统计代写|统计推断代写Statistical inference代考|Higher-order sample moments

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统计代写|统计推断代写Statistical inference代考|Higher-order sample moments

In Chapter 3 we defined the moments of a probability distribution as expectations; the $r^{\text {th }}$ moment of the random variable $X$ is the expectation of $X^r$, and so on. For a sample, the operation that is analogous to taking expectations is summing and dividing by the sample size. For example, the sample analogue of the mean, $\mu=\mathbb{E}(Y)$, is the sample mean, $\bar{Y}=\frac{1}{n} \sum_{i=1}^n Y_i$. We can use this idea to define sample moments and central sample moments.
Definition 7.2.1 (Sample moments and central sample moments) For a sample $Y_1, \ldots, Y_n$, the $r^{\text {th }}$ sample moment is
$$m_r^{\prime}=\frac{1}{n} \sum_{i=1}^n Y_i^r$$
and the $r^{\text {th }}$ central sample moment is
$$m_r=\frac{1}{n} \sum_{i=1}^n\left(Y_i-m_1^{\prime}\right)^r .$$
Note that the sample moments are random variables. The sample mean is the first sample moment, $\bar{Y}=m_1^{\prime}$, however, the sample variance is not quite the second central sample moment; if we define $S^2=\frac{1}{n-1} \sum_{i=1}^n\left(Y_i-\bar{Y}\right)^2$ then $S^2=\frac{n}{n-1} m_2$. Higherorder sample moments are used to measure properties such as skewness and kurtosis. The following claim is very useful in establishing the properties of central sample moments.
Claim 7.2.2 (Assuming zero mean)
When determining the properties of central sample moments, we may assume a population mean of zero.

统计代写|统计推断代写Statistical inference代考|Sample variance

For any sample of two or more observations, we can generate the sample variance.
Definition 7.2.3 (Sample variance)
For a sample $Y_1, \ldots, Y_n$ with $n>1$, the sample variance, $S^2$, is given by
$$S^2=\frac{1}{n-1} \sum_{i=1}^n\left(Y_i-\bar{Y}\right)^2 .$$
Two things to note about the sample variance:
i. As is usual practice, we write the sample variance as a function of the sample mean. However, it is a function of sample value alone, since
$$S^2=\frac{1}{n-1} \sum_{i=1}^n\left(Y_i-\frac{1}{n} \sum_{j=1}^n Y_j\right)^2,$$
which can be evaluated from a sample.
ii. The sample variance is generated by summing the squares of the differences between the individual sample members and the sample mean, then dividing by $n-1$. The reason for dividing by $n-1$ is made clear in Proposition 7.2.5.
The following lemma establishes useful facts about the sample variance that are simple consequences of the properties of summation.

统计推断代考

统计代写|统计推断代写Statistical inference代考|Higher-order sample moments

$\bar{Y}=\frac{1}{n} \sum_{i=1}^n Y_i$. 我们可以使用这个想法来定义样本矩 和中心样本矩。

$$m_r^{\prime}=\frac{1}{n} \sum_{i=1}^n Y_i^r$$

$$m_r=\frac{1}{n} \sum_{i=1}^n\left(Y_i-m_1^{\prime}\right)^r .$$

统计代写|统计推断代写Statistical inference代考|Sample variance

$$S^2=\frac{1}{n-1} \sum_{i=1}^n\left(Y_i-\bar{Y}\right)^2$$

i。按昭惯例，我们将样本方差写为样本均值的函数。然 而，它只是样本值的函数，因为
$$S^2=\frac{1}{n-1} \sum_{i=1}^n\left(Y_i-\frac{1}{n} \sum_{j=1}^n Y_j\right)^2$$

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MATLAB代写

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