Comparison of Some Exact Tests for a Common Location Parameter of Several Truncated Exponential Distributions with Different Scale Parameters

Abstract

In this paper we derive some exact tests and one conservative test for a common location parameter μ of several independent truncated exponential distributions with different scale parameters. Tests for μ = μ₀ against two-sided alternatives are suggested based on suitable combinations of natural pivots. The exact tests are based on non-minimal sufficient statistics while the conservative test is based on minimal sufficient statistics. Expressions for local power for alternatives μ > μ₀ are derived and compared. It turns out that the power for alternatives μ < μ₀ is the same for all proposed exact tests. We observe that the test based on a combination of inverse normal cdf of P-values is locally best in most cases. The performance of the conservative test is indeed remarkable in large samples. Some interesting features of a familiar test in the case of one population are also pointed out.

Appendix I: Proof of Eq. 31

That U is independent of (T₁,⋯ ,T_k) follows from a direct appeal to Basu’s theorem upon noting that for any fixed (σ₁,⋯ ,σ_k), U is complete for μ, (T₁,⋯ ,T_k) has a joint distribution which is free from μ. To derive the joint distribution of (T₁,⋯ ,T_k), we write

$$ \begin{array}{@{}rcl@{}} T_{i} &=& \sum\limits_{j=1}^{n_{i}} (x_{ij} - U) = \sum\limits_{j=1}^{n_{i}} (x_{ij}-U_{i}) + n_{i} (U_{i} - U) \\ &=& W_{i} + n_{i} (U_{i} - U) \end{array} $$

(8.1)

and note that W₁,⋯ ,W_k are independently distributed with $\frac {W_{i}}{\sigma _{i}} \sim Gamma$ (n_i − 1),i = 1,⋯ ,k, and W₁,⋯ ,W_k,U₁,⋯ ,U_k are all mutually independent. Then

$$ \begin{array}{@{}rcl@{}} Pr\{T_{1} > t_{1}, \cdots, T_{k} >t_{k} \} &=& Pr\{W_{1} + n_{1}(U_{1} - U) > t_{1}, \cdots, W_{k} + n_{k} (U_{k} - U) > t_{k} \} \\ &=& \sum\limits_{j=1}^{k} Pr\{W_{1} + n_{1}(U_{1} - U) > t_{1}, \cdots, W_{k} + n_{k} (U_{k} - U) > t_{k}, \\ && \quad \quad U_{i} > U_{j}, i \neq j \} \\ &=& \sum\limits_{j=1}^{k} B_{j} \end{array} $$

(8.2)

where

$$ \begin{array}{@{}rcl@{}} B_{j} &=& Pr\{W_{j}>t_{j}, W_{i} + n_{i} (U_{i} - U_{j}) > t_{i}, i \neq j , U_{i} > U_{j} \} \\ &=& Pr\{W_{j}>t_{j} \} Pr\{U_{i} > U_{j}, \forall i \neq j \} \\ && \times \bigg[\prod\limits_{i \neq j} Pr\{W_{i} + n_{i}(U_{i}-U_{j})>t_{i} \rvert U_{i} > U_{j} \}\bigg] . \end{array} $$

(8.3)

Note that

$$ \begin{array}{@{}rcl@{}} Pr\{U_{i} > U_{j}, i \neq j \} &=& Pr \big \{ \frac{n_{i}(U_{i} - \mu)}{\sigma_{i}} > \frac{(U_{j} - \mu) n_{i}}{\sigma_{i}}, \forall i \neq j \big \} \\ &=& E \big [ e^{-{\sum}_{i \neq j=1}^{k} \frac{(U_{j} - \mu) n_{i}}{\sigma_{i}} } \big] \\ &=& E \big [ e^{-\big({\sum}_{i \neq j=1}^{k} \frac{n_{i}}{\sigma_{i}}\big) (U_{j}-\mu) } \big] \\ &=& {\int}_{\mu}^{\infty} e^{-\big({\sum}_{i \neq j=1}^{k} \frac{n_{i}}{\sigma_{i}}\big) (u-\mu) } \big[\frac{n_{j}}{\sigma_{j}}\big] e^{-\frac{n_{j}}{\sigma_{j}}(u-\mu)} du \\ &=& \frac{n_{j} / \sigma_{j}} {\sum}_{i=1}^{k} \frac{n_{i}}{\sigma_{i}} . \end{array} $$

(8.4)

Also, from Ghosh and Razmpour (1984), it follows that

$$ Pr\{n_{i} (U_{i} - U_{j}) > x \rvert U_{i} > U_{j}\} = e^{-\frac{x}{\sigma_{i}}} . $$

(8.5)

Hence, conditional on U_i > U_j,

$$ \frac{W_{i} + n_{i} (U_{i} - U_{j})} {\sigma_{i}} \sim Gamma(n_{i}). $$

(8.6)

Combining (8.4), (8.5) and (8.6), we get

$$ B_{j} = \bigg [ \frac{n_{j} / \sigma_{j}} {\sum\limits_{i=1}^{k} \frac{n_{i}}{\sigma_{i}}} \bigg ] Pr\{W_{j} > t_{j} \} \bigg [ {\prod}_{i \neq j=1}^{k} Pr\big\{Gamma(n_{i}) > \frac{t_{i}}{\sigma_{i}}\big\} \bigg ] . $$

(8.7)

The result follows since $\frac {W_{j}} {\sigma _{j}} \sim Gamma(n_{j} -1)$.