Module6.pdf - Big Pic EDA 6.1 \u2014 6.3 MLE MoM 6.4 Regress 6.5 Large Methods of Mathematical Statistics 6 Point Estimation Tim Brown and Davide Ferrari

Module6.pdf - Big Pic EDA 6.1 — 6.3 MLE MoM 6.4 Regress...

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Big Pic EDA 6.1 — 6.3 MLE, MoM 6.4 Regress 6.5 Large Methods of Mathematical Statistics 6 Point Estimation Tim Brown and Davide Ferrari Methods of Math. Stats.: Estimation 1/113
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Big Pic EDA 6.1 — 6.3 MLE, MoM 6.4 Regress 6.5 Large Figure: From 3, Probability to 4, Inference via 2, Exploratory Data Analysis Methods of Math. Stats.: Estimation 2/113
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Big Pic EDA 6.1 — 6.3 MLE, MoM 6.4 Regress 6.5 Large Introduction The data is assumed to be (for the moment) numbers x 1 , . . . , x n . Methods of Math. Stats.: Estimation 3/113
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Big Pic EDA 6.1 — 6.3 MLE, MoM 6.4 Regress 6.5 Large Introduction The data is assumed to be (for the moment) numbers x 1 , . . . , x n . The model for the data is a random sample , that is a sequence of independent and identically distributed random variables X 1 , X 2 , . . . , X n . This model is equivalent to random selection from a hypothetical infinite population . Methods of Math. Stats.: Estimation 3/113
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Big Pic EDA 6.1 — 6.3 MLE, MoM 6.4 Regress 6.5 Large Introduction The data is assumed to be (for the moment) numbers x 1 , . . . , x n . The model for the data is a random sample , that is a sequence of independent and identically distributed random variables X 1 , X 2 , . . . , X n . This model is equivalent to random selection from a hypothetical infinite population . The goal is to use the data to learn about the distribution of the random variables from the data. Methods of Math. Stats.: Estimation 3/113
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Big Pic EDA 6.1 — 6.3 MLE, MoM 6.4 Regress 6.5 Large Introduction Ctd A Statistic T = φ ( X 1 , . . . , X n ) is a function of the sample and its realisation is denoted by t = φ ( x 1 , . . . , x n ) . Methods of Math. Stats.: Estimation 4/113
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Big Pic EDA 6.1 — 6.3 MLE, MoM 6.4 Regress 6.5 Large Introduction Ctd A Statistic T = φ ( X 1 , . . . , X n ) is a function of the sample and its realisation is denoted by t = φ ( x 1 , . . . , x n ) . A statistic has two purposes: I Describe/summarise the sample (Stage 2 in Figure 1) I Estimate the true (unknown) distribution generating the sample (Stage 4 in Figure 1 ) Methods of Math. Stats.: Estimation 4/113
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Big Pic EDA 6.1 — 6.3 MLE, MoM 6.4 Regress 6.5 Large Stress and cancer An experiment divided 10 mice randomly into control and stress groups, with the stress group receiving chronic stress . Methods of Math. Stats.: Estimation 5/113
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Big Pic EDA 6.1 — 6.3 MLE, MoM 6.4 Regress 6.5 Large Stress and cancer An experiment divided 10 mice randomly into control and stress groups, with the stress group receiving chronic stress . The biologists measured I Vascular endothelial growth factor C (VEGFC) — a protein involved in lymphangiogenesis — low or high levels of this are observed in many diseases I Prostaglandin-endoperoxide synthase 2 (COX2) — a protein involved in inflammatory processes related to cancer. Figure 2 shows cancer differences between the groups. Figure: Differences in cancer progression Methods of Math. Stats.: Estimation 5/113
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Big Pic EDA 6.1 — 6.3 MLE, MoM 6.4 Regress 6.5 Large Data: Stress and cancer From the widely reported study: "C. P. Le et al. Chronic stress in mice remodels lymph vasculature to promote tumour cell dissemination Nature Communications, 7, 2016".
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