统计代写|贝叶斯分析代写Bayesian Analysis代考|STATS3023

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

统计代写|贝叶斯分析代写Bayesian Analysis代考|Predicting Economic Growth

Table $2.1$ contains the (annualized) growth rate figures for the UK for each quarter from the start of 1993 to the end of 2007 (which was just prior to the start of the international economic collapse). So, for example, in the fourth quarter of 2007 the annual growth rate in the UK was $2.36 \%$.

Data such as this, especially given the length of time over which it has been collected, is considered extremely valuable for financial analysis and projections. Since so many aspects of the economy depend on the growth rate, we need our predictions of it for the coming months and years to be very accurate. So imagine that you were a financial analyst presented with this data in 2008. Although it would be nice to be able to predict the growth rate in each of the next few years, the data alone gives you little indication of how to do that. If you plot the growth over time as in Figure $2.1$ there is no obvious trend to spot.

But there is a lot that you can do other than making “point” predictions. What financial institutions would really like to know is the answer to questions like those in Sidebar 2.1.

Indeed, economic analysts feel that the kind of data provided enables them to answer such questions very confidently. The way they typically proceed is to “fit” the data to a standard curve (also called a statistical distribution). The answers to all the aforementioned questions can then be answered using standard statistical tables associated with that distribution.

In most cases the analysts assume that data of the kind seen here can be fitted by what is called a Normal distribution (also called a bell curve because that is its shape as shown in Figure 2.2).

The key thing about a Normal distribution is that it is an “idealized” view of a set of data. Imagine that, instead of trying to model annual growth rate, you were trying to model the height in centimeters of adults. Then, if you took a sample of, say, 1,000 adults and plotted the frequency of their heights within each 10 -centimeter interval you would get a graph that looks something like Figure 2.3. As you increase the sample size and decrease the interval size you would eventually expect to get something that looks like the Normal distribution in Figure 2.4.

The Normal distribution has some very nice mathematical properties (see Box 2.1), which makes it very easy for statisticians to draw inferences about the population that it is supposed to be modelling.

Unfortunately, it turns out that, for all its nice properties the Normal distribution is often a very poor model to use for most types of risk assessment. And we will demonstrate this by returning to our GDP growth rate data. In the period from 1993 to 2008 the average growth rate was $2.96 \%$ with a standard deviation of $0.75$.

统计代写|贝叶斯分析代写Bayesian Analysis代考|Patterns and Randomness

Take a look at Table 2.3. It shows the scores achieved (on an objective quality criteria) by the set of state schools in one council district in the UK. We have made the schools anonymous by using numbers rather than names. School 38 achieved a significantly higher score than the next best school, and its score (175) is over $52 \%$ higher than the lowest ranked school, number 41 (score 115). Tables like these are very important in the UK, since they are supposed to help provide informed “choice” for parents. Based on the impressive results of School 38 parents clamour to ensure that their child gets a place at this school. Not surprisingly, it is massively oversubscribed. Since these are the only available state schools in this district, imagine how you would feel if, instead of your child being awarded a place in School 38, he or she was being sent to school 41. You would be pretty upset, wouldn’t you? You should not be. We lied. The numbers do not represent schools at all. They are simply the numbers used in the UK National Lottery (1 to 49). And each “score” is the actual number of times that particular numbered ball had been drawn in the first 1,172 draws of the UK National Lottery. So the real question is: Do you believe that 38 is a “better” number than 41 ? Or, making the analogy with the school league table more accurate:
Do you believe the number 38 is more likely to be drawn next time than the number 41? (Since the usual interpretation of the school league table is that if your child attends the school at the top he or she will get better grades than if he or she attends the school at the bottom.)
The fact is that the scores are genuinely random. Although the “expected” number of times any one ball should have been drawn is about 144 you can see that there is a wide variation above and below this number (even though that is still the average score).

What many people fail to realise is that this kind of variation is inevitable. It turns out that in any sequence of 1,172 lottery draws there is about a $50 \%$ chance that at least half the numbers will be chosen either less than 136 times or more than 152 times. That indeed is roughly what happened in the real sample. Moreover, the probability that at least one number will be chosen more than 171 times is about 45\%. You may find it easier to think of rolling a die 60 times. You would almost certainly not get each of the six numbers coming up 10 times. You might get 16 threes and only 6 fours. That does that not make the number three “better” than the number four. The more times you roll the die, the closer in relative terms will be the frequencies of each number (specifically, if you roll the die $n$ times the frequency of each number will get closer to $n$ divided by 6 as $n$ gets bigger); but in absolute terms the frequencies will not be exactly the same. There will inevitably be some numbers with a higher count than others. And one number will be at the “top” of the table while another will be “bottom.”
We are not suggesting that all school league tables are purely random like this. But, imagine that you had a set of genuinely equal schools and you ranked them according to a suitable criteria like average exam scores. Then, in any given year, you would inevitably see variation like the earlier table. And you would be wrong to assume that the school at the top was better than the school at the bottom. In reality, there may be inherent quality factors that help determine where a school will come in a league table. But this does not disguise the fact that much of the variation in the results will be down to nothing more than pure and inevitable chance. See Box $2.2$ for another example.

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