As one of the most important distributions in probability and statistics, the gamma distribution is a flexible right-skewed distribution widely used in many areas such as reliability, environment, insurance and medicine. Burgin (1975) justified the applicability of the gamma distribution to inventory control. Vaz and Fortes (1988) discussed fitting a gamma distribution for grain sizes in a poly-crystal. Husak et al. (2007) used the gamma distribution to model rainfall data. Most recently, Johnson and Kliche (2020) compared seven estimation procedures for gamma parameters on raindrop size data. Since Ohlsson and Johansson (2010), the gamma distribution has become more or less the standard option in modeling claim cost in the insurance industry.

Many researchers have worked on the estimation of the parameters in two-parameter gamma distributions. Choi and Wette (1969) examined the numerical technique of the maximum likelihood method to estimate both parameters of a gamma distribution. Chen and Mi (1998) discussed the point estimation for the scale parameter of a gamma distribution based on grouped data, assuming that the shape parameter was known. Iliopoulos (2016) constructed exact confidence intervals for the shape parameter of a gamma distribution. The author compared the exact confidence intervals with bootstrap confidence intervals via simulation studies. Son and Oh (2006) developed a Gibbs sampling Bayesian estimator of the two-parameter gamma distribution under the non-informative prior.

Most of the literature on the parameter estimation of gamma distributions is based on fixed-sample-size procedures. That is, one aims to find the estimate for parameters of interest based on the obtained data, no matter how large or small it is. In certain situations, especially when data collecting process is time-consuming or costly, it is of great importance to understand what sample size is needed to obtain the value of estimators with prescribed accuracy, which, however, depends on some unknown nuisance parameter and cannot be fixed in advance. Thus, sequential sampling becomes necessary to solve such problems. Takada and Nagata (1995) considered a sequential procedure for building a fixed-width confidence interval for the mean of a gamma distribution. Isogai and Uno (1995) developed a sequential procedure for estimating the mean of a gamma distribution under a loss function of squared error plus linear cost. Liu (2001) worked on approximating the optimal fixed sample size expected reward through a two-stage sampling procedure. Recently, Zacks and Khan (2011) developed two-stage and sequential procedures of conducting fixed-width confidence interval estimation for the scale parameter when the shape parameter is known. Mahmoudi and Roughani (2015) studied a bounded risk two-stage sampling procedure for estimating the scale parameter of a gamma distribution with the shape parameter known. Roughani and Mahmoudi (2015) derived explicit formulas for the expected value and risk of the estimator of the scale parameter in a gamma distribution, where the shape parameter was assumed known. One may achieve a broad-ranging review in the field of sequential analysis by combining selected parts of interest from the following monographs and references therein: Stein (1949), Anscombe (1952, 1953), Chow and Robbins (1965), Woodroofe (1977), Ghosh and Mukhopadhyay (1981), Siegmund (1985), Ghosh et al. (1997) and Mukhopadhyay and de Silva (2009).

In this paper, we will focus on constructing a fixed-accuracy confidence interval for P(X > c) where X is a random variable that follows a gamma distribution, and c is a preassigned positive constant. In many applications, it is desired to estimate the probability of a random variable exceeding a specified value. For instance, in industrial hygiene, it is of interest to estimate the probability that the exposure level (level of exposure to a contaminant in a workplace) of a worker exceeds the occupational exposure limit (OEL; usually set by the Occupational Safety and Health Administration). See Krishnamoorthy and Mathew (2009). In lifetime data analysis, people are often interested in crucial events such as failure of equipment, death of a person, and development of symptoms of disease. The time to the occurrence of the event is called lifetime. There is a large volume of literature on modeling lifetime data using the gamma distribution. One is referred to Lawless (1982), Ansell and Phillips (1994), Balakrishnan and Ling (2014), Balakrishnan et al. (2019), etc. Moreover, the widely used exponential distribution is considered as a special case of the gamma distribution. One may see Nadarajah and Gupta (2007), Piegorsch (2015) and Dagpunar (2019) for more details. Therefore, it is essential to know the probability of a gamma random variable X exceeding a given value of interest, especially when at least one parameter is unknown. To the best of our knowledge, little literature has been focused on the confidence interval estimation of the rate parameter of a gamma distribution. And no literature exists in building a fixed-accuracy confidence interval for a function of the rate parameter (e.g., P(X > c)). Due to the fact that a fixed-width confidence interval is very likely to contain a negative lower bound or an upper bound being greater than 1, and it does not make sense for P(X > c) to be negative or to exceed 1, the problem for estimating P(X > c) will be better addressed using a fixed-accuracy confidence interval when the rate parameter is unknown.

Let us begin with a gamma random variable (X ~ Γ(α, β)) with the associated density function:

Here, β, the rate parameter, is of interest and remains unknown to us. We shall point out that in some research work, authors alternatively use θ = 1/β, the scale parameter, to characterize the gamma distribution. α is called the shape parameter, and can be either known or unknown.

The rest of the paper is organized as the following: In Section 2, we provide a purely sequential procedure to estimate P(X > c) with X coming from a two-parameter gamma population with known shape parameter α and unknown rate parameter β. We discuss some appealing properties of this procedure, and we support our findings through extensive simulations. In Section 3, we develop a nonparametric purely sequential procedure to estimate P(X > c) when both α and β are unknown. We also present some interesting properties of our proposed stopping rule followed by extensive simulations. Section 4 includes illustrations of the purely sequential procedures discussed in Sections 2 and 3 on three real-life data sets. Section 5 provides some concluding thoughts.

In the situation when α is known (for example, the exponential distribution is a special case of Γ(α, β) with α = 1), we can express the desired probability of X exceeding a constant c(> 0) with an incomplete gamma function as follows:

where F(·) stands for the distribution function of Γ(α, 1). Further, we define

and q has a range of (0, ∞).

Starting with a random sample, X₁, . . . , X_n, from a Γ(α, β) population, one can easily obtain the maximum likelihood estimator (MLE) of β, say β̂_n:

where ${\overline{X}}_n=n^{-1}{\sum }_{i=1}^n{X}_i$ is the sample mean. Further, using the invariant property of MLE, we have the MLEs of p and q given as follows, respectively:

and

The central limit theorem (CLT) for MLE tells us that as n→∞,

where $\stackrel{D}{\to }$ represents convergence in distribution. Applying the delta method, we have that as n→∞,

where “log” represents the natural logarithm, and

Note that by definition

so

And we have the expression of ${\sigma }_{\beta }^2$:

To estimate p = P_β(X > c), we first focus on q = p/(1 − p), and consider a fixed-accuracy confidence interval for q of the form given by

with d > 1 fixed. Obviously, J_n;1 is a subset of ℛ⁺, and thus takes into account the positivity of the parameter q. One may refer to Mukhopadhyay and Banerjee (2014), Banerjee and Mukhopadhyay (2016), Mukhopadhyay and Zhuang (2016), Bapat (2018) and other sources for additional background information on fixed-accuracy confidence interval estimation.

With a prescribed significance level 0 < γ < 1, J_n;1 should also satisfy the condition that the coverage probability should be at least 1 − γ or approximately 1 − γ; that is,

from which it follows that

where z ≡ z_γ/2 stands for the upper 100(γ/2) quantile of a standard normal distribution. Now, we define the optimal fixed sample size to be

The magnitude of $n_1^{*}$, unfortunately, remains unknown due to the fact that ${\sigma }_{\beta }^2$ depends on the unknown parameter β. Therefore, it is essential to estimate ${\sigma }_{\beta }^2$ by updating its estimator stage-wise as needed. Customarily, in the light of (2.3) and (2.11), one may use the MLE of ${\sigma }_{\beta }^2$ given by

2.1. A purely sequential procedure

In the light of Anscombe (1953) and Chow and Robbins (1965) who established fundamental theory of sequential estimation, we propose a purely sequential procedure to deal with the problem of constructing a fixed-accuracy confidence interval for P_β(X > c) under a gamma population when the rate parameter is unknown. The basic idea is that after a pilot sample, we draw observations one by one, and terminate sampling immediately when there are enough observations according to some predefined stopping rule.

Let

where m ≥ 1 indicates a pilot sample size, and ${\widehat{\sigma }}_{n;1}^2$ is defined in (2.15). That is, we first take a sample of size m, and check whether $m\ge {(z/\text{log\hspace{0.17em}}d)}^2{\widehat{\sigma }}_{m;1}^2$ is true or not. If it is true, we do not take any extra observations, and the final sample will be the pilot sample. Otherwise, we draw one observation at-a-time, check the stopping rule (2.16) successively with renewed ${\widehat{\sigma }}_{n;1}^2$, and stop sampling at the first time when $n\ge {(z/\text{log\hspace{0.17em}}d)}^2{\widehat{\sigma }}_{n;1}^2$ is observed and the terminated sample size is N₁ = n. We state Theorem 1 to show that the procedure ℘₁ terminates w.p.1.

For a two-parameter gamma distribution Γ(α, β) with both α and β unknown, we can no longer implement ℘₁ as per (2.16) to obtain a fixed-accuracy confidence interval for p that is defined in (2.1). Also, in this case, the MLEs of α and β have no closed-form expressions, and can only be solved numerically. In this section, we therefore develop a sequential fixed-accuracy confidence interval estimation methodology from a nonparametric perspective.

Let us define a new random variable Y based on the gamma random variable X: for a random sample of size n, X₁, . . . , X_n, let Y_i = I_{Xi>c} with i = 1, . . . , n, where I_A is the indicator function of an event A. Clearly, Y₁, . . . , Y_n are independent and identically distributed Bernoulli random variables with probability of success being p as is defined in (2.1).

Then,

serves as an unbiased and consistent estimator of p. Furthermore, it is also the MLE of p under circumstances where both α and β are unknown in a gamma distribution. By CLT, it further holds that as n→∞,

Similarly, we define

so following the delta method, as n→∞, we have

where ${\sigma }_p^2=1/\{p(1-p)\}$. In the light of (2.12) and (2.13), with preassigned d and γ, we consider the 100(1 − γ)% fixed-accuracy confidence interval for q:

which additionally satisfies the condition that

in order to obtain a confidence interval with 100(1 − γ)% confidence level.

Likewise, we derive an alternative optimal fixed sample size given by

which remains unknown, again. Therefore, it is essential to estimate ${\sigma }_p^2$ through sequential procedures. Here, we adopt the estimator

3.1. A purely sequential procedure

One can immediately identify a similarity between (2.14)–(2.15) and (3.7)–(3.8). For situations when both α and β are unknown, we can follow along the lines of Section 2, and develop a purely sequential fixed-accuracy confidence interval estimation procedure associated with the following stopping rule:

where m ≥ 1 continues to indicate a pilot sample size, and ${\widehat{\sigma }}_{n;2}^2$ is defined in (3.8). Different from the stopping rule (2.16), the term n⁻¹ is incorporated here to prevent unexpected early termination of sampling due to the fact that p̃_n is discrete and hence has a positive probability to be 0 or 1.

With the pilot sample data, if $m\ge {(z/\text{log\hspace{0.17em}}d)}^2({\widehat{\sigma }}_{m;2}^2+m^{-1})$, we do not take any extra observations, and the final sample is the pilot sample. Otherwise, we draw one observation at-a-time and check the stopping rule (3.9) successively with updated ${\widehat{\sigma }}_{n;2}^2$ until for the first time $n\ge {(z/\text{log\hspace{0.17em}}d)}^2({\widehat{\sigma }}_{n;2}^2+n^{-1})$ is observed. And the final sample size will be N₂ = n(> m). The following theorem states that the procedure ℘₂ also terminates w.p.1.

3.2. Simulated performances

Next, we include a simulation study by implementing the purely sequential fixed-accuracy confidence interval estimation procedures ℘₂ based on the stopping rule given by (3.9) in an analogous way as we did in Sections 2. We drew pseudo-random observations from a Γ(2, 2) population; that is, we had fixed α = 2 and β = 2, but pretended that they were unknown. Then, we set c = α/β = 1 so that P_α,β(X > c) represented the probability that an observation would turn out larger than the mean value. For brevity alone, we present in Table 2 the selected summaries when the pilot sample size m = 20, the significance level γ = 0.05, and d = 1.10, 1.15, 1.20, 1.25, 1.30, 1.35, 1.40, 1.45, 1.50, 1.55, 1.60, 1.65, 1.70.

Running 10,000 independent trials, we reported the mean and standard deviation of the terminated sample sizes N̄₂ and s(N₂), the ratio of N̄₂ to the optimal fixed sample size $n_2^{*}$ which should be close to 1, the average coverage probability ${\overline{\text{cp}}}_2$ which should be comparable with 1 − γ (= 0.95) along with its standard error $s({\overline{\text{cp}}}_2)$, and the mean of estimated values of p = P_α,β(X > c) with its standard error, ${\overline{\tilde{p}}}_{N_2}$ and $s({\overline{\tilde{p}}}_{N_2})$. The simulation results in Table 2 obviously double validate all the findings of the nonparametric procedure that we proposed for (3.9). Moreover, it is worth mentioning that the differences between N̄₂ and $n_2^{*}$ were empirically around 1.7–1.8 for different values of d, which further shows that our proposed procedure ℘₂ gives a very consistent and efficient determination of the required sample size for estimation. And that ${\overline{\tilde{p}}}_{N_2}$ values are close to p with tiny standard errors suggests the sampling procedure can provide perfect estimates for this probability of interest.

To demonstrate the practical applicability of our newly proposed fixed-accuracy confidence interval estimation procedures, we include illustrations using three real-life data sets: (i) the urine albumin-to-creatinine ratios (UACR, mg/g) of 5255 adolescent survey participants from NHANES 1999–2004, referred to as the “UACR data”; (ii) excess cycle times data in steel manufacturing; (iii) survival times data from a group of 97 female dementia patients diagnosed at age 70–74.

4.1. Illustration I: using UACR data

The reference population for our analysis is created using survey participants from NHANES 1999–2014 who met the following criteria: between 12 and 17 years old, not pregnant, blood pressure < 120/80 mmHg, without diabetes, no prescription medications used within the previous 30 days, and a Z-score for weight-to-height ratio ≤ 2. This yields a reference sample of size n = 5255. The UACR data were analyzed in Zou and Young (2020), where the authors had established a one-sided upper tolerance limit of 0.7358 mg/g and suggested to classify adolescents with UACR values falling below it as “strictly normal.” Interested readers may refer to the paper, and more background information is omitted here.

For illustrative purposes, we also treated the UACR data of size n = 5255 as our population, to which the exponential distribution gave a good fit (Zou and Young, 2020); that is, the shape parameter α = 1, but the rate parameter β was assumed unknown. We set c = 0.7358 and proceeded to estimate p = P_β(X > c), which can be interpreted as the proportion of healthy adolescents with mildly increased UACR. Implementing the procedures ℘₁ introduced earlier to draw observations from the UACR data set under simple random sampling without replacement, we constructed fixed-accuracy confidence intervals for q and p, respectively. The results are summarized in Table 3.

4.2. Illustration II: using excess cycle times data

The excess cycle times data in steel manufacturing was first given in Example 6.1 from Barnett and Lewis (1994) and it is assumed to be a sample from an exponential population. Kimber (1982) and Lin and Balakrishnan (2009) declared that the observed value 92 is an outlier. For this illustration, we use the sample data without the data point 92, and set the shape parameter α = 1 with an unknown rate parameter. We are interested in knowing the excess cycle time being greater than 35 as 35 is the largest normal value that is observed. We further assigned m = 5 and γ = 0.05. In the first step, we took the first five data points as our pilot sample data. Then, we checked with the stopping rule as per (2.16) and found it was not satisfied. Thus, we continued our sampling one-at-a-time according to (2.16). In the end, the sampling was terminated with 85 observations, which was recorded in Table 4 as the observations came in. The final 95% confidence interval estimation for P_β(X > 35) is (0.00214, 0.01897).

4.3. Illustration III: using survival time data from dementia patients

To illustrate the nonparametric procedure as per (3.9), we consider a data set on survival times from a group of 97 female dementia patients diagnosed at age 70–74. This data set was originally from Elandt-Johnson and Johnson (1999) and was recently discussed by Ozonur and Paul (2020) where they showed that the two-parameter gamma distribution adequately fits the dementia data. One may find the data set from Table 8 of Ozonur and Paul (2020).

According to the analysis by Xie et al. (2008), the estimated median survival time from onset of dementia to death was 4.6 years for women. Thus, it will be very helpful to know the probability of living exceeding the median time. We first randomly ordered dementia data and assuming that the data came in with that order. Then, we set γ = 0.05, d = 1.6, c = 4.6, m = 5. That is, we took the first five data points for initial analysis, and found that the stopping rule as per (3.9) was not satisfied. Thus, we continued our sampling one-at-a-time according to (3.9). The sampling was terminated after collecting the data from 71 patients, and the final data we collected are listed in Table 5.

In the end, using the final observations from Table 5, we were able to get a 95% confidence interval estimation for p = P_α,β(X > 4.6) to be (0.3263, 0.5535), where X denotes the survival time for a female dementia patient. Thus, using the data we obtained, we would conclude with 95% confidence that there is a 32.63% to 55.36% chance that women aged 70–74 can live more than 4.6 years from the onset of dementia.

Survival and reliability analysis are two of the most important scientific fields where the gamma distribution is often used to model data. And in these two fields, it is crucial to understand when the measurement, the random variable that is modeled using a gamma distribution, goes beyond a “dangerous” value. Therefore, in the paper, we focus on estimating P(X > c) from a two-parameter gamma population with an unknown rate parameter, and the shape parameter is either known or unknown. In cases of the known shape parameter, such as a sample from an exponential population, we provide an estimation strategy along with a purely sequential procedure for determining the necessary sample size of required accuracy. When the shape parameter is unknown, we come up with a nonparametric purely sequential procedure for achieving the prefixed accuracy. Both procedures perform well in terms of asymptotic efficiency and asymptotic consistency.

Finally, it is also worth mentioning that the nonparametric sequential fixed-accuracy confidence interval estimation procedure developed in Section 3 can be further extended to estimate P(X > c) in a general distribution-free case, since the idea does not depend on any specific population distribution. The gamma population with unknown shape parameter α and unknown rate parameter β can be viewed as an illustration.