Variance of statistic used in runs test

[DavideGenoa]

Hi, friends! Since this is my first post, I want to present myself as an Italian who is trying to teach himself mathematics and natural sciences, while having a strictly humanities-centered school background, and I am tempted very much to enrol in a university scientific course.
I read in the Italian language Wikipedia that the variance \text{Var}_{H_0}(R) of the statistic R used in the Wald-Wolfowitz test, under the null hypothesis that the X_1,...,X_n are independent, is(4N-6)p(1-p)-(12N-20)p^2(1-p)^2. It is worth to notice that, when discussing the test, it is common to give what I think to be approximations used in the case of the Gaussian approximation of R, rather than the real expectation and variance...
That statistic, as my book (S.M. Ross, Introduction to Probability and Statistics for Engineers and Scientists) explains, and as I find in the German language Wikipedia too, has the probability mass functionP_{H_0}(R=2k)=2\frac{\binom{N^+ -1}{k-1}\binom{N^- -1}{k-1}}{\binom{N^+ +N^-}{n}}P_{H_0}(R=2k+1)=\frac{\binom{N^+ -1}{k-1}\binom{N^- -1}{k}+\binom{N^+ -1}{k}\binom{N^- -1}{k-1}}{\binom{N^+ +N^-}{n}}
The Italian language Wikipedia considers the statistic R to be the same, under the null hypothesis of independence, as 1+\sum_{i=1}^{N-1}|X_i-X_{i+1}| where the X_i are Bernoulli random variables with expectation p, and the expectation E_{H_0}[R] given in that Wikipedia is the same talked about by a user here, who gives a short proof of the value of E_{H_0}[R]=1+2(N-1)p(1-p).

As to variance, I have tried a lot but my efforts to prove it by myself have been useless. I have tried to calculate the second moment by manipulating the sums \sum_{k=1}^{\min\{N^+ ,N^-\}}(4k^2 P_{H_0}(R=2k)+(2k+1)^2P_{H_0}(R=2k+1)) if N^+ \ne N^- and by similarly treating the case N^+ =N^- where I would say that the second moment is E_{H_0}[R^2]=\sum_{k=1}^{N^+ -1}(4k^2 P_{H_0}(R=2k)+(2k+1)^2 P_{H_0}(R=2k+1))+\frac{2(N^+)^2}{\binom{2N^+}{N^+}}, but I haven't been to simplify those sums with their factorials.
Does anybody knows or can link a proof of the formula for the variance \text{Var}_{H_0}(R)?
I \infty-ly thank you all!

 

[Stephen Tashi]

The Italian language Wikipedia considers the statistic R to be the same, under the null hypothesis of independence, as 1+\sum_{i=1}^{N-1}|X_i-X_{i+1}| where the X_i are Bernoulli random variables with expectation p

One thought is to let Y_i = |X_i - X_{i+1}|.

Only consecutive Y's are not independent, so

Var( 1 + \sum_{i=1}^{N-1} Y_i ) = \sum_{i=1}^{N-1} Var(Y_i) + 2 \sum_{i=1}^{N-2} Cov(Y_i,Y_{i+1})

Then you have to find formula for Var(Y_i) and Cov(Y_i,Y_{i+1}). That might not be easy, but iat least it focuses our attention on only three of the X's at a time.