From 90e8084f9c2ba38d4f72a22be0617656540d3b99 Mon Sep 17 00:00:00 2001 From: Hearot Date: Sun, 14 Jul 2024 22:54:43 +0200 Subject: [PATCH] feat(eps): aggiunge stimatore di massima verosomiglianza --- .../main.pdf | 6393 +++++++++-------- .../preamble.tex | 1 + .../sections/0-notazioni.tex | 1 + .../sections/4-statistica.tex | 158 +- 4 files changed, 3608 insertions(+), 2945 deletions(-) diff --git a/Secondo anno/Elementi di probabilità e statistica/main.pdf b/Secondo anno/Elementi di probabilità e statistica/main.pdf index ca38dfa..ecdadb9 100644 --- a/Secondo anno/Elementi di probabilità e statistica/main.pdf +++ b/Secondo anno/Elementi di probabilità e statistica/main.pdf @@ -553,54 +553,84 @@ endobj (\376\377\000I\000n\000d\000i\000c\000i\000\040\000s\000u\000\040\000c\000o\000p\000p\000i\000e\000\040\000d\000i\000\040\000d\000a\000t\000i) endobj 372 0 obj -<< /S /GoTo /D (subsection.4.1.3) >> +<< /S /GoTo /D (section.4.2) >> endobj 375 0 obj 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[(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 -1141 0 obj +1202 0 obj << -/Names [(theorem.1.39) 644 0 R (theorem.1.4) 583 0 R (theorem.1.40) 649 0 R (theorem.1.41) 650 0 R (theorem.1.42) 651 0 R (theorem.1.43) 652 0 R] -/Limits [(theorem.1.39) (theorem.1.43)] +/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 -1142 0 obj +1203 0 obj << -/Names [(theorem.1.44) 653 0 R (theorem.1.45) 654 0 R (theorem.1.46) 655 0 R (theorem.1.47) 656 0 R (theorem.1.5) 584 0 R (theorem.1.6) 585 0 R] -/Limits [(theorem.1.44) (theorem.1.6)] +/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 -1143 0 obj +1204 0 obj << -/Names [(theorem.1.7) 586 0 R (theorem.1.8) 587 0 R (theorem.1.9) 588 0 R (theorem.2.1) 662 0 R (theorem.2.10) 673 0 R (theorem.2.100) 821 0 R] -/Limits [(theorem.1.7) (theorem.2.100)] +/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 -1144 0 obj +1205 0 obj << -/Names [(theorem.2.101) 822 0 R (theorem.2.102) 831 0 R (theorem.2.103) 833 0 R (theorem.2.104) 834 0 R (theorem.2.105) 835 0 R (theorem.2.106) 836 0 R] -/Limits [(theorem.2.101) (theorem.2.106)] +/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)] >> endobj -1145 0 obj +1206 0 obj << -/Names [(theorem.2.107) 837 0 R (theorem.2.108) 838 0 R (theorem.2.11) 674 0 R (theorem.2.12) 675 0 R (theorem.2.13) 681 0 R (theorem.2.14) 682 0 R] -/Limits [(theorem.2.107) (theorem.2.14)] +/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)] >> endobj -1146 0 obj +1207 0 obj << -/Names [(theorem.2.15) 683 0 R (theorem.2.16) 684 0 R (theorem.2.17) 685 0 R (theorem.2.18) 686 0 R (theorem.2.19) 687 0 R (theorem.2.2) 663 0 R] -/Limits [(theorem.2.15) (theorem.2.2)] +/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)] >> endobj -1147 0 obj +1208 0 obj << -/Names [(theorem.2.20) 688 0 R (theorem.2.21) 689 0 R (theorem.2.22) 690 0 R (theorem.2.23) 691 0 R (theorem.2.24) 692 0 R (theorem.2.25) 693 0 R] -/Limits [(theorem.2.20) (theorem.2.25)] +/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)] >> endobj -1148 0 obj +1209 0 obj << -/Names [(theorem.2.26) 694 0 R (theorem.2.27) 701 0 R (theorem.2.28) 702 0 R (theorem.2.29) 703 0 R (theorem.2.3) 666 0 R (theorem.2.30) 704 0 R] -/Limits [(theorem.2.26) (theorem.2.30)] +/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)] >> endobj -1149 0 obj +1210 0 obj << -/Names [(theorem.2.31) 705 0 R (theorem.2.32) 706 0 R (theorem.2.33) 707 0 R (theorem.2.34) 708 0 R (theorem.2.35) 709 0 R (theorem.2.36) 710 0 R] -/Limits [(theorem.2.31) (theorem.2.36)] +/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)] >> endobj -1150 0 obj +1211 0 obj << -/Names [(theorem.2.37) 711 0 R (theorem.2.38) 713 0 R (theorem.2.39) 714 0 R (theorem.2.4) 667 0 R (theorem.2.40) 715 0 R (theorem.2.41) 716 0 R] -/Limits [(theorem.2.37) (theorem.2.41)] +/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)] >> endobj -1151 0 obj +1212 0 obj << -/Names [(theorem.2.42) 723 0 R (theorem.2.43) 724 0 R (theorem.2.44) 725 0 R (theorem.2.45) 726 0 R (theorem.2.46) 729 0 R (theorem.2.47) 730 0 R] -/Limits [(theorem.2.42) (theorem.2.47)] +/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)] >> endobj -1152 0 obj +1213 0 obj << -/Names [(theorem.2.48) 731 0 R (theorem.2.49) 732 0 R (theorem.2.5) 668 0 R (theorem.2.50) 733 0 R (theorem.2.51) 734 0 R (theorem.2.52) 735 0 R] -/Limits [(theorem.2.48) (theorem.2.52)] +/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)] >> endobj -1153 0 obj +1214 0 obj << -/Names [(theorem.2.53) 738 0 R (theorem.2.54) 746 0 R (theorem.2.55) 753 0 R (theorem.2.56) 754 0 R (theorem.2.57) 755 0 R (theorem.2.58) 756 0 R] -/Limits [(theorem.2.53) (theorem.2.58)] +/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)] >> endobj -1154 0 obj +1215 0 obj << -/Names [(theorem.2.59) 757 0 R (theorem.2.6) 669 0 R (theorem.2.60) 758 0 R (theorem.2.61) 759 0 R (theorem.2.62) 760 0 R (theorem.2.63) 761 0 R] -/Limits [(theorem.2.59) (theorem.2.63)] +/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)] >> endobj -1155 0 obj +1216 0 obj << -/Names [(theorem.2.64) 762 0 R (theorem.2.65) 763 0 R (theorem.2.66) 764 0 R (theorem.2.67) 772 0 R (theorem.2.68) 773 0 R (theorem.2.69) 774 0 R] -/Limits [(theorem.2.64) (theorem.2.69)] +/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)] >> endobj -1156 0 obj +1217 0 obj << -/Names [(theorem.2.7) 670 0 R (theorem.2.70) 775 0 R (theorem.2.71) 776 0 R (theorem.2.72) 777 0 R (theorem.2.73) 778 0 R (theorem.2.74) 779 0 R] -/Limits [(theorem.2.7) (theorem.2.74)] +/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)] >> endobj -1157 0 obj +1218 0 obj << -/Names [(theorem.2.75) 780 0 R (theorem.2.76) 781 0 R (theorem.2.77) 782 0 R (theorem.2.78) 783 0 R (theorem.2.79) 784 0 R (theorem.2.8) 671 0 R] -/Limits [(theorem.2.75) (theorem.2.8)] +/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)] >> endobj -1158 0 obj +1219 0 obj << -/Names [(theorem.2.80) 785 0 R (theorem.2.81) 786 0 R (theorem.2.82) 796 0 R (theorem.2.83) 797 0 R (theorem.2.84) 798 0 R (theorem.2.85) 799 0 R] -/Limits [(theorem.2.80) (theorem.2.85)] +/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)] >> endobj -1159 0 obj +1220 0 obj << -/Names [(theorem.2.86) 800 0 R (theorem.2.87) 801 0 R (theorem.2.88) 802 0 R (theorem.2.89) 803 0 R (theorem.2.9) 672 0 R (theorem.2.90) 804 0 R] -/Limits [(theorem.2.86) (theorem.2.90)] +/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)] >> endobj -1160 0 obj +1221 0 obj << -/Names [(theorem.2.91) 805 0 R (theorem.2.92) 808 0 R (theorem.2.93) 809 0 R (theorem.2.94) 810 0 R (theorem.2.95) 816 0 R (theorem.2.96) 817 0 R] -/Limits [(theorem.2.91) (theorem.2.96)] +/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)] >> endobj -1161 0 obj +1222 0 obj << -/Names [(theorem.2.97) 818 0 R (theorem.2.98) 819 0 R (theorem.2.99) 820 0 R (theorem.3.1) 850 0 R (theorem.3.10) 874 0 R (theorem.3.11) 881 0 R] -/Limits [(theorem.2.97) (theorem.3.11)] +/Names [(theorem.2.9) 700 0 R (theorem.2.90) 832 0 R (theorem.2.91) 833 0 R (theorem.2.92) 836 0 R (theorem.2.93) 837 0 R (theorem.2.94) 838 0 R] +/Limits [(theorem.2.9) (theorem.2.94)] >> endobj -1162 0 obj +1223 0 obj << -/Names [(theorem.3.12) 882 0 R (theorem.3.13) 885 0 R (theorem.3.14) 886 0 R (theorem.3.15) 887 0 R (theorem.3.16) 888 0 R (theorem.3.17) 889 0 R] -/Limits [(theorem.3.12) (theorem.3.17)] +/Names [(theorem.2.95) 844 0 R (theorem.2.96) 845 0 R (theorem.2.97) 846 0 R (theorem.2.98) 847 0 R (theorem.2.99) 848 0 R (theorem.3.1) 878 0 R] +/Limits [(theorem.2.95) (theorem.3.1)] >> endobj -1163 0 obj +1224 0 obj << -/Names [(theorem.3.18) 890 0 R (theorem.3.19) 891 0 R (theorem.3.2) 851 0 R (theorem.3.20) 892 0 R (theorem.3.21) 893 0 R (theorem.3.22) 906 0 R] -/Limits [(theorem.3.18) (theorem.3.22)] +/Names [(theorem.3.10) 902 0 R (theorem.3.11) 909 0 R (theorem.3.12) 910 0 R (theorem.3.13) 913 0 R (theorem.3.14) 914 0 R (theorem.3.15) 915 0 R] +/Limits [(theorem.3.10) (theorem.3.15)] >> endobj -1164 0 obj +1225 0 obj << -/Names [(theorem.3.23) 911 0 R (theorem.3.24) 912 0 R (theorem.3.25) 913 0 R (theorem.3.26) 914 0 R (theorem.3.27) 915 0 R (theorem.3.28) 916 0 R] -/Limits [(theorem.3.23) (theorem.3.28)] +/Names [(theorem.3.16) 916 0 R (theorem.3.17) 917 0 R (theorem.3.18) 918 0 R (theorem.3.19) 919 0 R (theorem.3.2) 879 0 R (theorem.3.20) 920 0 R] +/Limits [(theorem.3.16) (theorem.3.20)] >> endobj -1165 0 obj +1226 0 obj << -/Names [(theorem.3.29) 917 0 R (theorem.3.3) 855 0 R (theorem.3.30) 922 0 R (theorem.3.31) 923 0 R (theorem.3.32) 924 0 R (theorem.3.33) 925 0 R] -/Limits [(theorem.3.29) (theorem.3.33)] +/Names [(theorem.3.21) 921 0 R (theorem.3.22) 934 0 R (theorem.3.23) 939 0 R (theorem.3.24) 940 0 R (theorem.3.25) 941 0 R (theorem.3.26) 942 0 R] +/Limits [(theorem.3.21) (theorem.3.26)] >> endobj -1166 0 obj +1227 0 obj << -/Names [(theorem.3.34) 926 0 R (theorem.3.35) 927 0 R (theorem.3.36) 930 0 R (theorem.3.37) 933 0 R (theorem.3.38) 934 0 R (theorem.3.39) 935 0 R] -/Limits [(theorem.3.34) (theorem.3.39)] +/Names [(theorem.3.27) 943 0 R (theorem.3.28) 944 0 R (theorem.3.29) 945 0 R (theorem.3.3) 883 0 R (theorem.3.30) 950 0 R (theorem.3.31) 951 0 R] +/Limits [(theorem.3.27) (theorem.3.31)] >> endobj -1167 0 obj +1228 0 obj << -/Names [(theorem.3.4) 861 0 R (theorem.3.40) 936 0 R (theorem.3.41) 937 0 R (theorem.3.42) 938 0 R (theorem.3.43) 939 0 R (theorem.3.44) 940 0 R] -/Limits [(theorem.3.4) (theorem.3.44)] +/Names [(theorem.3.32) 952 0 R (theorem.3.33) 953 0 R (theorem.3.34) 954 0 R (theorem.3.35) 955 0 R (theorem.3.36) 958 0 R (theorem.3.37) 961 0 R] +/Limits [(theorem.3.32) (theorem.3.37)] >> endobj -1168 0 obj +1229 0 obj << -/Names [(theorem.3.45) 947 0 R (theorem.3.46) 948 0 R (theorem.3.47) 949 0 R (theorem.3.48) 950 0 R (theorem.3.49) 951 0 R (theorem.3.5) 862 0 R] -/Limits [(theorem.3.45) (theorem.3.5)] +/Names [(theorem.3.38) 962 0 R (theorem.3.39) 963 0 R (theorem.3.4) 889 0 R (theorem.3.40) 964 0 R (theorem.3.41) 965 0 R (theorem.3.42) 966 0 R] +/Limits [(theorem.3.38) (theorem.3.42)] >> endobj -1169 0 obj +1230 0 obj << -/Names [(theorem.3.50) 952 0 R (theorem.3.51) 953 0 R (theorem.3.52) 954 0 R (theorem.3.53) 955 0 R (theorem.3.54) 956 0 R (theorem.3.55) 957 0 R] -/Limits [(theorem.3.50) (theorem.3.55)] +/Names [(theorem.3.43) 967 0 R (theorem.3.44) 968 0 R (theorem.3.45) 975 0 R (theorem.3.46) 976 0 R (theorem.3.47) 977 0 R (theorem.3.48) 978 0 R] +/Limits [(theorem.3.43) (theorem.3.48)] >> endobj -1170 0 obj +1231 0 obj << -/Names [(theorem.3.56) 958 0 R (theorem.3.57) 959 0 R (theorem.3.58) 960 0 R (theorem.3.59) 961 0 R (theorem.3.6) 863 0 R (theorem.3.60) 962 0 R] -/Limits [(theorem.3.56) (theorem.3.60)] +/Names [(theorem.3.49) 979 0 R (theorem.3.5) 890 0 R (theorem.3.50) 980 0 R (theorem.3.51) 981 0 R (theorem.3.52) 982 0 R (theorem.3.53) 983 0 R] +/Limits [(theorem.3.49) (theorem.3.53)] >> endobj -1171 0 obj +1232 0 obj << -/Names [(theorem.3.61) 969 0 R (theorem.3.62) 970 0 R (theorem.3.63) 971 0 R (theorem.3.64) 972 0 R (theorem.3.65) 976 0 R (theorem.3.66) 977 0 R] -/Limits [(theorem.3.61) (theorem.3.66)] +/Names [(theorem.3.54) 984 0 R (theorem.3.55) 985 0 R (theorem.3.56) 986 0 R (theorem.3.57) 987 0 R (theorem.3.58) 988 0 R (theorem.3.59) 989 0 R] +/Limits [(theorem.3.54) (theorem.3.59)] >> endobj -1172 0 obj +1233 0 obj << -/Names [(theorem.3.67) 978 0 R (theorem.3.68) 979 0 R (theorem.3.69) 980 0 R (theorem.3.7) 864 0 R (theorem.3.70) 981 0 R (theorem.3.71) 982 0 R] -/Limits [(theorem.3.67) (theorem.3.71)] +/Names [(theorem.3.6) 891 0 R (theorem.3.60) 990 0 R (theorem.3.61) 997 0 R (theorem.3.62) 998 0 R (theorem.3.63) 999 0 R (theorem.3.64) 1000 0 R] +/Limits [(theorem.3.6) (theorem.3.64)] >> endobj -1173 0 obj +1234 0 obj << -/Names [(theorem.3.72) 983 0 R (theorem.3.73) 990 0 R (theorem.3.74) 991 0 R (theorem.3.75) 992 0 R (theorem.3.76) 993 0 R (theorem.3.77) 994 0 R] -/Limits [(theorem.3.72) (theorem.3.77)] +/Names [(theorem.3.65) 1004 0 R (theorem.3.66) 1005 0 R (theorem.3.67) 1006 0 R (theorem.3.68) 1007 0 R (theorem.3.69) 1008 0 R (theorem.3.7) 892 0 R] +/Limits [(theorem.3.65) (theorem.3.7)] >> endobj -1174 0 obj +1235 0 obj << -/Names [(theorem.3.78) 995 0 R (theorem.3.79) 996 0 R (theorem.3.8) 870 0 R (theorem.3.9) 873 0 R (theorem.4.1) 1002 0 R (theorem.4.2) 1003 0 R] -/Limits [(theorem.3.78) (theorem.4.2)] +/Names [(theorem.3.70) 1009 0 R (theorem.3.71) 1010 0 R (theorem.3.72) 1011 0 R (theorem.3.73) 1018 0 R (theorem.3.74) 1019 0 R (theorem.3.75) 1020 0 R] +/Limits [(theorem.3.70) (theorem.3.75)] >> endobj -1175 0 obj +1236 0 obj << -/Names [(theorem.4.3) 1004 0 R (theorem.4.4) 1005 0 R (theorem.4.5) 1006 0 R (theorem.4.6) 1007 0 R (theorem.4.7) 1013 0 R] -/Limits [(theorem.4.3) (theorem.4.7)] +/Names [(theorem.3.76) 1021 0 R (theorem.3.77) 1022 0 R (theorem.3.78) 1023 0 R (theorem.3.79) 1024 0 R (theorem.3.8) 898 0 R (theorem.3.9) 901 0 R] +/Limits [(theorem.3.76) (theorem.3.9)] >> endobj -1176 0 obj +1237 0 obj +<< +/Names [(theorem.4.1) 1030 0 R (theorem.4.10) 1044 0 R (theorem.4.11) 1047 0 R (theorem.4.12) 1048 0 R (theorem.4.13) 1049 0 R (theorem.4.14) 1050 0 R] +/Limits [(theorem.4.1) (theorem.4.14)] +>> +endobj +1238 0 obj +<< +/Names [(theorem.4.15) 1051 0 R (theorem.4.16) 1052 0 R (theorem.4.17) 1053 0 R (theorem.4.18) 1054 0 R (theorem.4.19) 1055 0 R (theorem.4.2) 1031 0 R] +/Limits [(theorem.4.15) (theorem.4.2)] +>> +endobj +1239 0 obj +<< +/Names [(theorem.4.20) 1056 0 R (theorem.4.21) 1061 0 R (theorem.4.22) 1062 0 R (theorem.4.3) 1032 0 R (theorem.4.4) 1033 0 R (theorem.4.5) 1034 0 R] +/Limits [(theorem.4.20) (theorem.4.5)] +>> +endobj +1240 0 obj +<< +/Names [(theorem.4.6) 1035 0 R (theorem.4.7) 1041 0 R (theorem.4.8) 1042 0 R (theorem.4.9) 1043 0 R] +/Limits [(theorem.4.6) (theorem.4.9)] +>> +endobj +1241 0 obj << -/Kids [1101 0 R 1102 0 R 1103 0 R 1104 0 R 1105 0 R 1106 0 R] +/Kids [1162 0 R 1163 0 R 1164 0 R 1165 0 R 1166 0 R 1167 0 R] /Limits [(Doc-Start) (Item.32)] >> endobj -1177 0 obj +1242 0 obj << -/Kids [1107 0 R 1108 0 R 1109 0 R 1110 0 R 1111 0 R 1112 0 R] +/Kids [1168 0 R 1169 0 R 1170 0 R 1171 0 R 1172 0 R 1173 0 R] /Limits [(Item.33) (Item.65)] >> endobj -1178 0 obj +1243 0 obj << -/Kids [1113 0 R 1114 0 R 1115 0 R 1116 0 R 1117 0 R 1118 0 R] -/Limits [(Item.7) (page.29)] +/Kids [1174 0 R 1175 0 R 1176 0 R 1177 0 R 1178 0 R 1179 0 R] +/Limits [(Item.66) (page.27)] >> endobj -1179 0 obj +1244 0 obj << -/Kids [1119 0 R 1120 0 R 1121 0 R 1122 0 R 1123 0 R 1124 0 R] -/Limits [(page.3) (section.2.8)] +/Kids [1180 0 R 1181 0 R 1182 0 R 1183 0 R 1184 0 R 1185 0 R] +/Limits [(page.28) (section.2.5)] >> endobj -1180 0 obj +1245 0 obj << -/Kids [1125 0 R 1126 0 R 1127 0 R 1128 0 R 1129 0 R 1130 0 R] -/Limits [(section.3.1) (subsection.2.5.1)] +/Kids [1186 0 R 1187 0 R 1188 0 R 1189 0 R 1190 0 R 1191 0 R] +/Limits [(section.2.6) (subsection.2.3.1)] >> endobj -1181 0 obj +1246 0 obj << -/Kids [1131 0 R 1132 0 R 1133 0 R 1134 0 R 1135 0 R 1136 0 R] -/Limits [(subsection.2.5.2) (theorem.1.16)] +/Kids [1192 0 R 1193 0 R 1194 0 R 1195 0 R 1196 0 R 1197 0 R] +/Limits [(subsection.2.3.2) (table.4.1)] >> endobj -1182 0 obj +1247 0 obj << -/Kids [1137 0 R 1138 0 R 1139 0 R 1140 0 R 1141 0 R 1142 0 R] -/Limits [(theorem.1.17) (theorem.1.6)] +/Kids [1198 0 R 1199 0 R 1200 0 R 1201 0 R 1202 0 R 1203 0 R] +/Limits [(theorem.1.1) (theorem.1.41)] >> endobj -1183 0 obj +1248 0 obj << -/Kids [1143 0 R 1144 0 R 1145 0 R 1146 0 R 1147 0 R 1148 0 R] -/Limits [(theorem.1.7) (theorem.2.30)] +/Kids [1204 0 R 1205 0 R 1206 0 R 1207 0 R 1208 0 R 1209 0 R] +/Limits [(theorem.1.42) (theorem.2.23)] >> endobj -1184 0 obj +1249 0 obj << -/Kids [1149 0 R 1150 0 R 1151 0 R 1152 0 R 1153 0 R 1154 0 R] -/Limits [(theorem.2.31) (theorem.2.63)] +/Kids [1210 0 R 1211 0 R 1212 0 R 1213 0 R 1214 0 R 1215 0 R] +/Limits [(theorem.2.24) (theorem.2.56)] >> endobj -1185 0 obj +1250 0 obj << -/Kids [1155 0 R 1156 0 R 1157 0 R 1158 0 R 1159 0 R 1160 0 R] -/Limits [(theorem.2.64) (theorem.2.96)] +/Kids [1216 0 R 1217 0 R 1218 0 R 1219 0 R 1220 0 R 1221 0 R] +/Limits [(theorem.2.57) (theorem.2.89)] >> endobj -1186 0 obj +1251 0 obj << -/Kids [1161 0 R 1162 0 R 1163 0 R 1164 0 R 1165 0 R 1166 0 R] -/Limits 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statistica/preamble.tex +++ b/Secondo anno/Elementi di probabilità e statistica/preamble.tex @@ -20,6 +20,7 @@ \usepackage{nicefrac} \usepackage{longtable} \usepackage{pdflscape} +\usepackage{mathtools} \newtheorem*{warn}{\warning \; Attenzione} diff --git a/Secondo anno/Elementi di probabilità e statistica/sections/0-notazioni.tex b/Secondo anno/Elementi di probabilità e statistica/sections/0-notazioni.tex index 3cf156b..0f2005c 100644 --- a/Secondo anno/Elementi di probabilità e statistica/sections/0-notazioni.tex +++ b/Secondo anno/Elementi di probabilità e statistica/sections/0-notazioni.tex @@ -131,6 +131,7 @@ 0 & \text{altrimenti}. \end{cases} \] + \item $1_{\texttt{exp}}$ -- $1$ se $\texttt{exp}$ è vera, $0$ altrimenti. \item $\groupto$ -- simbolo utilizzato al posto $\to$ quando si elencano più funzioni che condividono o lo stesso dominio o lo stesso codominio (e.g.~$f$, $g : A$, $B \groupto C$ elenca una funzione $f : A \to C$ e una $g : B \to C$; $f$, $g : A \groupto B$, $C$ elenca una funzione $f : A \to B$ e una $g : A \to C$). diff --git a/Secondo anno/Elementi di probabilità e statistica/sections/4-statistica.tex b/Secondo anno/Elementi di probabilità e statistica/sections/4-statistica.tex index 28d106b..ceb0766 100644 --- a/Secondo anno/Elementi di probabilità e statistica/sections/4-statistica.tex +++ b/Secondo anno/Elementi di probabilità e statistica/sections/4-statistica.tex @@ -87,7 +87,7 @@ Analogamente definiamo per dei dati $y_1$, ..., $y_n \in \RR$ la v.a.~$Y$. che, per la disuguaglianza di Cauchy-Schwarz, appartiene all'intervallo $[-1, 1]$. \end{definition} -\subsection{Modello statistico} +\section{Modello statistico} Come già osservato, la statistica inferenziale parametrica studia situazioni in cui è necessario ricavare o stimare un singolo parametro su un dato modello di probabilità al fine @@ -113,4 +113,160 @@ di ricavare la distribuzione di probabilità dei dati $x_1$, ..., $x_n$. $Q_\theta \sim B(\theta)$, con $\Theta = [0, 1]$, dove $1$ identifica la testa e $0$ la croce. \end{example} +\section{Teoria degli stimatori su campioni di taglia \texorpdfstring{$n$}{n}} + +\subsection{Campione, statistica e stimatore} + +D'ora in avanti, sottintenderemo di star lavorando sul modello +statistico $(S, \cS, (Q_\theta)_{\theta \in \Theta})$. + +\begin{definition}[Campione i.i.d.~di taglia $n$] + Dato un modello statistico, si dice + che una famiglia di v.a.~$(X_i : \Omega \to S)_{i \in [n]}$ i.i.d.~è un \textbf{campione i.i.d.~di taglia $n$} + se per ogni $\sigma \in \Sigma$ esiste uno spazio di probabilità $(\Omega, \FF, P_\sigma)$ tale per cui + $(P_\sigma)^{X_i}$ è uguale in legge a $Q_\theta$. +\end{definition} + +Dato un campione di taglia $n$, useremo $P_\sigma$ per riferirci alla misura di probabilità +su $(\Omega, \FF)$ appena descritta. Scriveremo +come apice $\sigma$ per indicare di star lavorando nello spazio +di probabilità $(\Omega, \FF, P_\theta)$ (e.g.~$\EE^\sigma$ è riferito +a $P_\theta$). + +\begin{definition}[Statistica e stimatore] + Dato un campione i.i.d.~$(X_i)_{i \in [n]}$, si dice \textbf{statistica} + una v.a.~dipendente dalle v.a.~$X_i$ ed eventualmente dal parametro $\sigma$. + Si dice \textbf{stimatore} una statistica non dipendente direttamente da $\sigma$. +\end{definition} + +\subsection{Correttezza di uno stimatore} + +\begin{definition}[Stimatore corretto] + Si dice che uno stimatore $U$ è \textbf{corretto} (o \textit{non distorto}) rispetto + a $h : \Sigma \to \RR$ se per ogni $\sigma \in \Sigma$ vale che: + \begin{enumerate}[(i.)] + \item $U$ è $P_\sigma$-integrabile (i.e.~ammette valore atteso), + \item $\EE^\sigma[U] = h(\sigma)$. + \end{enumerate} +\end{definition} + +\begin{remark} + La media campionaria è uno stimatore corretto del valore atteso ($h : \sigma \mapsto \EE^\sigma[X_1]$). Infatti: + \[ + \EE^\sigma\!\left[\overline{X}\right] = \EE^\sigma[X_1]. + \] +\end{remark} + +\begin{remark} + La varianza campionaria è uno stimatore corretto della varianza ($h : \sigma \mapsto \Var^\sigma(X_1)$). Infatti: + \[ + \EE^\sigma[S^2] = \frac{1}{n-1} \left( n \EE^\sigma[X_1^2] - \EE^\sigma[X_1^2] - (n-1) \EE^\sigma[X_1]^2 \right) = \Var^\sigma(X_1). + \] + Si verifica analogamente che il coeff.~di correlazione campionario è uno stimatore corretto del + coeff.~di correlazione tra $X_i$ e $X_j$. +\end{remark} + +\subsection{Consistenza e non distorsione di una successione di stimatori} + +\begin{definition}[Successione non distorta di stimatori] + Una successione di stimatori $(U_k)_{k \in \NN^+}$ di $h(\sigma)$ si dice + \textbf{asintoticamente non distorta} se $U_k$ è $P_\sigma$-integrabile + (i.e.~ammette valore atteso) e: + \[ + \lim_{k \to \infty} \EE^\sigma[U_k] = h(\sigma). + \] +\end{definition} + +\begin{definition}[Successione consistente di stimatori] + Una successione di stimatori $(U_k)_{k \in \NN^+}$ di $h(\sigma)$ si dice + \textbf{consistente} se: + \[ + \lim_{k \to \infty} P_\sigma(\abs{U_k - h(\sigma)} > \eps) = 0, \quad \forall \eps > 0, + \] + ovverosia se $U_k$ converge in $P_\sigma$-probabilità a $h(\sigma)$. +\end{definition} + +\begin{remark} + La successione di stimatori $(\overline{X_n})_{n \in \NN^+}$, corretti per + il valore atteso, è sia consistente che + asintoticamente non distorta, per la LGN. +\end{remark} + +\begin{remark} + La successione di stimatori $(S^2_n)_{n \in \NN^+}$, corretti per la + varianza, consistente, sempre per la LGN. +\end{remark} + +\subsection{Stimatore di massima verosomiglianza} + +D'ora in avanti sottintenderemo di star lavorando sullo +spazio misurabile $(\RR, \BB(\RR))$. + +\begin{notation} + Data la famiglia di probabilità $(Q_\sigma)_{\sigma \in \Sigma})$, usiamo + scrivere $m_\sigma$ per riferirci alla densità discreta $q_\sigma$ (o $p_\sigma$) + di $Q_\sigma$, qualora sia discreta, oppure alla sua funzione di densità + $f_\sigma$, qualora $Q_\sigma$ sia assolutamente continua. +\end{notation} + +\begin{definition}[Funzione di verosomiglianza] + Dato un campione $(X_i)_{i \in [n]}$ i.i.d.~, si definisce + \textbf{funzione di verosomiglianza} la funzione $L : \Sigma \times \RR^n$ + tale per cui: + \[ + (\sigma, (x_i)_{i \in [n]}) \xmapsto{L} L_\sigma(x_1, \ldots, x_n) \defeq m_\sigma(x_1) \cdots m_\sigma(x_n). + \] + Equivalentemente, $L_\sigma(x_1, \ldots, x_n)$ rappresenta la densità congiunta su $Q_\sigma$ + di $x_1$, ..., $x_n$. +\end{definition} + +\begin{notation} + Scriveremo $L_U(X_1, \ldots, X_n)$ con $U$ v.a. e + $(X_i)_{i \in [n]}$ famiglia di v.a.~reali sottintendendo + l'insieme $L_{U(\omega)}(X_1(\omega), \ldots, X_n(\omega))$, + assumendo $U(\omega) \in \Sigma$. +\end{notation} + +\begin{definition}[Stimatore di massima verosomiglianza di $\sigma$] + Si dice che uno stimatore $U$ è di \textbf{massima verosomiglianza di $\sigma$} + su un campione i.i.d.~$(X_i)_{i \in [n]}$ se: + \[ + L_U(X_1, \ldots, X_n) = \sup_{\theta \in \Theta} L_\theta(X_1, \ldots, X_n), \quad \forall \omega \in S. + \] + In altre parole, uno stimatore $U$ è di massima verosomiglianza su un campione se + per dei dati $x_1$, ..., $x_n$ restituisce il parametro $\theta$ che massimizza + $L_\theta(x_1, \ldots, x_n)$, ovverosia la densità consiunta dei dati + $x_1$, ..., $x_n$ (i.e.~la probabilità che si ottenga $x_1$, ..., $x_n$). +\end{definition} + +\begin{example}[Prova di Bernoulli] + Sia $Q_\theta \sim B(\theta)$. Dati gli esiti $x_1$, ..., $x_n$ di $n$ prove, + ricaviamo che: + \[ + L_\theta(x_1, \ldots, x_n) = \theta^{\sum_i x_i} (1 - \theta)^{n - \theta^{\sum_i x_i}}, + \] + da cui: + \[ + \log L_\theta(x_1, \ldots, x_n) = n \overline{x} \log(\theta) + n (1 - \overline{x}) \log(1 - \theta). + \] + Tale funzione ha massimo per $\theta = \overline{x}$, e dunque + $\overline{X}$ è uno stimatore di massima verosomiglianza di $\theta$. \smallskip + + In altre parole, la migliore stima di $\sigma$ data una sequenza di $n$ prove di Bernoulli è + la frequenza relativa di successi. +\end{example} + +\begin{example} + Sia $Q_\theta \sim U([0, \theta])$ con $\theta > 0$. Dati gli esiti $x_1$, ..., $x_n$ ricaviamo che: + \[ + L_\theta(x_1, \ldots, x_n) = \frac{1}{\theta^n} \prod_i 1_{[0, \theta]}(x_i) = + \frac{1}{\theta^n} 1_{0 \leq \min_i x_i \leq \max_i x_i \leq \theta}, + \] + che ha massimo per $\theta = \max_i x_i$. Pertanto $\max\{X_1, \ldots, X_n\}$ è uno stimatore + di massima somiglianza di $\theta$. \smallskip + + In altre parole, dati degli esiti $x_1$, ..., $x_n$, una delle migliori stime che possiamo fare + su $\theta$ è $\max_i x_i$. +\end{example} + \end{multicols*}