The Huggins-Roy family of effective sample sizes

When we work with weighted samples, it’s essential to introduce adjustments for the sample size. Indeed, let’s consider two following weighted samples:

\[\mathbf{x}_1 = \{ x_1, x_2, \ldots, x_n \}, \quad \mathbf{w}_1 = \{ w_1, w_2, \ldots, w_n \}, \]

\[\mathbf{x}_2 = \{ x_1, x_2, \ldots, x_n, x_{n+1} \}, \quad \mathbf{w}_2 = \{ w_1, w_2, \ldots, w_n, 0 \}. \]

Since the weight of \(x_{n+1}\) in the second sample is zero, it’s natural to expect that both samples have the same set of properties. However, there is a major difference between \(\mathbf{x}_1\) and \(\mathbf{x}_2\): their sample sizes which are \(n\) and \(n+1\). In order to eliminate this difference, we typically introduce the effective sample size (ESS) which is estimated based on the list of weights.

There are various ways to estimate the ESS. In this post, we briefly discuss the Huggins-Roy’s family of ESS.

For a list of weights \(\mathbf{w} = \{ w_1, w_2, \ldots, w_n \}\), we can consider the corresponding normalized weights (or standardized weights):

\[\overline{\mathbf{w}} = \frac{\mathbf{w}}{\sum_{i=1}^n w_i}. \]

For any non-degenerate weighted sample, the sum of all weights is always positive so that the normalized weights are defined.

The Huggins-Roy’s family is given by:

\[\operatorname{ESS}_\beta(\overline{\mathbf{w}}) = \Bigg( \frac{1}{\sum_{i=1}^n \overline{w}_i^\beta } \Bigg)^{\frac{1}{\beta - 1}} = \Bigg( \sum_{i=1}^n \overline{w}_i^\beta \Bigg)^{\frac{1}{1 - \beta}}. \]

This family is proposed in [Huggins2019] and discussed in [Elvira2021] and [Elvira2022].

In order to understand this approach, let’s consider several special cases.

  • \(\beta = 0\):

\[\operatorname{ESS}_0(\overline{\mathbf{w}}) = n - n_z(\overline{\mathbf{w}}), \]

where \(n_z(\overline{\mathbf{w}})\) is the number of zeros in \(\overline{\mathbf{w}}\). This approach is quite straightforward: we just omit elements with zero weights.

  • \(\beta = 1/2\):

\[\operatorname{ESS}_{1/2}(\overline{\mathbf{w}}) = \Bigg( \sum_{i=1}^n \sqrt{\overline{w}_i} \Bigg)^2. \]

  • \(\beta = 1\):

\[\operatorname{ESS}_{1}(\overline{\mathbf{w}}) = \operatorname{exp} \Bigg( -\sum_{i=1}^n \overline{w}_i \log \overline{w}_i \Bigg)^2. \]

This approach is also known as perplexity

  • \(\beta = 2\):

\[\operatorname{ESS}_{2}(\overline{\mathbf{w}}) = \frac{1}{\sum_{i=1}^n \overline{w}_i^2 }. \]

This approach is often referenced as the Kish’s effective sample size (see [Kish1965]).

  • \(\beta = \infty\):

\[\operatorname{ESS}_{\infty}(\overline{\mathbf{w}}) = \frac{1}{\max [\overline{w}_1, \overline{w}_2, \ldots, \overline{w}_n] }. \]

This approach is also popular and straightforward: we define the sample size based on the maximum weight.