Interval Estimation for Gumbel Distribution Using Climate Records

Abstract

The Gumbel distribution is one of the most popular widely used distributions in climate modeling. In this paper, we present exact confidence intervals (CI) and joint confidence regions (JCR) for the parameters of Gumbel distribution based on record data. Exact CI and JCR for the parameters of inverse Weibull distribution are also discussed. Three numerical examples with climate data are presented to illustrate the proposed methods. A simulation study is conducted to study the performance of the proposed CI and region.

1 Introduction

There are several situations pertaining to meteorology, hydrology, largest insurance claims, seismology, and athletic events in which only observations that exceed or only those that fall below the current extreme value are recorded and the complete data are not available. For example, an electronic component ceases to function in an environment of too high temperature, a battery dies under the stress of time, and a wooden beam breaks when sufficient perpendicular force is applied to it. Moreover, in some reliability experiments, the units that are experimented on are destroyed. If units are expensive, the cost of the experiment is mainly the cost of the destroyed units. In such cases, it is possible to set up the experiment in such a way that only units whose life lengths are (lower) record values are destroyed (see [1]). Hence, in such experiments, measurements may be made sequentially and only the record values are observed. Data of this type are called record data or records. The statistical study of record values started with [9] and has now spread in different directions. For more details on records and its applications, see [4, 18], and [7].

A random variable X is said to have Gumbel distribution, if its cumulative distribution function (cdf) is

$$\begin{aligned} F(x;\mu , \sigma )=\mathrm{e}^{-\mathrm{e}^{-\frac{x-\mu }{\sigma }}},~~~x\in R,~~\mu \in R,~~ \sigma >0, \end{aligned}$$

(1.1)

where $\mu$ and $\sigma$ are the location and scale parameters, respectively. The Gumbel probability density function (pdf) has the form:

$$\begin{aligned} f(x;\mu ,\sigma ) = \frac{1}{\sigma }\mathrm{e}^{-\frac{x-\mu }{\sigma }-\mathrm{e}^{-\frac{x-\mu }{\sigma }}}. \end{aligned}$$

(1.2)

The Gumbel distribution was introduced by [12] and since then it received a considerable attention in the literature. It is frequently used in engineering and climate modeling. The book by [15], which describes the Gumbel distribution, presents some of its application areas in engineering including flood frequency analysis, network engineering, nuclear engineering, offshore engineering, space engineering, software reliability engineering, structural engineering, and wind engineering. See also [16, 19], and [10] for some generalizations of the Gumbel distribution.

Many authors have studied statistical inference based on record data from the Gumbel distribution. Balakrishnan et al. [8] derived some recurrence relations for the single and product moments of record values from Gumbel distribution. Nagaraja [17] proved that some inference procedures based on asymptotic theory of extreme order statistics are equivalent to those based on record values from the Gumbel distribution. Based on record data, [2, 3] derived the maximum likelihood, best linear invariant and minimum variance unbiased estimators of the Gumbel location, and scale parameters $\mu$ and $\sigma$. Also, he presented two types of predictors of the $s\mathrm{th}$ record value based on the first m $(m<s)$ record values. Ali Mousa et al. [6] considered Bayesian estimation, prediction, and characterization for Gumbel distribution based on record data. Recently, [5] have studied the characterizations of Rayleigh distribution based on order statistics and record values.

The purpose of this paper is to construct the interval estimation for the parameters of the Gumbel distribution based on lower record values. The rest of this paper is organized as follows. Sect. 2 provides some preliminaries. In Sect. 3, we present an exact confidence interval (CI) for parameter $\sigma$ and an exact joint confidence region (JCR) for the parameters $\mu$ and $\sigma$. In Sect. 4, the exact CI and JCR for the parameters of inverse Weibull distribution are discussed. Section 5 discusses three numerical examples with climate data for illustration. In Sect. 6, a Monte Carlo simulation is conducted to study the performance of the proposed confidence interval and region.

2 Preliminaries

Let $X_{1},~X_{2},...$ be a sequence of independent and identically distributed (iid) continuous random variables with cdf F(x) and pdf f(x). An observation $X_j$ is called an upper (lower) record value of this sequence if its value exceeds (is lower than) that of all previous observations. Generally, let us define $T_{1}=1,~ U_{1}=X_{1}$, and for $n\ge 2$

$$\begin{aligned} T_{n}=\hbox { min}\big \{ j>T_{n-1}:X_{j}>X_{T_{n-1}}\big \}, \qquad U_{n}=X_{T_{n}}. \end{aligned}$$

Then the sequence $\{U_{n}\} (\{T_{n}\})$ is known as upper record statistics (upper record times). Similarly, the lower record times $S_{n}$ and the lower record values $L_{n}$ are defined as follows: $S_{1}=1,~L_{1}=X_{1}$, and for $n\ge 2,~S_{n}=\hbox { min}\{j>S_{n-1}:X_{j}<X_{S_{n-1}}\},~L_{n}=X_{S_{n}}$. The following lemmas are useful in this paper.

Lemma 2.1

Let $L_1>L_2> \dots >L_m$ be the first m observed lower record values from a population with cdf F(.). Define

$$\begin{aligned} U_{i} = -\ln [F(L_i)], \qquad i = 1,2,\dots ,m. \end{aligned}$$

Then $U_{1}<U_{2}<\dots <U_{m}$ are the first m upper record values from a standard exponential distribution.

Proof

From the joint pdf of $L_1, L_2,..., L_m$ and using a simple Jacobian argument, we can easily obtain the joint pdf of $U_1, U_2,..., U_m$ as

$$\begin{aligned} f_{U_1,U_2,...,U_m}(u_1,u_2,...,u_m)=\mathrm{e}^{-u_m}, \qquad 0<u_1<u_2<\cdots <u_m, \end{aligned}$$

which is the joint pdf of the first m upper record values from a standard exponential distribution (see [7]). The proof is thus obtained. $\square$

Lemma 2.2

If $U_{1}<U_{2}<\dots <U_{m}$ are the first m upper record values from a standard exponential distribution. Then the spacings $U_1, U_2-U_1,\dots ,U_m-U_{m-1}$ are iid random variables from a standard exponential distribution.

Proof

The proof can be found in [7]. $\square$

3 Confidence Interval and Joint Confidence Region

Let $L_1>L_2>\dots >L_m$ be the first m observed lower record values from the Gumbel distribution. In this section, a $100(1-\alpha )\,\%$ CI for scale parameter $\sigma$ and a $100(1-\alpha )\,\%$ JCR for $(\mu ,\sigma )$ are constructed based on the observed lower records $\underline{L}=(L_1,L_2,\dots ,L_m)$. A $100(1-\alpha )\,\%$ joint confidence region (or set) for $(\mu ,\sigma )$ is a random set $C(\underline{L})$ such that

$$\begin{aligned} P\left[ (\mu ,\sigma )\in C(\underline{L}) \right] =1-\alpha . \end{aligned}$$

One of the applications of the JCR of the parameters is to find confidence bounds for the functions of the two parameters $\mu$ and $\sigma$.

The most popular approach in obtaining CI and JCR is to use pivotal quantities (functions of data and of unknown parameters whose distributions are known and parameter free). Another useful approach is to use the approximate normality of maximum likelihood estimates (MLEs) and obtain the large sample CI and JCR. Since the asymptotic theory of the MLEs of the parameters is not practical in case of records data [see [7], P. 24)], we shall use here the first approach to obtain CI and JCR.

Let us define

$$\begin{aligned} Y_{i} = \exp \left( -\frac{L_i-\mu }{\sigma }\right) , \qquad i = 1,2,\dots ,m. \end{aligned}$$

(3.1)

Then, by Lemma 2.1, $Y_{1}<~Y_{2}< \dots < Y_{m}$ are the first m upper record values from a standard exponential distribution. Moreover, by Lemma 2.2, we can observe that

$$\begin{aligned} \left\{ \begin{array}{l} Z_{1}= Y_{1}\\ Z_{2}= Y_{2} - Y_{1}\\ \vdots \\ Z_{m}= Y_{m} - Y_{m-1}\\ \end{array}\right. \end{aligned}$$

(3.2)

are iid random variables from a standard exponential distribution. Hence

$$\begin{aligned} V=2Z_{1}=2Y_{1} \end{aligned}$$

has a chi-square distribution with 2 degrees of freedom and

$$\begin{aligned} U= 2\sum _{i=2}^{m} Z_{i}= 2~(Y_{m} - Y_{1}) \end{aligned}$$

has a chi-square distribution with $2m-2$ degrees of freedom. We can also find that U and V are independent random variables. Let

$$\begin{aligned} P_1=\frac{U/2(m-1)}{V/2}=\frac{U}{(m-1)V}=\frac{1}{m-1} \left( \frac{Y_{m}-Y_{1}}{Y_{1}}\right) , \end{aligned}$$

(3.3)

and

$$\begin{aligned} P_2= U+V = 2 Y_{m}. \end{aligned}$$

(3.4)

It is easy to show that $P_1$ has an F distribution with $2m-2$ and 2 degrees of freedom and $P_2$ has a chi-square distribution with 2m degrees of freedom. Furthermore, $P_1$ and $P_2$ are independent, see [13, p. 350].

Let $F_{\alpha }(\upsilon _1,\upsilon _2)$ be the percentile of F distribution with right-tail probability $\alpha$ and $\upsilon _1$ and $\upsilon _2$ degrees of freedom. Next theorem gives an exact CI for the scale parameter $\sigma$ based on lower record values.

Theorem 3.1

Suppose that $L_1>~L_2>\dots >~L_m$ be the first m observed lower record values from the Gumbel distribution in (1.1). Then, for any $0<\alpha <1$,

$$\begin{aligned} \frac{L_1-L_m}{\ln \big [1+(m-1)F_{\frac{\alpha }{2}}(2m-2,2)\big ]}<\sigma < \frac{L_1-L_m}{\ln \big [1+(m-1)F_{1-\frac{\alpha }{2}}(2m-2,2)\big ]} \end{aligned}$$

is a $100(1-\alpha )\,\%$ CI for $\sigma$.

Proof

From (3.3), we know that the pivot

$$\begin{aligned} P_1(\sigma )=\frac{1}{m-1}\left[ \frac{\mathrm{e}^{-\frac{L_m-\mu }{\sigma }}-\mathrm{e}^{-\frac{L_1-\mu }{\sigma }} }{\mathrm{e}^{-\frac{L_1-\mu }{\sigma }}} \right] =\frac{1}{m-1}\left[ \mathrm{e}^{\frac{L_1-L_m}{\sigma }}-1 \right] , \end{aligned}$$

has an F distribution with $2m-2$ and 2 degrees of freedom. Hence, for $0<\alpha <1$, we obtain

$$\begin{aligned} P\left( F_{1-\frac{\alpha }{2}}(2m-2,2)<P_1<F_{\frac{\alpha }{2}} (2m-2,2)\right) =1-\alpha , \end{aligned}$$

which is equivalent to

$$\begin{aligned}&P\left( \frac{L_1-L_m}{\ln \big [1+(m-1)F_{\frac{\alpha }{2}} (2m-2,2)\big ]}\right. \\&\quad \left. <\sigma <\frac{L_1-L_m}{\ln \big [1+(m-1)F_{1-\frac{\alpha }{2}} (2m-2,2)\big ]}\right) =1-\alpha . \end{aligned}$$

This completes the proof. $\square$

It should be mentioned here that we can also use $P_{1}(\sigma )$ to test null hypothesis $H_{0}:\sigma =\sigma _{0}$.

Let $\chi ^2_{\alpha }(\upsilon )$ denote the percentile of $\chi ^{2}$ distribution with right-tail probability $\alpha$ and $\upsilon$ degrees of freedom. Next theorem gives an exact JCR for the parameters $\mu$ and $\sigma$.

Theorem 3.2

Suppose that $L_1>~L_2>\dots >~L_m$ be the first m observed lower record values from Gumbel distribution. Then, the following inequalities determine $100(1-\alpha )\,\%$ JCR for $\mu$ and $\sigma$:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{L_1-L_m}{\ln \Big [1+(m-1)F_{\frac{1-\sqrt{1-\alpha }}{2}}(2m-2,2)\Big ]}<\sigma < \frac{L_1-L_m}{\ln \Big [1+(m-1)F_{\frac{1+\sqrt{1-\alpha }}{2}}(2m-2,2)\Big ]},\\ \\ L_m+\sigma \ln \left( \frac{\chi ^2_{\frac{1+\sqrt{1-\alpha }}{2}}(2m)}{2}\right) <\mu < L_m+\sigma \ln \left( \frac{\chi ^2_{\frac{1-\sqrt{1-\alpha }}{2}}(2m)}{2}\right) . \\ \end{array}\right. \end{aligned}$$

Proof

From (3.4), we know that

$$\begin{aligned} P_2(\mu ,\sigma )=2~\mathrm{e}^{-\frac{L_m-\mu }{\sigma }} \end{aligned}$$

has a $\chi ^2$ distribution with 2m degrees of freedom, and it is independent of $P_1$. Hence, for $0<\alpha <1$, we have

$$\begin{aligned} P\left[ F_{\frac{1+\sqrt{1-\alpha }}{2}}(2m-2,2) <P_1<F_{\frac{1-\sqrt{1-\alpha }}{2}}(2m-2,2)\right] =\sqrt{1-\alpha }, \end{aligned}$$

and

$$\begin{aligned} P\left[ \chi ^2_{\frac{1+\sqrt{1-\alpha }}{2}} (2m)<P_2<\chi ^2_{\frac{1-\sqrt{1-\alpha }}{2}} (2m)\right] =\sqrt{1-\alpha }. \end{aligned}$$

From these relationships, we conclude that

$$\begin{aligned}&P\left[ F_{\frac{1+\sqrt{1-\alpha }}{2}} (2m-2,2)< \frac{\mathrm{e}^{\frac{L_1-L_m}{\sigma }}-1}{m-1}<F_{\frac{1- \sqrt{1-\alpha }}{2}}(2m-2,2),\right. \\&\quad \qquad \left. \chi ^2_{\frac{1+\sqrt{1-\alpha }}{2}} (2m)<2~\mathrm{e}^{-\frac{L_m-\mu }{\sigma }} <\chi ^2_{\frac{1-\sqrt{1-\alpha }}{2}}(2m)\right] \\&\quad = 1-\alpha , \end{aligned}$$

or equivalently

$$\begin{aligned} \left\{ \begin{array}{l} \frac{L_1-L_m}{\ln \Big [1+(m-1)F_{\frac{1-\sqrt{1-\alpha }}{2}}(2m-2,2)\Big ]}<\sigma < \frac{L_1-L_m}{\ln \Big [1+(m-1)F_{\frac{1+\sqrt{1-\alpha }}{2}}(2m-2,2)\Big ]},\\ \\ L_m+\sigma \ln \left( \frac{\chi ^2_{\frac{1+\sqrt{1-\alpha }}{2}}(2m)}{2}\right) <\mu < L_m+\sigma \ln \left( \frac{\chi ^2_{\frac{1-\sqrt{1-\alpha }}{2}}(2m)}{2}\right) . \\ \end{array}\right. \end{aligned}$$

As an application, Theorem 3.2 can be used to obtain a lower confidence bound for the expected value and reliability function of a Gumbel distribution. Here, we only obtain a lower confidence bound for the expected value. The expected value of the Gumbel distribution is $E(X)=\mu +\gamma \sigma$, where $\gamma =0.57722$ is the Euler’s constant. The following corollary will be used to obtain the lower confidence bound for E(X). The proof is easy and omitted.

Corollary 3.3

Suppose that $L_1>L_2>\dots >L_m$ be the first m observed lower record values from Gumbel distribution. Then, the following inequalities determine $100(1-\alpha )\,\%$ JCR for the parameters $\mu$ and $\sigma \mathrm{:}$

$$\begin{aligned} \left\{ \begin{array}{l} \dfrac{L_1-L_m}{\ln \Big [1+(m-1)F_{\frac{1-\sqrt{1-\alpha }}{2}}(2m-2,2)\Big ]}<\sigma < \dfrac{L_1-L_m}{\ln \Big [1+(m-1)F_{\frac{1+\sqrt{1-\alpha }}{2}}(2m-2,2)\Big ]},\\ \\ \mu >L_m+\sigma \ln \left( \frac{\chi ^2_{\sqrt{1-\alpha }}(2m)}{2}\right) \\ \end{array}\right. \end{aligned}$$

where $0<\alpha <1$.

Theorem 3.4

Suppose that $L_1>L_2>\dots >L_m$ be the first m observed lower record values from Gumbel distribution. Then for any $0<\alpha <1$,

$$\begin{aligned} \inf _\sigma \left\{ L_m+\sigma \left[ \ln \left( \frac{\chi ^2_{\sqrt{1-\alpha }} (2m)}{2}\right) +\gamma \right] \right\} \end{aligned}$$

is a $(1-\alpha )100\,\%$ lower confidence bound for the expected value E(X), where the infimum is taken over the interval

$$\begin{aligned} \left\{ \sigma : \frac{L_1-L_m}{\ln [1+(m-1)F_{\frac{1-\sqrt{1-\alpha }}{2}}(2m-2,2)]}<\sigma < \frac{L_1-L_m}{\ln [1+(m-1)F_{\frac{1+\sqrt{1-\alpha }}{2}}(2m-2,2)]}\right\} . \end{aligned}$$

Proof

For fixed t and $\sigma$, $E(X)=\mu +\gamma \sigma$ is increasing in $\mu$, then

$$\begin{aligned}&P\left[ E(X)> \inf _\sigma \left[ L_m+\sigma \left[ \ln \left( \frac{\chi ^2_{\sqrt{1-\alpha }}(2m)}{2}\right) +\gamma \right] \right] :\right. \\&\quad \quad \left. \frac{L_1-L_m}{\ln \Big [1+(m-1)F_{\frac{1-\sqrt{1-\alpha }}{2}}(2m-2,2)\Big ]}<\sigma < \frac{L_1-L_m}{\ln \Big [1+(m-1)F_{\frac{1+\sqrt{1-\alpha }}{2}}(2m-2,2)\Big ]}\right] \\&\quad =P\left[ E(X)> \inf _\sigma [\mu +\gamma \sigma ]:\mu =L_m+\sigma \ln \left( \frac{\chi ^2_{\sqrt{1-\alpha }}(2m)}{2}\right) , \right. \\&\quad \quad \left. \frac{L_1-L_m}{\ln \Big [1+(m-1)F_{\frac{1-\sqrt{1-\alpha }}{2}}(2m-2,2)\Big ]}<\sigma < \frac{L_1-L_m}{\ln \Big [1+(m-1)F_{\frac{1+\sqrt{1-\alpha }}{2}}(2m-2,2)\Big ]} \right] \\&\quad =P\left[ E(X)> \inf _\sigma [\mu +\gamma \sigma ]:\mu >L_m+\sigma \ln \left( \frac{\chi ^2_{\sqrt{1-\alpha }}(2m)}{2}\right) , \right. \\&\quad \quad \left. \frac{L_1-L_m}{\ln \Big [1+(m-1)F_{\frac{1-\sqrt{1-\alpha }}{2}}(2m-2,2)\Big ]}<\sigma < \frac{L_1-L_m}{\ln \Big [1+(m-1)F_{\frac{1+\sqrt{1-\alpha }}{2}}(2m-2,2)\Big ]} \right] \\&\quad =P\left[ \mu >L_m+\sigma \ln \left( \frac{\chi ^2_{\sqrt{1-\alpha }}(2m)}{2}\right) ,\right. \\&\quad \quad \left. \frac{L_1-L_m}{\ln \Big [1+(m-1)F_{\frac{1-\sqrt{1-\alpha }}{2}}(2m-2,2)\Big ]}<\sigma < \frac{L_1-L_m}{\ln \Big [1+(m-1)F_{\frac{1+\sqrt{1-\alpha }}{2}}(2m-2,2)\Big ]} \right] \\&\quad = 1-\alpha . \end{aligned}$$

The proof is thus completed. $\square$

4 Results for Inverse Weibull distribution

The results in Theorems 3.1 and 3.2 can be used for constructing exact CI and JCR for the parameters of inverse Weibull distribution. It is known that, if the random variable X has a Gumbel distribution in (1.1), then $Z=\exp (X)$ has the inverse Weibull distribution with cdf as

$$\begin{aligned} G(z)=\mathrm{e}^{-\lambda z^{-\beta }}, \end{aligned}$$

where $\beta =1/\sigma$ and $\lambda =\exp (\beta \mu )$.

In the next theorem, an exact CI for $\beta$ and an exact JCR for the parameters $\beta$ and $\lambda$ are given. The proof is easy and omitted.

Theorem 4.1

Suppose that $Z_{1}>~Z_{2}>\dots >~Z_{m}$ be the first m observed lower record values from inverse Weibull distribution. Then, a $100(1-\alpha )\,\%$ CI for $\beta$ is

$$\begin{aligned} \frac{\ln \Big [1+(m-1)F_{1-\frac{\alpha }{2}}(2m-2,2)\Big ]}{\ln Z_{1}-\ln Z_{m}}<\beta < \frac{\ln \Big [1+(m-1)F_{\frac{\alpha }{2}}(2m-2,2)\Big ]}{\ln Z_{1}-\ln Z_{m}}, \end{aligned}$$

and a $100(1-\alpha )\,\%$ JCR for parameters $\beta$ and $\lambda$ is determined by the following inequalities

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\ln \Big [1+(m-1)F_{\frac{1+\sqrt{1-\alpha }}{2}}(2m-2,2)\Big ]}{\ln Z_{1}-\ln Z_{m}}<\beta < \frac{\ln \Big [1+(m-1)F_{\frac{1-\sqrt{1-\alpha }}{2}}(2m-2,2)\Big ]}{\ln Z_{1}-\ln Z_{m}},\\ \\ \frac{\chi ^2_{\frac{1+\sqrt{1-\alpha }}{2}}(2m)}{2}\mathrm{e}^{\beta \ln Z_{m} }<\lambda < \frac{\chi ^2_{\frac{1-\sqrt{1-\alpha }}{2}}(2m)}{2}\mathrm{e}^{\beta \ln Z_{m}}. \\ \end{array}\right. \end{aligned}$$

5 Applications

In this section, three examples with climate record data are given to illustrate the proposed CI and JCR.

5.1 Example 1 (Air Temperature Data )

The following data represent the first six lower records of the average annual mean monthly air temperatures (in degrees centigrade) at Babolsar city in north of Iran from 1951 to 2000 (see the website: http://www.iranhydrology.com/meteo.asp):

The correlation coefficient of these six lower records and the expected values of the first Gumbel lower record values in Table 1 is 0.9693. This indicates that the Gumbel model provides a good fit to these record values.

Table 1 Expected value of the first Gumbel lower record values

To find a $95\,\%$ CI for $\sigma$ and a JCR for $\mu$ and $\sigma$, we need the following percentiles:

$$\begin{aligned}&F_{0.025(10,2)}= 39.39797 ,~~~F_{0.975(10,2)}=0.1832712, \\&F_{0.0127(10,2)}=78.13913 ,~~~F_{0.9873(10,2)}=0.1434067 , \end{aligned}$$

and

$$\begin{aligned} \chi ^{2}_{0.0127(12)}= 25.4812\; , \;\chi ^{2}_{0.9873(12)}=3.765882. \end{aligned}$$

Using the methods described in Sect. 3, we can obtain CI and JCR for the parameters. By Theorem 3.1, the 95 % CI for $\sigma$ is (0.4728, 3.8436), with confidence length 3.3808. By Theorem 3.2, the 95 % JCR for $\mu$ and $\sigma$ is determined by the following inequalities:

$$\begin{aligned}& 0.4187 < \sigma <4.6245 \\&14.9+\sigma \ln \left( \frac{3.7659}{2}\right) <\mu < 14.9+\sigma \ln \left( \frac{25.4812}{2}\right) , \end{aligned}$$

with area

$$\begin{aligned} \int _{0.4187}^{4.6245} \int _{14.9+\sigma \ln \left( \frac{3.7659}{2} \right) }^{14.9+\sigma \ln \left( \frac{25.4812}{2}\right) }\mathrm{d}\mu ~ \mathrm{d}\sigma =20.27702. \end{aligned}$$

Figure 1 shows the above JCR. Also a $95\,\%$ lower confidence bound for the expected value E(X) is

$$\begin{aligned} \inf _{\sigma }\Big [14.9+\sigma (\ln (4.420471/2)+0.57722)\Big ] = \inf _{\sigma }\Big [14.9+1.370319\sigma \Big ], \end{aligned}$$

where the infimum is taken over $0.4187<\sigma <4.6245$. Thus, the $95\,\%$ lower confidence bound for E(X) is $14.9+(1.370319)(0.4187)=15.47378$.

Fig. 1

Joint confidence region for parameters in Example 1

5.2 Example 2 (Rainfall Data)

In this example, we present a data analysis and illustrate application of the results in Sect. 3 to the seasonal (July 1–June 30) rainfall in inches recorded at Los Angeles Civic Center from 1962 to 2012 (see the website of Los Angeles Almanac: http://www.laalmanac.com/weather/we13.htm). The data are as follows:

Here, we checked the validity of the Gumbel based on the parameters ${\widehat{\mu }}=11.604, {\widehat{\sigma }}=6.2657$, using the Kolmogorov–Smirnov (K–S) test. It is observed that the K–S distance is K–S $=0.09227$ with a corresponding $\textit{p}\,\mathrm{value}=0.76863$. So, the Gumbel model provides a good fit to the above data.

If only the lower record values of the seasonal rainfall have been observed, these are

By Theorem 3.1, the 95 % CI for $\sigma$ is (0.9776454, 7.948644 ), with confidence length 6.970999. By Theorem 3.2, the 95 % JCR for $\mu$ and $\sigma$ is determined by the following inequalities:

$$\begin{aligned}& 0.8659263 <\sigma <9.56348 , \\&3.21+\sigma \ln \left( \frac{3.7659}{2}\right) <\mu < 3.21+\sigma \ln \left( \frac{25.4812}{2}\right) \end{aligned}$$

with area 86.71719. Figure 2 shows the above joint confidence region. In this example, a $95\,\%$ lower confidence bound for the expected value E(X) is

$$\begin{aligned} \inf _{\sigma }\Big [3.21+\sigma (\ln (4.420471/2)+0.57722)\Big ] = \inf _{\sigma }\Big [3.21+1.370319\sigma \Big ], \end{aligned}$$

where the infimum is taken over $0.8659263 <\sigma <9.56348$. Thus, we obtain the $95\,\%$ lower confidence bound for E(X) as $3.21+(1.370319)(0.8659263)=4.396657$.

Fig. 2

Joint confidence region for parameters in Example 2

5.3 Example 3 (Floods Data)

Consider the data given by [11] represent the maximum flood levels (in millions of cubic feet per second) of the Susquehenna River at Harrisburg, Pennsylvenia, over 20 4-year periods (1890–1969) as:

Maswadah [14] showed that the inverse Weibull distribution with cdf and pdf

$$\begin{aligned}&G(y,\beta ,\lambda ) = \mathrm{e}^{-\lambda y^{-\beta }}, \qquad y>0,~~\lambda >0,~~\beta >0, \\&g(y,\beta ,\lambda ) = \lambda \beta y^{-\beta -1}\mathrm{e}^{-\lambda y^{-\beta }} \end{aligned}$$

provides a very good fit to the given data set. Using the method described in Sect. 4, we can obtain CI and JCR for the parameters $\beta$ and $\lambda$.

Now, the lower records of the maximum flood level are as follows:

By Theorem 4.1, the 95 % CI for $\beta$ is (0.7200, 5.8548), with confidence length 5.1338, and the 95 % JCR for $\beta$ and $\lambda$ is determined by the following inequalities:

$$\begin{aligned}&0.5984<\beta <6.6091, \\&\frac{3.7659}{2}\mathrm{e}^{\beta \ln 0.265}<\lambda < \frac{25.4812}{2}\mathrm{e}^{\beta \ln 0.265}, \end{aligned}$$

with area 3.6918. Figure 3 shows the above JCR.

Fig. 3

Joint confidence region for parameters in Example 3

6 Simulation Study

In this section, a Monte Carlo simulation is conducted to study the performance of the proposed CI and region. In this simulation, we randomly generate lower record sample $L_{1}, L_{2},\dots , L_{m}$ from the Gumbel distribution with the value of parameters $(\mu =0,\sigma =1)$ and then computed 95 % confidence intervals and regions using the results presented in Sect. 3. We then replicated the process 10,000 times. We presented, the simulated average confidence length for parameter $\sigma$, confidence area for the parameters ($\mu ,\sigma$), and the 95 % coverage probabilities of the proposed CI and regions in Table 2.

Table 2 The simulated average confidence length (CL), confidence area (CA), and $95\,\%$ coverage probabilities (CP) for the parameters

From Table 2, we observe that when m increases, the average confidence length for $\sigma$ and the average confidence area for ($\mu ,\sigma$) are decreased.

The simulation results show that the coverage probabilities of the exact CI for parameter $\sigma$ and joint confidence regions for parameters ($\mu ,\sigma$) are close to the desired level of 0.95 for different sample sizes. Hence, our proposed methods for constructing exact CI and JCR can be used reliably.