lecture 2 note - Lecture 2 statistical inference and...

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Lecture 2: statistical inference and hypothesis testing Xin Tong * 1 Statistical Inference As we mentioned in previous lecture, this class focuses on how to construct a quantitative stochastic model for financial time series. Just like any scientific discipline, this involves 1)in- ference of model parameters, e.g. the average return 2) test of model/hypothesis: e.g. certain trading strategy will make money. Both can be achieved by exploiting the financial data, and are core questions of Statistics . This lecture will review some basic and fundamental statistical ideas, methods and concepts. It is worth noticing, this lecture will be solely in the frequencists’ point of view. Bayesian treatments exist but are not very standard. Basic concepts In statistics, a population is a collection of all objects or people to be studied. e.g. All students of NUS. Let say they belong to a set E . Because of practical concerns, we can only have a smaller finite group of observations, X 1 , X 2 , . . . , X n . They are called the samples . e.g. we can pick 50 students randomly on campus. This collection of samples is often called the data in practice. In most applications, we are interested in certain properties of the population, which can often be written as a parameter θ . The statistical inference question is finding a function ˆ θ ( X 1 , . . . , X n ) to estimate θ . ˆ θ will be called the estimator of θ . Generally speaking, functions of samples are called statistics . In most applications, the samples are drawn from the population randomly. But how to model this randomness is a very broad question. As a starter, we will take one popular assumption/perspective. That is to think the population as a probability distribution on the set E , and the samples are randomly drawn from this distribution, and they are independent to each other. This is usually written as “ X 1 , X 2 , . . . , X n are independent and identically distributed (i.i.d.) samples”. If each one has a density p ( x i | θ ), the joint joint density is the product p ( x 1 , . . . , x n | θ ) = p ( x 1 | θ ) p ( x 2 | θ ) · · · p ( x n | θ ) . For time series, we usually use T to denote the number of samples, which is the overall time length. * [email protected] 1
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X. Tong Statistical inference and testing In the following we will first construct/derive some estimators, then discuss how to judge them. But usually these discussion require some probabilistic assumptions to begin, so the logic can flow. 1.1 Principles of estimator construction Plug-in formulation In most applications, the interested quantity θ can be written as θ = E f ( X ), where X follows the distribution representing the population. e.g. the average age of NUS students. Then a natural way to construct an estimator is to use the sample average ˆ θ n = 1 n n X i =1 f ( X i ) . For example, the plug-in estimators for the mean, covariance, skewness and Kurtosis are ˆ μ X = 1 n n X i =1 X i , ˆ Σ X (= ˆ σ 2 X ) = n i =1 ( X i - ˆ μ X ) · ( X i - ˆ μ X ) T n - 1 , b S X = n i =1 ( X i - ˆ μ X ) 3 ( n - 1)ˆ σ 3 X , b K X = n i =1 ( X i - ˆ μ X ) 3 ( n - 1)ˆ σ 4 X .
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