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endobj -1177 0 obj +1185 0 obj << -/Names [(page.11) 657 0 R (page.12) 676 0 R (page.13) 689 0 R (page.14) 708 0 R (page.15) 728 0 R (page.16) 750 0 R] +/Names [(page.11) 662 0 R (page.12) 681 0 R (page.13) 694 0 R (page.14) 713 0 R (page.15) 733 0 R (page.16) 755 0 R] /Limits [(page.11) (page.16)] >> endobj -1178 0 obj +1186 0 obj << -/Names [(page.17) 773 0 R (page.18) 799 0 R (page.19) 823 0 R (page.2) 557 0 R (page.20) 843 0 R (page.21) 858 0 R] +/Names [(page.17) 778 0 R (page.18) 804 0 R (page.19) 828 0 R (page.2) 562 0 R (page.20) 848 0 R (page.21) 863 0 R] /Limits [(page.17) (page.21)] >> endobj -1179 0 obj +1187 0 obj << -/Names [(page.22) 872 0 R (page.23) 877 0 R (page.24) 897 0 R (page.25) 933 0 R (page.26) 949 0 R (page.27) 974 0 R] +/Names [(page.22) 877 0 R (page.23) 882 0 R (page.24) 902 0 R (page.25) 938 0 R (page.26) 954 0 R (page.27) 979 0 R] /Limits [(page.22) (page.27)] >> endobj -1180 0 obj +1188 0 obj << -/Names [(page.28) 996 0 R (page.29) 1017 0 R (page.3) 571 0 R (page.30) 1029 0 R (page.31) 1040 0 R (page.32) 1060 0 R] +/Names [(page.28) 1001 0 R (page.29) 1022 0 R (page.3) 576 0 R (page.30) 1034 0 R (page.31) 1045 0 R (page.32) 1067 0 R] /Limits [(page.28) (page.32)] >> endobj -1181 0 obj +1189 0 obj << -/Names [(page.33) 1067 0 R (page.34) 1072 0 R (page.35) 1076 0 R (page.4) 578 0 R (page.5) 585 0 R (page.6) 589 0 R] +/Names [(page.33) 1075 0 R (page.34) 1080 0 R (page.35) 1084 0 R (page.4) 583 0 R (page.5) 590 0 R (page.6) 594 0 R] /Limits [(page.33) (page.6)] >> endobj -1182 0 obj +1190 0 obj << -/Names [(page.7) 594 0 R (page.8) 599 0 R (page.9) 604 0 R (section*.10) 37 0 R (section*.11) 41 0 R (section*.12) 45 0 R] +/Names [(page.7) 599 0 R (page.8) 604 0 R (page.9) 609 0 R (section*.10) 37 0 R (section*.11) 41 0 R (section*.12) 45 0 R] /Limits [(page.7) (section*.12)] >> endobj -1183 0 obj +1191 0 obj << -/Names [(section*.13) 49 0 R (section*.15) 600 0 R (section*.3) 9 0 R (section*.4) 13 0 R (section*.5) 17 0 R (section*.6) 21 0 R] +/Names [(section*.13) 49 0 R (section*.15) 605 0 R (section*.3) 9 0 R (section*.4) 13 0 R (section*.5) 17 0 R (section*.6) 21 0 R] /Limits [(section*.13) (section*.6)] >> endobj -1184 0 obj +1192 0 obj << /Names [(section*.7) 25 0 R (section*.8) 29 0 R (section.1.1) 61 0 R (section.1.2) 81 0 R (section.1.3) 85 0 R (section.1.4) 101 0 R] /Limits [(section*.7) (section.1.4)] >> endobj -1185 0 obj +1193 0 obj << /Names [(section.1.5) 117 0 R (section.2.1) 125 0 R (section.2.2) 141 0 R (section.2.3) 165 0 R (section.2.4) 189 0 R (section.2.5) 193 0 R] /Limits [(section.1.5) (section.2.5)] >> endobj -1186 0 obj +1194 0 obj << /Names [(section.2.6) 209 0 R (section.2.7) 221 0 R (section.2.8) 233 0 R (section.3.1) 249 0 R (section.3.10) 345 0 R (section.3.11) 353 0 R] /Limits [(section.2.6) (section.3.11)] >> endobj -1187 0 obj +1195 0 obj << /Names [(section.3.2) 265 0 R (section.3.3) 277 0 R (section.3.4) 281 0 R (section.3.5) 285 0 R (section.3.6) 297 0 R (section.3.7) 317 0 R] /Limits [(section.3.2) (section.3.7)] >> endobj -1188 0 obj +1196 0 obj << /Names [(section.3.8) 333 0 R (section.3.9) 337 0 R (section.4.1) 361 0 R (section.4.2) 373 0 R (section.4.3) 377 0 R (subsection.1.1.1) 65 0 R] /Limits [(section.3.8) (subsection.1.1.1)] >> endobj -1189 0 obj +1197 0 obj << /Names [(subsection.1.1.2) 69 0 R (subsection.1.1.3) 73 0 R (subsection.1.1.4) 77 0 R (subsection.1.3.1) 89 0 R (subsection.1.3.2) 93 0 R (subsection.1.3.3) 97 0 R] /Limits [(subsection.1.1.2) (subsection.1.3.3)] >> endobj -1190 0 obj +1198 0 obj << /Names [(subsection.1.4.1) 105 0 R (subsection.1.4.2) 109 0 R (subsection.1.4.3) 113 0 R (subsection.2.1.1) 129 0 R (subsection.2.1.2) 133 0 R (subsection.2.1.3) 137 0 R] /Limits [(subsection.1.4.1) (subsection.2.1.3)] >> endobj -1191 0 obj +1199 0 obj << /Names [(subsection.2.2.1) 145 0 R (subsection.2.2.2) 149 0 R (subsection.2.2.3) 153 0 R (subsection.2.2.4) 157 0 R (subsection.2.2.5) 161 0 R (subsection.2.3.1) 169 0 R] /Limits [(subsection.2.2.1) (subsection.2.3.1)] >> endobj -1192 0 obj +1200 0 obj << /Names [(subsection.2.3.2) 173 0 R (subsection.2.3.3) 177 0 R (subsection.2.3.4) 181 0 R (subsection.2.3.5) 185 0 R (subsection.2.5.1) 197 0 R (subsection.2.5.2) 201 0 R] /Limits [(subsection.2.3.2) (subsection.2.5.2)] >> endobj -1193 0 obj +1201 0 obj << /Names [(subsection.2.5.3) 205 0 R (subsection.2.6.1) 213 0 R (subsection.2.6.2) 217 0 R (subsection.2.7.1) 225 0 R (subsection.2.7.2) 229 0 R (subsection.2.8.1) 237 0 R] /Limits [(subsection.2.5.3) (subsection.2.8.1)] >> endobj -1194 0 obj +1202 0 obj << /Names [(subsection.2.8.2) 241 0 R (subsection.3.1.1) 253 0 R (subsection.3.1.2) 257 0 R (subsection.3.1.3) 261 0 R (subsection.3.10.1) 349 0 R (subsection.3.2.1) 269 0 R] /Limits [(subsection.2.8.2) (subsection.3.2.1)] >> endobj -1195 0 obj +1203 0 obj << /Names [(subsection.3.2.2) 273 0 R (subsection.3.5.1) 289 0 R (subsection.3.5.2) 293 0 R (subsection.3.6.1) 301 0 R (subsection.3.6.2) 305 0 R (subsection.3.6.3) 309 0 R] /Limits [(subsection.3.2.2) (subsection.3.6.3)] >> endobj -1196 0 obj +1204 0 obj << /Names [(subsection.3.6.4) 313 0 R (subsection.3.7.1) 321 0 R (subsection.3.7.2) 325 0 R (subsection.3.7.3) 329 0 R (subsection.3.9.1) 341 0 R (subsection.4.1.1) 365 0 R] /Limits [(subsection.3.6.4) (subsection.4.1.1)] >> endobj -1197 0 obj -<< -/Names [(subsection.4.1.2) 369 0 R (subsection.4.3.1) 381 0 R (subsection.4.3.2) 385 0 R (subsection.4.3.3) 389 0 R (subsection.4.3.4) 393 0 R (table.4.1) 868 0 R] -/Limits [(subsection.4.1.2) (table.4.1)] ->> -endobj -1198 0 obj -<< -/Names [(theorem.1.1) 605 0 R (theorem.1.10) 617 0 R (theorem.1.11) 618 0 R (theorem.1.12) 626 0 R (theorem.1.13) 627 0 R (theorem.1.14) 631 0 R] -/Limits [(theorem.1.1) (theorem.1.14)] ->> -endobj -1199 0 obj -<< -/Names [(theorem.1.15) 632 0 R (theorem.1.16) 641 0 R (theorem.1.17) 642 0 R (theorem.1.18) 645 0 R (theorem.1.19) 646 0 R (theorem.1.2) 606 0 R] -/Limits [(theorem.1.15) (theorem.1.2)] ->> -endobj -1200 0 obj -<< -/Names [(theorem.1.20) 647 0 R (theorem.1.21) 648 0 R (theorem.1.22) 649 0 R (theorem.1.23) 650 0 R (theorem.1.24) 651 0 R (theorem.1.25) 652 0 R] -/Limits [(theorem.1.20) (theorem.1.25)] ->> -endobj -1201 0 obj -<< -/Names [(theorem.1.26) 658 0 R (theorem.1.27) 659 0 R (theorem.1.28) 660 0 R (theorem.1.29) 661 0 R (theorem.1.3) 607 0 R (theorem.1.30) 662 0 R] -/Limits [(theorem.1.26) (theorem.1.30)] ->> -endobj -1202 0 obj -<< -/Names [(theorem.1.31) 663 0 R (theorem.1.32) 664 0 R (theorem.1.33) 665 0 R (theorem.1.34) 666 0 R (theorem.1.35) 667 0 R (theorem.1.36) 668 0 R] -/Limits [(theorem.1.31) (theorem.1.36)] ->> -endobj -1203 0 obj -<< -/Names [(theorem.1.37) 670 0 R (theorem.1.38) 671 0 R (theorem.1.39) 672 0 R (theorem.1.4) 611 0 R (theorem.1.40) 677 0 R (theorem.1.41) 678 0 R] -/Limits [(theorem.1.37) (theorem.1.41)] ->> -endobj -1204 0 obj -<< -/Names [(theorem.1.42) 679 0 R (theorem.1.43) 680 0 R (theorem.1.44) 681 0 R (theorem.1.45) 682 0 R (theorem.1.46) 683 0 R (theorem.1.47) 684 0 R] -/Limits [(theorem.1.42) (theorem.1.47)] ->> -endobj 1205 0 obj << -/Names [(theorem.1.5) 612 0 R (theorem.1.6) 613 0 R (theorem.1.7) 614 0 R (theorem.1.8) 615 0 R (theorem.1.9) 616 0 R (theorem.2.1) 690 0 R] -/Limits [(theorem.1.5) (theorem.2.1)] +/Names [(subsection.4.1.2) 369 0 R (subsection.4.3.1) 381 0 R (subsection.4.3.2) 385 0 R (subsection.4.3.3) 389 0 R (subsection.4.3.4) 393 0 R (subsection.4.3.5) 397 0 R] +/Limits [(subsection.4.1.2) (subsection.4.3.5)] >> endobj 1206 0 obj << -/Names [(theorem.2.10) 701 0 R (theorem.2.100) 849 0 R (theorem.2.101) 850 0 R (theorem.2.102) 859 0 R (theorem.2.103) 861 0 R (theorem.2.104) 862 0 R] -/Limits [(theorem.2.10) (theorem.2.104)] +/Names [(table.4.1) 873 0 R (theorem.1.1) 610 0 R (theorem.1.10) 622 0 R (theorem.1.11) 623 0 R (theorem.1.12) 631 0 R (theorem.1.13) 632 0 R] +/Limits [(table.4.1) (theorem.1.13)] >> endobj 1207 0 obj << -/Names [(theorem.2.105) 863 0 R (theorem.2.106) 864 0 R (theorem.2.107) 865 0 R (theorem.2.108) 866 0 R (theorem.2.11) 702 0 R (theorem.2.12) 703 0 R] -/Limits [(theorem.2.105) (theorem.2.12)] +/Names [(theorem.1.14) 636 0 R (theorem.1.15) 637 0 R (theorem.1.16) 646 0 R (theorem.1.17) 647 0 R (theorem.1.18) 650 0 R (theorem.1.19) 651 0 R] +/Limits [(theorem.1.14) (theorem.1.19)] >> endobj 1208 0 obj << -/Names [(theorem.2.13) 709 0 R (theorem.2.14) 710 0 R (theorem.2.15) 711 0 R (theorem.2.16) 712 0 R (theorem.2.17) 713 0 R (theorem.2.18) 714 0 R] -/Limits [(theorem.2.13) (theorem.2.18)] +/Names [(theorem.1.2) 611 0 R (theorem.1.20) 652 0 R (theorem.1.21) 653 0 R (theorem.1.22) 654 0 R (theorem.1.23) 655 0 R (theorem.1.24) 656 0 R] +/Limits [(theorem.1.2) (theorem.1.24)] >> endobj 1209 0 obj << -/Names [(theorem.2.19) 715 0 R (theorem.2.2) 691 0 R (theorem.2.20) 716 0 R (theorem.2.21) 717 0 R (theorem.2.22) 718 0 R (theorem.2.23) 719 0 R] -/Limits [(theorem.2.19) (theorem.2.23)] +/Names [(theorem.1.25) 657 0 R (theorem.1.26) 663 0 R (theorem.1.27) 664 0 R (theorem.1.28) 665 0 R (theorem.1.29) 666 0 R (theorem.1.3) 612 0 R] +/Limits [(theorem.1.25) (theorem.1.3)] >> endobj 1210 0 obj << -/Names [(theorem.2.24) 720 0 R (theorem.2.25) 721 0 R (theorem.2.26) 722 0 R (theorem.2.27) 729 0 R (theorem.2.28) 730 0 R (theorem.2.29) 731 0 R] -/Limits [(theorem.2.24) (theorem.2.29)] +/Names [(theorem.1.30) 667 0 R (theorem.1.31) 668 0 R (theorem.1.32) 669 0 R (theorem.1.33) 670 0 R (theorem.1.34) 671 0 R (theorem.1.35) 672 0 R] +/Limits [(theorem.1.30) (theorem.1.35)] >> endobj 1211 0 obj << -/Names [(theorem.2.3) 694 0 R (theorem.2.30) 732 0 R (theorem.2.31) 733 0 R (theorem.2.32) 734 0 R (theorem.2.33) 735 0 R (theorem.2.34) 736 0 R] -/Limits [(theorem.2.3) (theorem.2.34)] +/Names [(theorem.1.36) 673 0 R (theorem.1.37) 675 0 R (theorem.1.38) 676 0 R (theorem.1.39) 677 0 R (theorem.1.4) 616 0 R (theorem.1.40) 682 0 R] +/Limits [(theorem.1.36) (theorem.1.40)] >> endobj 1212 0 obj << -/Names [(theorem.2.35) 737 0 R (theorem.2.36) 738 0 R (theorem.2.37) 739 0 R (theorem.2.38) 741 0 R (theorem.2.39) 742 0 R (theorem.2.4) 695 0 R] -/Limits [(theorem.2.35) (theorem.2.4)] +/Names [(theorem.1.41) 683 0 R (theorem.1.42) 684 0 R (theorem.1.43) 685 0 R (theorem.1.44) 686 0 R (theorem.1.45) 687 0 R (theorem.1.46) 688 0 R] +/Limits [(theorem.1.41) (theorem.1.46)] >> endobj 1213 0 obj << -/Names [(theorem.2.40) 743 0 R (theorem.2.41) 744 0 R (theorem.2.42) 751 0 R (theorem.2.43) 752 0 R (theorem.2.44) 753 0 R (theorem.2.45) 754 0 R] -/Limits [(theorem.2.40) (theorem.2.45)] +/Names [(theorem.1.47) 689 0 R (theorem.1.5) 617 0 R (theorem.1.6) 618 0 R (theorem.1.7) 619 0 R (theorem.1.8) 620 0 R (theorem.1.9) 621 0 R] +/Limits [(theorem.1.47) (theorem.1.9)] >> endobj 1214 0 obj << -/Names [(theorem.2.46) 757 0 R (theorem.2.47) 758 0 R (theorem.2.48) 759 0 R (theorem.2.49) 760 0 R (theorem.2.5) 696 0 R (theorem.2.50) 761 0 R] -/Limits [(theorem.2.46) (theorem.2.50)] +/Names [(theorem.2.1) 695 0 R (theorem.2.10) 706 0 R (theorem.2.100) 854 0 R (theorem.2.101) 855 0 R (theorem.2.102) 864 0 R (theorem.2.103) 866 0 R] +/Limits [(theorem.2.1) (theorem.2.103)] >> endobj 1215 0 obj << -/Names [(theorem.2.51) 762 0 R (theorem.2.52) 763 0 R (theorem.2.53) 766 0 R (theorem.2.54) 774 0 R (theorem.2.55) 781 0 R (theorem.2.56) 782 0 R] -/Limits [(theorem.2.51) (theorem.2.56)] +/Names [(theorem.2.104) 867 0 R (theorem.2.105) 868 0 R (theorem.2.106) 869 0 R (theorem.2.107) 870 0 R (theorem.2.108) 871 0 R (theorem.2.11) 707 0 R] +/Limits [(theorem.2.104) (theorem.2.11)] >> endobj 1216 0 obj << -/Names [(theorem.2.57) 783 0 R (theorem.2.58) 784 0 R (theorem.2.59) 785 0 R (theorem.2.6) 697 0 R (theorem.2.60) 786 0 R (theorem.2.61) 787 0 R] -/Limits [(theorem.2.57) (theorem.2.61)] +/Names [(theorem.2.12) 708 0 R (theorem.2.13) 714 0 R (theorem.2.14) 715 0 R (theorem.2.15) 716 0 R (theorem.2.16) 717 0 R (theorem.2.17) 718 0 R] +/Limits [(theorem.2.12) (theorem.2.17)] >> endobj 1217 0 obj << -/Names [(theorem.2.62) 788 0 R (theorem.2.63) 789 0 R (theorem.2.64) 790 0 R (theorem.2.65) 791 0 R (theorem.2.66) 792 0 R (theorem.2.67) 800 0 R] -/Limits [(theorem.2.62) (theorem.2.67)] +/Names [(theorem.2.18) 719 0 R (theorem.2.19) 720 0 R (theorem.2.2) 696 0 R (theorem.2.20) 721 0 R (theorem.2.21) 722 0 R (theorem.2.22) 723 0 R] +/Limits [(theorem.2.18) (theorem.2.22)] >> endobj 1218 0 obj << -/Names [(theorem.2.68) 801 0 R (theorem.2.69) 802 0 R (theorem.2.7) 698 0 R (theorem.2.70) 803 0 R (theorem.2.71) 804 0 R (theorem.2.72) 805 0 R] -/Limits [(theorem.2.68) (theorem.2.72)] +/Names [(theorem.2.23) 724 0 R (theorem.2.24) 725 0 R (theorem.2.25) 726 0 R (theorem.2.26) 727 0 R (theorem.2.27) 734 0 R (theorem.2.28) 735 0 R] +/Limits [(theorem.2.23) (theorem.2.28)] >> endobj 1219 0 obj << -/Names [(theorem.2.73) 806 0 R (theorem.2.74) 807 0 R (theorem.2.75) 808 0 R (theorem.2.76) 809 0 R (theorem.2.77) 810 0 R (theorem.2.78) 811 0 R] -/Limits [(theorem.2.73) (theorem.2.78)] +/Names [(theorem.2.29) 736 0 R (theorem.2.3) 699 0 R (theorem.2.30) 737 0 R (theorem.2.31) 738 0 R (theorem.2.32) 739 0 R (theorem.2.33) 740 0 R] +/Limits [(theorem.2.29) (theorem.2.33)] >> endobj 1220 0 obj << -/Names [(theorem.2.79) 812 0 R (theorem.2.8) 699 0 R (theorem.2.80) 813 0 R (theorem.2.81) 814 0 R (theorem.2.82) 824 0 R (theorem.2.83) 825 0 R] -/Limits [(theorem.2.79) (theorem.2.83)] +/Names [(theorem.2.34) 741 0 R (theorem.2.35) 742 0 R (theorem.2.36) 743 0 R (theorem.2.37) 744 0 R (theorem.2.38) 746 0 R (theorem.2.39) 747 0 R] +/Limits [(theorem.2.34) (theorem.2.39)] >> endobj 1221 0 obj << -/Names [(theorem.2.84) 826 0 R (theorem.2.85) 827 0 R (theorem.2.86) 828 0 R (theorem.2.87) 829 0 R (theorem.2.88) 830 0 R (theorem.2.89) 831 0 R] -/Limits [(theorem.2.84) (theorem.2.89)] +/Names [(theorem.2.4) 700 0 R (theorem.2.40) 748 0 R (theorem.2.41) 749 0 R (theorem.2.42) 756 0 R (theorem.2.43) 757 0 R (theorem.2.44) 758 0 R] +/Limits [(theorem.2.4) (theorem.2.44)] >> endobj 1222 0 obj << -/Names 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a/Secondo anno/Elementi di probabilità e statistica/sections/4-statistica.tex +++ b/Secondo anno/Elementi di probabilità e statistica/sections/4-statistica.tex @@ -197,6 +197,27 @@ a $P_\theta$). varianza, consistente, sempre per la LGN. \end{remark} +\subsection{Rischio quadratico e preferibilità} + +\begin{definition}[Rischio quadratico di uno stimatore] + Dato uno stimatore $U$ di $h : \Sigma \to \RR$, si definisce + \textbf{rischio quadratico} di $U$ per $\sigma$ il seguente valore: + \[ + R_\sigma(U) = \EE[(U - h(\sigma))^2]. + \] +\end{definition} + +\begin{remark} + Se $U$ è uno stimatore corretto di $h$, allora + $R_\sigma(U) = \Var^\sigma(U)$. +\end{remark} + +\begin{definition}[Preferibilità] + Dati due stimatori $U$, $V$ di $h : \Sigma \to \RR$, si dice + che $U$ è \textbf{preferibile} rispetto a $V$ se + $R_\sigma(U) \leq R_\sigma(V)$ per ogni $\sigma \in \Sigma$. +\end{definition} + \subsection{Stimatore di massima verosomiglianza} D'ora in avanti sottintenderemo di star lavorando sullo