ABSTRACT
This chapter deals with statistical testing of hypotheses on parameters, either the parameters
a, b and c appearing in the WEIBULL distribution function (= function parameters) (see Sect. 21.1) or parameters which depend on a, b and/or c such as the mean, the variance, a reliability or a percentile (= functional parameters) (see Sect. 21.2). In some cases a test
of a function parameter is equivalent to a test of a functional parameter, for example,
• when b and c are known a hypothesis on a also is a hypothesis on the mean µ = a+b Γ(1 + 1/c),
• when c is known a hypothesis on b also is a hypothesis on the variance σ2 = b2[ Γ(1 + 2/c)− Γ2(1 + 1/c)],
or a test of a functional parameter is equivalent to a test on a function parameter, e.g., a test
on the percentile xP = b [ − ln(1 − P )]1/c is also a test on the scale parameter b when
P = 1− 1/e ≈ 0.6321. Testing a hypothesis on only one parameter using data of only one sample (= one-sample
problem) is intimately related to setting up a confidence interval for that parameter. A
(1− α) confidence interval for a parameter θ contains all those values of θ which — when put under the null hypothesis — are not significant, i.e., cannot be rejected, at level α. Thus,
• a two-sided (1 − α) level confidence interval for θ, θ̂ℓ ≤ θ ≤ θ̂u, will not reject H0 : θ = θ0 in favor of HA : θ 6= θ0 for all θ0 ∈
[ θ̂ℓ, θ̂u
] when α is chosen as level
of significance or
• a one-sided (1− α) confidence interval θ ≤ θ̂u (θ ≥ θ̂ℓ) will not reject H0 : θ ≤ θ0 (H0 : θ ≥ θ0) for all θ0 ≤ θ̂u (θ0 ≥ θ̂ℓ) when the level of significance is α.
