R Program MLE and log MLE Parameter Advanced Statistics Worksheet 4 problems on excel spreadsheet involved the topics mentioned above. TIME LIMITED (3 hour

R Program MLE and log MLE Parameter Advanced Statistics Worksheet 4 problems on excel spreadsheet involved the topics mentioned above. TIME LIMITED (3 hours) Data on # of hurricanes hitting 4 islands (the islands are A, B, C, D)
Year
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
Island
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
Total
3
0
2
0
0
0
1
0
1
0
3
3
0
0
2
0
2
1
0
2
0
0
0
0
3
0
0
0
2
0
0
3
0
0
0
0
1
1
1
0
0
0
0
0
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
0
0
1
1
0
2
0
0
1
0
0
1
1
1
0
1
0
1
1
0
0
2
0
6
0
2
1
1
0
2
0
1
2
4
0
0
0
0
1
1
1
0
0
0
2
0
1
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
0
0
1
1
0
0
0
1
0
0
1
0
1
0
0
1
4
0
2
4
3
0
0
1
1
0
3
5
4
1
2
4
1
0
1
2
3
1
1
3
1
6
1
3
3
4
2
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
2
3
1
1
1
3
3
2
4
0
2
2
4
1
1
2
2
1
3
7
2
2
5
1
2
3
3
1
0
1
9
1
0
1
3
1
0
0
4
1
1
2
2
3
0
5
2
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
1800
1801
1802
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
C
C
C
2
3
4
1
3
2
5
0
3
3
1
3
0
1
1
2
1
1
4
0
4
3
1
0
1
0
2
3
0
10
0
0
3
1
1
3
0
3
2
2
3
2
4
2
4
3
1
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
0
0
0
2
5
3
2
10
5
6
3
3
4
1
2
0
2
7
1
1
1
5
7
1
1
1
1
2
4
5
3
1
1
3
0
3
1
5
0
2
2
2
3
2
4
1
3
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
5
1
0
5
0
1
8
1
3
1
3
1
2
3
1
2
6
2
0
0
1
2
3
3
3
1
2
2
1
2
1
5
1
1
1
3
1
0
5
1
4
1
2
2
7
2
0
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
4
4
3
1
4
1
2
2
3
0
7
4
0
2
2
6
4
2
3
1
2
1
3
2
3
3
5
5
6
1
3
1
0
0
2
3
1
1
3
6
2
7
5
7
0
3
1
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1800
1801
1802
1803
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
D
D
D
D
1
4
1
2
2
1
2
2
4
2
7
2
1
1
1
1
3
5
3
7
2
3
2
2
2
6
3
0
2
4
1
4
3
1
6
4
0
3
5
4
2
3
5
4
2
1
3
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
4
5
5
1
1
4
4
7
7
6
5
1
3
4
4
4
3
3
2
3
4
2
2
5
1
1
6
3
0
9
1
2
3
3
5
1
3
2
3
3
2
1
3
4
6
4
4
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
0
5
1
2
3
4
6
5
9
5
3
0
1
3
3
5
2
2
5
7
3
3
2
3
1
3
2
2
5
4
1
4
5
5
2
7
2
1
4
2
0
3
1
5
7
5
9
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
2
2
1
3
2
4
6
3
6
5
7
2
2
2
1
4
3
5
4
1
6
3
3
4
5
5
6
2
4
3
4
0
2
7
1
4
5
2
9
1
4
4
5
0
1
0
4
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
D
5
3
6
4
3
4
3
4
3
0
6
3
4
2
5
5
4
5
2
0
6
1
4
5
3
6
9
3
1
1
7
1
6
1
2
2
3
4
5
We desire a Poisson model of annual hurricane count, assuming that the four islands are identical
Determine the maximum likelihood Poisson parameter (per year) and the corresponding log likelihood of the sample
λ
log like
Obs
3
0
2
0
0
0
1
0
1
0
3
3
0
0
2
0
2
1
0
2
0
0
0
0
3
0
0
0
2
0
0
3
0
0
0
0
1
1
1
Like
Log(Like)
0
0
0
0
0
0
0
1
1
0
2
0
0
1
0
0
1
1
1
0
1
0
1
1
0
0
2
0
6
0
2
1
1
0
2
0
1
2
4
0
0
0
0
1
1
1
0
0
0
2
0
1
0
0
1
1
0
0
0
1
0
0
1
0
1
0
0
1
4
0
2
4
3
0
0
1
1
0
3
5
4
1
2
4
1
0
1
2
3
1
1
3
1
6
1
3
3
4
2
2
3
1
1
1
3
3
2
4
0
2
2
4
1
1
2
2
1
3
7
2
2
5
1
2
3
3
1
0
1
9
1
0
1
3
1
0
0
4
1
1
2
2
3
0
5
2
2
3
4
1
3
2
5
0
3
3
1
3
0
1
1
2
1
1
4
0
4
3
1
0
1
0
2
3
0
10
0
0
3
1
1
3
0
3
2
2
3
2
4
2
4
3
1
0
0
0
2
5
3
2
10
5
6
3
3
4
1
2
0
2
7
1
1
1
5
7
1
1
1
1
2
4
5
3
1
1
3
0
3
1
5
0
2
2
2
3
2
4
1
3
5
1
0
5
0
1
8
1
3
1
3
1
2
3
1
2
6
2
0
0
1
2
3
3
3
1
2
2
1
2
1
5
1
1
1
3
1
0
5
1
4
1
2
2
7
2
0
4
4
3
1
4
1
2
2
3
0
7
4
0
2
2
6
4
2
3
1
2
1
3
2
3
3
5
5
6
1
3
1
0
0
2
3
1
1
3
6
2
7
5
7
0
3
1
1
4
1
2
2
1
2
2
4
2
7
2
1
1
1
1
3
5
3
7
2
3
2
2
2
6
3
0
2
4
1
4
3
1
6
4
0
3
5
4
2
3
5
4
2
1
3
4
5
5
1
1
4
4
7
7
6
5
1
3
4
4
4
3
3
2
3
4
2
2
5
1
1
6
3
0
9
1
2
3
3
5
1
3
2
3
3
2
1
3
4
6
4
4
0
5
1
2
3
4
6
5
9
5
3
0
1
3
3
5
2
2
5
7
3
3
2
3
1
3
2
2
5
4
1
4
5
5
2
7
2
1
4
2
0
3
1
5
7
5
9
2
2
1
3
2
4
6
3
6
5
7
2
2
2
1
4
3
5
4
1
6
3
3
4
5
5
6
2
4
3
4
0
2
7
1
4
5
2
9
1
4
4
5
0
1
0
4
5
3
6
4
3
4
3
4
3
0
6
3
4
2
5
5
4
5
2
0
6
1
4
5
3
6
9
3
1
1
7
1
6
1
2
2
3
4
5
elihood of the sample
We desire an improved Poisson model of annual hurricane count
We assume that hurricane risk is partly explained by geographic location, measured in degrees latitude
Island
Degrees Latitude
A
38
B
21
C
16
D
10
Our link will be Xiβ = ln(μi) where μi is the Poisson parameter for island i
The matrix X will have 2 columns: a column of 1’s and a column of degrees latitude
β will be a 2×1 matrix
Using the test β shown, determine the likelihood of the sample
test β
constant
1.1
test β
multiplier
-0.01
log like
Use these suggested column headings if you like
Xiβ
deg lat
hurri
μ
Like
Log(Like)
Statistical Testing
Likelihood Ratio Test
null hypothesis
test statistic
dof
critical value (5% one-tail)
decision (accept or reject)
the problem 1 model is just as explanatory as the problem 2 model
oblem 2 model
Logistic Regression
We define islands A and B to be “northern”
We define islands C and D to be “southern”
explanatory
response
hurricane damage in $
southern = 1 and northern = 0
Training Data
hurricane
damage
in $
northern
or
southern
15,052
362,558
362,671
496,074
569,594
578,437
638,864
665,207
689,868
723,559
813,208
860,335
868,164
898,243
932,456
1,264,564
1,323,522
1,396,776
1,478,409
1,539,630
1,622,920
1,655,014
1,657,944
1,705,953
1,714,141
1,762,577
1,805,366
1,813,509
1,897,340
1,946,262
1,960,008
1,971,299
2,092,213
2,095,609
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
Test Data
hurricane
damage
in $
northern
or
southern
25,000,000
50,000,000
75,000,000
100,000,000
125,000,000
150,000,000
0
0
0
1
1
1
P(southern)
using your
model
2,099,061
2,112,060
2,182,729
2,587,689
2,600,123
2,734,087
2,741,496
2,754,279
2,796,121
2,858,261
3,238,561
3,252,375
3,419,392
3,464,465
3,465,665
3,476,214
3,534,223
3,645,442
3,964,369
4,111,107
4,282,375
4,313,105
4,503,740
4,691,666
4,705,152
4,724,249
4,752,276
4,768,972
4,836,066
4,924,304
4,947,385
4,962,667
4,969,816
5,174,014
5,231,800
5,318,437
5,896,674
5,900,307
5,950,151
5,962,062
5,974,527
6,066,146
6,218,173
6,480,959
6,494,663
6,860,414
7,050,602
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
1
0
0
1
1
0
0
0
0
0
1
1
1
1
0
0
0
0
0
1
0
0
0
0
0
7,278,174
7,303,727
7,332,218
7,337,351
7,422,160
7,684,297
7,758,991
7,911,831
7,945,775
8,104,444
8,141,735
8,404,870
8,435,729
8,551,713
8,723,384
8,954,984
8,956,673
8,999,062
9,094,738
9,126,874
9,223,854
9,434,851
9,439,464
9,687,435
9,815,748
9,898,984
9,949,820
9,950,233
10,143,354
10,187,334
10,201,109
10,233,633
10,523,630
10,653,008
10,842,424
11,020,011
11,297,213
12,042,209
12,421,977
12,680,157
12,771,427
13,581,950
13,640,349
14,360,026
14,568,168
14,572,814
14,741,554
0
0
0
0
0
1
0
1
0
0
1
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0
1
0
0
0
0
0
15,140,022
15,814,291
16,511,896
16,747,458
17,014,138
17,079,187
17,391,761
17,681,676
18,099,378
18,289,583
18,573,453
18,614,750
18,708,803
18,829,931
18,942,966
19,347,698
19,994,246
20,007,323
20,476,446
20,839,536
20,862,356
20,928,729
21,025,808
21,204,767
21,352,123
21,375,811
21,763,014
22,116,128
22,274,258
22,544,618
22,615,528
24,006,622
24,062,321
24,273,820
24,638,533
24,799,923
24,907,337
26,280,531
26,551,623
26,604,421
27,784,727
28,337,425
29,551,694
29,837,991
30,471,132
30,851,970
31,698,006
0
1
0
0
0
0
1
1
0
0
1
0
1
1
0
0
0
1
0
1
1
1
0
0
0
0
0
1
0
1
1
0
1
0
0
0
1
1
0
1
0
1
0
1
1
0
1
31,868,790
32,051,353
32,073,622
32,353,749
32,869,327
33,875,468
33,902,356
34,802,072
36,247,156
36,296,520
36,434,257
36,523,462
36,909,001
38,151,532
38,351,151
38,637,255
39,358,347
39,373,291
39,408,828
40,166,534
41,278,544
42,167,808
42,210,017
43,111,832
43,124,770
43,482,860
44,194,406
47,337,123
48,342,981
48,539,262
49,537,070
49,892,458
50,233,271
50,789,039
53,102,575
54,409,661
56,839,952
57,338,588
57,385,603
57,588,292
58,670,435
61,207,040
61,463,089
63,481,905
67,201,643
67,899,484
71,796,172
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
1
0
0
1
1
1
1
0
1
1
0
1
1
0
1
0
1
0
1
1
1
0
0
1
1
73,557,869
73,625,524
74,491,676
79,635,098
79,664,173
82,155,166
87,463,035
91,252,079
93,690,521
94,867,693
95,240,780
102,884,000
106,396,625
110,634,815
131,100,679
138,519,031
151,682,841
196,931,119
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1

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