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Some parametric tests based on sample spacings

Rahul Singh, Neeraj Misra

Abstract

Assume that we have a random sample from an absolutely continuous distribution (univariate, or multivariate) with a known functional form and some unknown parameters. In this paper, we have studied several parametric tests based on statistics that are symmetric functions of $m$-step disjoint sample spacings. Asymptotic properties of these tests have been investigated under the simple null hypothesis and under a sequence of local alternatives converging to the null hypothesis. The asymptotic properties of the proposed tests have also been studied under the composite null hypothesis. We observed that these tests have similar asymptotic properties as the likelihood ratio test. Finite sample performances of the proposed tests are assessed numerically. A data analysis based on real data is also reported. The proposed tests provide alternative to similar tests based on simple spacings (i.e., $m=1$), that were proposed earlier in the literature. These tests also provide an alternative to likelihood ratio tests in situations where likelihood function may be unbounded and hence, likelihood ratio tests do not exist.

Some parametric tests based on sample spacings

Abstract

Assume that we have a random sample from an absolutely continuous distribution (univariate, or multivariate) with a known functional form and some unknown parameters. In this paper, we have studied several parametric tests based on statistics that are symmetric functions of -step disjoint sample spacings. Asymptotic properties of these tests have been investigated under the simple null hypothesis and under a sequence of local alternatives converging to the null hypothesis. The asymptotic properties of the proposed tests have also been studied under the composite null hypothesis. We observed that these tests have similar asymptotic properties as the likelihood ratio test. Finite sample performances of the proposed tests are assessed numerically. A data analysis based on real data is also reported. The proposed tests provide alternative to similar tests based on simple spacings (i.e., ), that were proposed earlier in the literature. These tests also provide an alternative to likelihood ratio tests in situations where likelihood function may be unbounded and hence, likelihood ratio tests do not exist.

Paper Structure

This paper contains 10 sections, 9 theorems, 75 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Tests for Univariate Distributions
  3. Main Results
  4. Composite Null Hypothesis
  5. Tests for Multivariate Distributions
  6. Numerical Study
  7. Univariate Case
  8. Multivariate Case
  9. Real Data Analysis
  10. Discussion

Key Result

Theorem 1

Let $X_1,X_2,\ldots,X_{n}$ be iid observations from $F_{\eta}$. Suppose that $m=o(n)$, and assume that conditions (A1)-(A7) hold. Then, there exists a sequence $\{\hat{\theta}^{(m)}_{\phi,n}=$$\arg \inf_{\theta\in\Theta}S^{(m)}_{\phi,n}(\theta)\}_{n\geq1}$ such that $\hat{\theta}^{(m)}_{\phi,n}$ is

Figures (5)

  • Figure 1: The empirical powers of $\tilde{T}_{-\log,n}^{(1)}(1)$ (hollow circles), $\tilde{T}_{-\log,n}^{(m_{opt}(n))}(1)$ (solid circles) and LRT (hollow squares), for the alternative $\eta=\theta$ and $n=200$.
  • Figure 2: Q-Q plot of 1000 replicates of $\tilde{T}_{-\log,n}^{(m)}(\theta_0)$, $n=225$ under $H_0$. Empirical distribution quantiles on the horizontal axis and the limiting $\chi^2_2$ distribution quantiles on the vertical axis.
  • Figure 3: Q-Q plot of 1000 replicates of $\tilde{T}_{-\log,n}^{(m)}(\theta_0)$, $n=225$ under the alternative $\eta_n=\theta_0+\Delta n^{-1/2}$ with $\Delta=(3,3)$. Empirical distribution quantiles on the horizontal axis and the limiting $\chi^2_2(\Delta^tI(\theta_0)\Delta)$ distribution quantiles on the vertical axis.
  • Figure 4: Left plot is Q-Q plot of 1000 replicates of $\tilde{T}_{\phi,n}(\theta_0)$, $n=100$, under $H_0$; and $\phi(x)=-\log(x)+x-1$. Empirical distribution quantiles on the horizontal axis and the limiting $\chi^2_2$ distribution quantiles on the vertical axis. The right plot is corresponding plot under the alternative $\eta_n=\theta_0+\Delta n^{-1/2}$, with $\Delta=(1,1)$, and the limiting distribution $\chi^2_2\left(\frac{b_\phi^2}{\sigma_q^2}\Delta^t I(\theta_0)\Delta\right)$.
  • Figure 5: The empirical powers of $\tilde{T}_{\phi,n}(\theta_0)$, with $\phi(x)=-\log(x)+x-1$ (hollow circles) and the LRT (hollow squares), for $n=100$.

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Corollary 2
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • proof : Theorem \ref{['thm2']}
  • proof : Corollary \ref{['cor2']}
  • ...and 7 more