Sample standard deviation serially correlated normal data

[rhz]

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

 

[mathman]

( ∑ X_k)² = ∑∑X_kX_j
Take expectation and subtract out the mean squared and you will have:

nσ² + ∑∑(k≠j) cov(k,j)

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

 

[rhz]

( ∑ X_k)² = ∑∑X_kX_j
Take expectation and subtract out the mean squared and you will have:

nσ² + ∑∑(k≠j) cov(k,j)

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

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.

 

[mathman]

Fix your latex!

 

[blue_raver22]

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!