Navruz-Özdemir Quantile Estimator

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The Navruz-Özdemir quantile estimator suggests the following equation to estimate the $p^\textrm{th}$ quantile of sample $X$:

$$ \begin{split} \operatorname{NO}_p = & \Big( (3p-1)X_{(1)} + (2-3p)X_{(2)} - (1-p)X_{(3)} \Big) B_0 +\\ & +\sum_{i=1}^n \Big((1-p)B_{i-1}+pB_i\Big)X_{(i)} +\\ & +\Big( -pX_{(n-2)} + (3p-1)X_{(n-1)} + (2-3p)X_{(n)} \Big) B_n \end{split} $$

where $B_i = B(i; n, p)$ is probability mass function of the binomial distribution $B(n, p)$, $X_{(i)}$ are order statistics of sample $X$.

In this post, I derive these equations following the paper “A new quantile estimator with weights based on a subsampling approach” (2020) by Gözde Navruz and A. Fırat Özdemir. Also, I add some additional explanations, simplify the final equation, and provide reference implementations in C# and R.

Preparation

The first steps exactly match the preparation performed for the Sfakianakis-Verginis quantile estimator. Consider sample $X = \{ X_1, X_2, \ldots, X_n \}$ ($n \geq 3$). Let $X_{(1)}, X_{(2)}, \ldots, X_{(n)}$ be the order statistics of this sample ($X_{(k)}$ is the $k^\textrm{th}$ smallest element). Now let’s build the following intervals:

$$ S_0 = \big(L, X_{(1)} \big),\; S_1 = \big[X_{(1)}, X_{(2)} \big), \ldots, S_{n-1} = \big[X_{(n-1)},X_{n} \big),\; S_{(n)} = \big[X_{(n)}, U \big) $$

where $L$ and $U$ are lower and upper bounds for $X$ values ($L$ and $U$ could be equal to $-\infty$ and $\infty$).

We want to estimate $p^\textrm{th}$ quantile $Q_p$. Obviously, $Q_p$ should belong to one of the $S_i$ intervals (because they cover all possible $Q_p$ values). With the help of $Q_p$, we can introduce variables $\delta_i$:

$$ \delta_i = \begin{cases} 1 & \textrm{if}\; X_i \leq Q_p,\\ 0 & \textrm{if}\; X_i > Q_p. \end{cases} $$

Since $Q_p$ is the $p^\textrm{th}$ quantile, the probability of $(X_i \leq Q_p)$ is $p$ and the probability of $(X_i > Q_p)$ is $1-p$. Thus, $\delta_i$ belongs to the bernoulli distribution: $\delta_i \sim \textrm{Bernoulli}(p)$.

Next, consider the sum of $\delta_i$ values:

$$ N = \delta_1 + \delta_2 + \ldots + \delta_{n-1} + \delta_n. $$

Since $\delta_i$ are independent variables from the Bernoulli distribution, their sum belongs to the binomial distribution: $N \sim \textrm{Binomial(n, p)}$. Now we can get the probability of $Q_p \in S_i$:

$$ P(Q_p \in S_i) = P(N = i) = B(i; n, p) = {n \choose k} p^k (1-p)^{(n-k)}. $$

Let $Q'_{p,i}$ be a point estimator of $Q_p$ conditioned on the event $Q_p \in S_i$. With the help of $Q'_{p,i}$, we could introduce a quantile estimator:

$$ Q_p \approx \operatorname{E}(Q'_p) = \sum_{i=0}^n P(Q_p \in S_i) \cdot Q'_{p,i}. $$

The Navruz-Özdemir quantile estimator

In [sfakianakis2008], Michael E. Sfakianakis and Dimitris G. Verginis described three options to choose $Q'_p$:

$$ \begin{split} \operatorname{SV1}_p:\quad & Q'_{p,i} = (X_{(i)}+X_{(i+1)}) / 2\\ \operatorname{SV2}_p:\quad & Q'_{p,i} = X_{(i+1)}\\ \operatorname{SV3}_p:\quad & Q'_{p,i} = X_{(i)}\\ \end{split} $$

Based on these assumptions, they got three different quantile estimators:

$$ \begin{split} \operatorname{SV1}_p =& \frac{B_0}{2} \big( X_{(1)}+X_{(2)}-X_{(3)} \big) + \sum_{i=1}^{n} \frac{B_i+B_{i-1}}{2} X_{(i)} + \frac{B_n}{2} \big(- X_{(n-2)}+X_{(n-1)}-X_{(n)} \big),\\ \operatorname{SV2}_p =& \sum_{i=1}^{n} B_{i-1} X_{(i)} + B_n \cdot \big(2X_{(n)} - X_{(n-1)}\big),\\ \operatorname{SV3}_p =& \sum_{i=1}^n B_i X_{(i)} + B_0 \cdot \big(2X_{(1)}-X_{(2)}\big). \end{split} $$

In [navruz2020], Gözde Navruz and A. Fırat Özdemir suggested another way to choose $Q'_{p,i}$:

$$ \operatorname{NO}_p:\quad Q'_{p,i} = pX_{(i)} + (1-p) X_{(i+1)} $$

Also, following [sfakianakis2008], they used the following assumptions for $Q'_{p,0}$ and $Q'_{p,n}$:

$$ Q'_{p,0}-Q'_{p,1} = Q'_{p,1} - Q'_{p,2}; \quad Q'_{p,n} - Q'_{p,n-1} = Q'_{p,n-1}-Q'_{p,n-2}. $$

Thus, we have:

$$ \begin{split} Q'_{p,0} = & 2Q'_{p,1} - Q'_{p,2} &= 2pX_{(1)}+2(1-p)X_{(2)} - pX_{(2)} - (1-p)X_{(3)} \\ & &= 2pX_{(1)} + (2-3p)X_{(2)} - (1-p)X_{(3)},\\ Q'_{p,n} = & 2Q'_{p,n-1} - Q'_{p,n-2} &= 2pX_{(n-1)}+2(1-p)X_{(n)} - pX_{(n-2)} - (1-p)X_{(n-1)}\\ & &= -pX_{(n-2)} + (3p-1)X_{(n-1)} + 2(1-p)X_{(n)}. \end{split} $$

Let’s also denote $B(i; n, p)$ (the probability mass function os the binomial distribution $B(n, p)$) as $B_i$ to make the equations more readable.

Let’s start to derive the equation for the Navruz-Özdemir quantile estimator. Here is the first step:

$$ \operatorname{NO}_p = \sum_{i=0}^n P(Q_p \in S_i) \cdot Q'_{p,i} =B_0 Q'_{p,0} + \sum_{i=1}^{n-1} B_i \cdot (pX_{(i)} + (1-p)X_{(i+1)}) + B_n Q'_{p,n} $$

It gives us the following expression:

$$ \begin{split} \operatorname{NO}_p = & \big( 2pX_{(1)} + (2-3p)X_{(2)} - (1-p)X_{(3)} \big) B_0 + \\ & pX_{(1)}B_1 + (1-p)X_{(2)}B_1 + pX_{(2)}B_2 + (1-p)X_{(3)}B_2 \ldots + pX_{(n-1)}B_n + (1-p)X_{(n)}B_n +\\ & \big( -pX_{(n-2)} + (3p-1)X_{(n-1)} + 2(1-p)X_{(n)} \big) B_n \end{split} $$

By regrouping the equation members, we get the final equation:

$$ \begin{split} \operatorname{NO}_p = & \Big( (3p-1)X_{(1)} + (2-3p)X_{(2)} - (1-p)X_{(3)} \Big) B_0 +\\ & +\sum_{i=1}^n \Big((1-p)B_{i-1}+pB_i\Big)X_{(i)} +\\ & +\Big( -pX_{(n-2)} + (3p-1)X_{(n-1)} + (2-3p)X_{(n)} \Big) B_n \end{split} $$

In [navruz2020], the estimator has slightly different representation:

$$ \begin{split} NO_q = & \big(B(0;n,q)2q + B(1;n,q)q\big)X_{(1)} + B(0;n,q)(2-3q)X_{(2)} - B(0;n,q)(1-q)X_{(3)} \\ & + \sum_{i=1}^{n-2} \big( B(i;n,q)(1-q) + B(i+1;n,q)q \big) X_{(i+1)} - B(n;n,q)qX_{(n-2)} \\ & + B(n;n,q)(3q-1)X_{(n-1)} + \big(B(n-1;n,q)(1-q)+B(n;n,q)(2-2q)\big)X_{(n)}. \end{split} $$

Reference implementation

If you use R, here is the function that you can use in your scripts:

noquantile <- function(x, probs) {
  n <- length(x)
  if (n <= 2)
    return(quantile(x, probs))
  x <- sort(x)
  sapply(probs, function(p) {
    B <- function(x) dbinom(x, n, p)
    (B(0) * 2 * p + B(1) * p) * x[1] +
      B(0) * (2 - 3 * p) * x[2] -
      B(0) * (1 - p) * x[3] +
      sum(sapply(1:(n-2), function(i)
        (B(i) * (1 - p) + B(i + 1) * p) * x[i + 1])) -
      B(n) * p * x[n - 2] +
      B(n) * (3 * p - 1) * x[n - 1] +
      (B(n - 1) * (1 - p) + B(n) * (2 - 2 * p)) * x[n]
  })
}

If you use C#, you can take an implementation from the latest nightly version (0.3.0-nightly.90+) of Perfolizer (you need NavruzOzdemirQuantileEstimator).

References

  • [Sfakianakis2008]
    Sfakianakis, Michael E., and Dimitris G. Verginis. “A new family of nonparametric quantile estimators.” Communications in Statistics—Simulation and Computation® 37, no. 2 (2008): 337-345.
    https://doi.org/10.1080/03610910701790491
  • [Navruz2020]
    Navruz, Gözde, and A. Fırat Özdemir. “A new quantile estimator with weights based on a subsampling approach.” British Journal of Mathematical and Statistical Psychology 73, no. 3 (2020): 506-521.
    https://doi.org/10.1111/bmsp.12198