Abstract
The maximum likelihood method is the most widely used estimation method. On the other hand, it can produce substantial bias, and an approximate confidence interval based on the maximum likelihood estimator cannot be valid when the sample size is small. Because the sizes of the record values are considerably smaller than the original sequence observed in the majority of cases, a method appropriate for this situation is required for precise inference. This paper provides the exact confidence intervals for unknown parameters and exact predictive intervals for the future upper record values by providing some pivotal quantities in the two-parameter Rayleigh distribution based on the upper record values. Finally, the validity of the proposed inference methods was examined from Monte Carlo simulations and real data.
Highlights
IntroductionThe cumulative distribution function (cdf) and probability density function (pdf) of the random variable (RV), X, with the Rayleigh distribution are given, respectively, by (x − μ)
The cumulative distribution function and probability density function of the random variable (RV), X, with the Rayleigh distribution are given, respectively, by (x − μ)2F (x) = 1 − exp [− 2σ2 ], (1) f (x) = x−μ σ2(x − μ)2 exp [− 2σ2 ], x > μ, σ > 0, (2)where μ is the location parameter and σ is the scale parameter
Dyer and Whisenand [1] examined the properties of the k-optimum best linear unbiased estimators (BLUEs) of the scale parameter in the Rayleigh distribution and provided an approximate k-optimum BLUE based on k order statistics
Summary
The cumulative distribution function (cdf) and probability density function (pdf) of the random variable (RV), X, with the Rayleigh distribution are given, respectively, by (x − μ). This paper constructs exact CIs for unknown parameters (μ, σ) of the Rayleigh distribution based on the upper record values by providing some pivotal quantities, which are much more efficient than the maximum likelihood method in terms of computation cost. Another aim of this paper is to construct exact predictive intervals (PIs) for the future upper record values based on the past upper record values from the Rayleigh distribution because it is very important to correctly predict in many fields such as earthquakes, flood, and rainfall.
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