\)`, - `\(\sigma_h^*=\frac{1}{N_h-1}\sum_{i\in\mathcal{P}_h}(y_i-\mu_h)^2\)`: corrected variance of stratum `\(h\)`. ] -- .pull-right[ .blue[Stratified sampling] Perform .alert[one SRS per stratum] where - `\(n_h\)`: number of sampled individuals per stratum, - `\(n=\sum_h n_h\)`: total sample size, - `\(\overline{Y}_h=\frac{1}{n_h}\sum_{i\in\mathcal{P}_h}Y_i\)`: empirical mean of stratum `\(h\)`. ] -- One then estimates the population mean `\(\mu\)` using `$$\overline{Y}_{St} = \sum_{h=1}^{H}\frac{N_h}{N}\overline{Y}_h$$` --- ## Properties of stratified sampling Consider a stratified sampling strategy. Then `$$\begin{eqnarray*} E[\overline{Y}_{St}] &=& \mu \\ V[\overline{Y}_{St}] &=& \sum_{h=1}^H \left(\frac{N_h}{N}\right)^2\times \frac{1}{n_h}\left(1-\frac{n_h}{N_h}\right) \times (\sigma_h^*)^2 \\ V[\overline{Y}_{St}] &\leq& V[\overline{Y}] \end{eqnarray*}$$` .blue[Proof]: based on the properties of SRS (equalities) and on the decomposition of the population variance + assuming `\(\frac{N_h-1}{N_h}\approx1\)` (inequality). --- ## Proof --- class: middle, inverse, center # IID sampling in infinite populations --- ## Sampling in an infinite population Consider a population `\(\mathcal{P}\)` where variable `\(Y\)` has distribution `\(\mathcal{L}\)`. Define $$E[Y] = \mu, \quad V[Y] =\sigma^2 $$ Assume `\(\mu\)` is estimated by applying the empirical mean to sample `\(Y_1,...,Y_n\)`, where `$$Y_i \sim \mathcal{L}, \text{ i.i.d.}$$` Then $$E\left[\overline{Y}\right] = \mu,\quad V\left[\overline{Y}\right] = \frac{\sigma^2}{n} $$ .question[quizz4] .blue[Proof:] Your turn! --- class: middle, center, inverse # Exercises --- ## Exercise 1: Bulb lifetime A bulb manufacturer claims that 90% of the produced bulbs have a lifetime of 500 hours or higher. An investigator bought 12 bulbs and obtained the following lifetimes: ``` 536 490 510 572 564 537 549 535 549 528 569 533 ``` Assuming bulb lifetimes have a Poisson distribution: - What is the average lifetime of the produced bulbs? - Is the manufacturer honest ? -- .blue[Answers] `\(\widehat{E[L]} =\widehat{\lambda}=\)` 539.333, `\(\quad \widehat{P}_{\widehat{\lambda}}(L\geq 500)=\)` 0.954 --- ## Exercise 2: Exponential distribution A a positive real valued random variable `\(X\)` has a exponential distribution `\(\mathcal{E}(\lambda)\)` with parameter `\(\lambda\)`, noted `\(X\sim\mathcal{E}(\lambda)\)`, if $$f_X(x) = \lambda e^{-\lambda x}, \ x\geq0, \ \lambda\geq0. $$ - Check that `\(\int_0^\infty f(t)dt = 1\)` (i.e. `\(f\)` is a density function). - Compute `\(E[X]\)`, where `\(X\sim\mathcal{E}(\lambda)\)`. - Assuming `\(X_1,...X_n\sim\mathcal{E}(\lambda)\)` i.i.d., suggest an estimator for `\(\lambda\)`. --- ## Summary .blue[3 quantities to distinguish] - `\(\mu=E[X]\)`: .alert[true] population mean. - `\(\overline{Y}=\frac{1}{n}\sum_{i=1}^nY_i\)`: estimator. - `\(\widehat{\mu}=\frac{1}{n}\sum_{i=1}^ny_i\)`: estimate. Remember that estimator `\(\neq\)` estimate -- .blue[Sampling] - In the following, we will always assume either SRS (finite pop.) or i.i.d. sampling (infinite pop.). -- .blue[Performance] - Different estimators can be compared through their MSE. - Keep in mind that `\(MSE(T) = B^2(T) + V(T)\)` -- .blue[What's next ?] - What if intuition fails to provide an estimator (or a good one)? - `\(\Rightarrow\)` Need for a *systematic strategy* to obtain a .alert[satisfactory] estimator!