Sample standard deviation serially correlated normal data

In summary, the conversation discusses the statistical properties of the sample standard deviation of a sequence of identically distributed normal random variables with serial correlation. The formula for calculating the sample standard deviation is mentioned and the concept of covariance is also brought up. The conversation ends with a request to fix an error in the LaTeX code.
  • #1
rhz
12
0
Hi,

Can anyone point me to a reference for the statistical properties of the sample standard deviation of a sequence of identically distributed normal random variables subject to some form of serial correlation?

Thanks,

rhz
 
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  • #2
( ∑ Xk)2 = ∑∑XkXj
Take expectation and subtract out the mean squared and you will have:

2 + ∑∑(k≠j) cov(k,j)

cov(k,j) is the covariance of XkXj.
 
  • #3
mathman said:
( ∑ Xk)2 = ∑∑XkXj
Take expectation and subtract out the mean squared and you will have:

2 + ∑∑(k≠j) cov(k,j)

cov(k,j) is the covariance of XkXj.

Hi,

OK, but I'm interested in the statistical properties of the sample standard deviation:

\sqrt{\hat\sigma^2} = \sqrt \left ( \frac{1}{N-1}\sum^{N-1}_{i=0}(x_i-\hat{\mu})^2 \right )
\hat\mu = \frac{1}{N}\sum^{N-1}_{i=0}x_i

Thanks.
 
  • #4
Fix your latex!
 
  • #5


I would be happy to provide some information on this topic. The sample standard deviation is a commonly used measure of variability in a dataset, and it is important to understand its statistical properties in different situations. In the case of serially correlated normal data, the standard deviation may be affected by the correlation between the data points.

One useful reference for this topic is the book "Time Series Analysis" by James D. Hamilton. In Chapter 5, the author discusses the properties of the sample standard deviation in the presence of serial correlation. He notes that when the data is normally distributed and has a constant mean and variance, the sample standard deviation will be biased and underestimate the true standard deviation if there is positive serial correlation. On the other hand, if there is negative serial correlation, the sample standard deviation will be biased and overestimate the true standard deviation.

Another important aspect to consider is the sample size. As the sample size increases, the bias in the sample standard deviation decreases. This means that for larger sample sizes, the sample standard deviation will be a better estimate of the true standard deviation, even in the presence of serial correlation.

In addition to Hamilton's book, there are many other resources available that discuss the statistical properties of the sample standard deviation in different scenarios. It may also be helpful to consult with a statistician or conduct some simulations to better understand the specific effects of serial correlation on the sample standard deviation in your dataset.

I hope this information is helpful in addressing your question. Best of luck with your research!
 

1. What is a sample standard deviation?

A sample standard deviation is a measure of how spread out a set of data points are from the mean or average. It is calculated by taking the square root of the sum of squared differences between each data point and the mean, divided by the number of data points minus one.

2. What does it mean for data to be serially correlated?

Serial correlation, also known as autocorrelation, refers to the relationship between consecutive data points in a time series. If data points are positively serially correlated, it means that a higher value in one data point is likely to be followed by a higher value in the next data point. If data points are negatively serially correlated, it means that a higher value in one data point is likely to be followed by a lower value in the next data point.

3. How does serial correlation impact the sample standard deviation?

Serial correlation can impact the sample standard deviation by artificially inflating or deflating its value. In a positively serially correlated dataset, the standard deviation may be higher than it should be, as consecutive data points are more similar to each other. In a negatively serially correlated dataset, the standard deviation may be lower than it should be, as consecutive data points are more dissimilar from each other.

4. Is it possible to have a normal distribution with serially correlated data?

Yes, it is possible to have a normal distribution with serially correlated data. The presence of serial correlation does not necessarily mean that the data is not normally distributed. However, it is important to take into account the serial correlation when analyzing and interpreting the data, as it can affect the validity of statistical tests and conclusions.

5. How can serial correlation in normal data be addressed?

There are several methods for addressing serial correlation in normal data. One approach is to use statistical tests such as the Durbin-Watson test to detect the presence of serial correlation and adjust the standard errors in statistical models accordingly. Another approach is to use time series analysis techniques, such as autoregressive models, to account for the serial correlation in the data. The best approach will depend on the specific dataset and research question at hand.

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