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Linear mixed models (LMMs) are among the most commonly used tools for genetic association studies. However, the standard method for estimating variance components in LMMs -- the restricted maximum likelihood estimation method (REML) -- suffers from several important drawbacks: REML is computationally slow, requires individual-level genotypes and phenotypes, and produces biased estimates in case control studies. To remedy these drawbacks, we present an alternative framework for variance component estimation, which we refer to as MQS. MQS is based on the method of moments (MoM) and the minimal norm quadratic unbiased estimation (MINQUE) criteria, and brings two seemingly unrelated methods -- the renowned Haseman-Elston (HE) regression and the recent LD score regression (LDSC) -- into the same unified framework. With this new framework, we provide an alternative but mathematically equivalent form of HE that allows for the use of summary statistics and is faster to compute. We also provide an exact estimation form of LDSC to yield unbiased and more accurate estimates with calibrated confidence intervals. A key feature of our method is that it can effectively use a small random subset of individuals for computation while still producing estimates that are almost as accurate as if the full data were used. As a result, our method produces unbiased and accurate estimates with calibrated standard errors, while it is computationally efficient for large data sets. Using simulations and applications to 33 phenotypes from 7 real data sets, we illustrate the benefits of our method for estimating and partitioning chip heritability. Our method is implemented in the GEMMA software package, freely available at www.xzlab.org/software.html.