Asymptotic distributions of the signal-to-interference ratios of LMMSE detection in multiuser communications

Let ${\mathbf{s}}_k=\frac{1}{\sqrt{N}}(v_{1k},...,v_{Nk})^T,$ $k=1,...,K$, where $\{v_{ik},i,k$ $=1,...\}$ are independent and identically distributed random variables with $Ev_{11}=0$ and $Ev_{11}^2=1$. Let ${\mathbf{S}}_k=({\mathbf{s}}_1,...,{\mathbf{s}}_{k-1},$ ${\mathbf{s}}_{k+1},...,{\mathbf{s}}_K)$, ${\mathbf{P}}_k=\operatorname {diag}(p_1,...,$ $p_{k-1},p_{k+1},...,p_K)$ and \beta_k=p_k{\mathbf{s}}_k^T({\mathb f{S}}_k{\mathbf{P}}_k{\mathbf{S}}_k^T+\sigma^2{\mathbf{I}})^{-1}{\math bf{s}}_k, where $p_k\geq 0$ and the $\beta_k$ is referred to as the signal-to-interference ratio (SIR) of user $k$ with linear minimum mean-square error (LMMSE) detection in wireless communications. The joint distribution of the SIRs for a finite number of users and the empirical distribution of all users' SIRs are both investigated in this paper when $K$ and $N$ tend to infinity with the limit of their ratio being positive constant. Moreover, the sum of the SIRs of all users, after subtracting a proper value, is shown to have a Gaussian limit.Comment: Published at http://dx.doi.org/10.1214/105051606000000718 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Assessment of diagnostic value of age for meridional amblyopia with Logistic regression and receiver operating characteristic curve

AIM: To assess the diagnostic value of age for meridional amblyopia by Logistic regression and receiver operating characteristic(ROC)curve.<p>METHODS: A total of 1 005 children(1 910 eyes)with unknown ocular abnormalities other than with-the-rule astigmatism(aged 4-8 years)were recruited. Astigmatism >or =1.00D and sphere < or = 3.00D were present in one or both eyes. The difference of sphere between both eyes was less 1.50D. The difference of astigmatism between both eyes was less 1.00D. All astigmatism was calculated by the absolute value. By analyzing age, sex, astigmatism type, diopter of cylinder and diopter of sphere with Logistic regression, two mathematical models were established. Then the diagnostic efficacy of the model was assessed using the ROC curve.<p>RESULTS: The model 1 included 4 parameters(sex, astigmatism type, diopter of cylinder and diopter of sphere). The model 2 included 5 parameters(the 4 parameters of the model 1 adding age). Using Logistic regression, the diopter of cylinder had an influence on the diagnosis of meridional amblyopia in two models. In model 2, age was another influencing factor on the diagnosis of meridional amblyopia. The model 1 area under ROC curve(AUC)was 0.64, and the model 2 was 0.74. The area of model 2 was greater than the model 1. There was statistical difference in the AUC of two models(<i>P</i><0.05).<p>CONCLUSION: Age might be an influential factor on the diagnosis of meridional amblyopia using Logistic regression and ROC curve