Question: Suppose that random variable X 0, representing the annual rainfall (in metres) at your weather station, has a Gamma distribution, i.e., X Gamma(, ), where
Suppose that random variable X 0, representing the annual rainfall (in metres) at your weather station, has a Gamma distribution, i.e., X Gamma(, ), where and are
positive parameters. Below, we re-parameterize by using the parameter = 1/.
(a) Use Python to describe your rainfall data series, by computing summary statistics, and plotting the kernel density function. Explain and discuss these results, including the nature and implications of the empirical distribution for rainfall at your weather station.
(b) Estimate the parameters of the Gamma distribution by the method of maximum likelihood, presenting your results (parameter estimates, standard errors, etc.) in tabular form.
(c) Plot the estimated (using the maximum likelihood estimates) Gamma density for
rainfall. Using the estimated Gamma density for rainfall, compute estimates of the probabilities of rainfall being (i) less than 70% of average annual rainfall (very dry), (ii) more than 130% of average annual rainfall (very wet) and (iii) either (i) or (ii). Discuss these results and their implications.
(d) Use the likelihood ratio test procedure to test the null hypothesis that = 1 (meaning
that the population mean and variance are equal) against the alternative hypothesis that = 1 using a 5% significance level. Your answer should provide full details of the logic used and your conclusion. [Hint: To estimate the model with = 1, replace '{theta}' by '1' in the estimation command.]
Include your Python program(s) at the end of your answers to this question.
The data:
| Year | Rainfall16 |
| 1966 | 0.86043618 |
| 1967 | 0.78498955 |
| 1968 | 2.7866608 |
| 1969 | 2.7167775 |
| 1970 | 2.3972599 |
| 1971 | 0.38684201 |
| 1972 | 2.6825074 |
| 1973 | 5.0955186 |
| 1974 | 4.4039672 |
| 1975 | 2.1769829 |
| 1976 | 1.899169 |
| 1977 | 1.1354679 |
| 1978 | 1.54926 |
| 1979 | 1.2363492 |
| 1980 | 2.4929891 |
| 1981 | 6.2224455 |
| 1982 | 1.2718447 |
| 1983 | 3.7862099 |
| 1984 | 1.3253747 |
| 1985 | 0.93920953 |
| 1986 | 1.3986398 |
| 1987 | 1.1841027 |
| 1988 | 0.98393821 |
| 1989 | 1.8156087 |
| 1990 | 3.1067913 |
| 1991 | 1.0512532 |
| 1992 | 0.94449276 |
| 1993 | 0.4766173 |
| 1994 | 1.7962279 |
| 1995 | 1.0900812 |
| 1996 | 0.86254085 |
| 1997 | 3.4354329 |
| 1998 | 0.52220215 |
| 1999 | 3.4198457 |
| 2000 | 1.3095363 |
| 2001 | 1.7745219 |
| 2002 | 1.9668938 |
| 2003 | 5.6520617 |
| 2004 | 1.6694899 |
| 2005 | 3.1429521 |
| 2006 | 0.66600744 |
| 2007 | 1.0695483 |
| 2008 | 3.1826559 |
| 2009 | 0.52570474 |
| 2010 | 5.8281976 |
| 2011 | 0.83969946 |
| 2012 | 0.71746577 |
| 2013 | 0.91720641 |
| 2014 | 2.9881486 |
| 2015 | 3.5251173 |
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