2.3 Counting Statistics

The simplest of all non-parametric distribution-free statistics are those based on counts of whether a particular event happens in each of the \(n\) independent trials. Let \(X_1, \, X_2, \cdots , \, X_n\) be independent random variables such that \(X_i\) is continuous random variable cdf \(F_i(x), \,\, \, i = 1, \, 2, \, \cdots , \, n\).
Let \(F_i(\theta ) = P(X_i < \theta ) = P_0\, , \, \, 0 < P_0 < 1\, \hspace {0.2cm} i = 1, \, 2, \, \cdots , \, n\) that is each \(X_i\) has the same unknown \(P_0^{\text {th}}\) percentile or Quantile.
Let \(\theta _0\) be a known real number and define a statistic \(\Psi _i\) by \[\Psi _i = \Psi (X_i - \theta _0) = \begin {cases} 1 & X_i - \theta _0 > 0\\ & \iff X_i > \theta _0\\\\ 0 & X_i - \theta _0 \leq 0\\ & X_i \leq \theta _0\\ \end {cases}\hspace {0.5cm}\cdots \cdots \hspace {0.3cm} 2.2.1 \]

\[\Psi _i = \begin {cases} 1 & X_i > \theta _0\\ 0 & X_i \leq \theta _0\\ \end {cases} \hspace {0.5cm} i = 1,\, 2, \, \cdots , \, n\]

Theorem 2.3.1. Let \(\Psi _1, \, \Psi _2, \, \cdots \, , \, \Psi _n\) be defined by \(2.2.1\) and let \(T(\Psi _1, \, \Psi _2, \, \cdots , \, \Psi _n)\) be any statistic based on \(\Psi _1, \, \Psi _2, \, \cdots , \, \Psi _n\) only. Then if \(\theta = \theta _0\)

(i): the statistic \(\Psi _1, \, \Psi _2, \, \cdots , \, \Psi _n\) are \(iid\) Bernoulli with parameter \(1 - P_0\).
(ii): \(T(\Psi _1, \, \Psi _2, \, \cdots , \, \Psi _n)\) is non-parametric distribution-free statistic over the class \(\mathscr {Z}_4\), consisting of all joint distribution of independent continuous random variables each with \(P_0^{\text {th}}\) quantile equal to \(\theta _0\).

Proof. The proof is trivial. □

The particular function of \(\Psi _i's\) that are useful in hypothesis testing situation is \[T = T(\Psi _1, \, \Psi _2, \, \cdots , \, \Psi _n) = \sum ^n_{i = 1} \Psi _i.\] \(T\thicksim B(n, 1 - P_0)\), provided the joint distribution is in class \(\mathscr {Z}_4\). In particular when
\(P_0 = \frac {1}{2}\, \implies \, \theta _0\) is the median.
\(H_0: \) each \(X_i \hspace {0.2cm} i = 1, \, 2, \, \cdots ,\, n\) has median \(\theta _0\) then \(T = \sum ^n_{i = 1}\Psi _i\) is known as the sign test statistic.

[next] [prev] [prev-tail] [front] [up]

Questions on this section

Stuck on something here? Ask below and it stays attached to this topic.