Abstract

Background. A country’s agriculture reflects a backbone and performs a vital part in the betterment of the economy and individuals. Facts and figures of the agriculture sector offer a solid foundation and factual pathway intended for upcoming decisions in favor of a country. Accordingly, the probability models have a more significant influence not only in reliability engineering, hydrology, ecology, and medicine but also in agriculture sciences. Objective. The primary objective of this study is to propose a reliable and efficient model for pearl millet yield analysis, thereby empowering decision-makers to make informed decisions about their farming practices. With the successful implementation of this model, farmers can potentially increase their pearl millet yield, leading to higher incomes and improved livelihoods for the rural population of Pakistan. Model. This study proposes a novel probability model, namely, the alpha transformed odd exponential power function (ATOE-PF) distribution, for analyzing pearl millet yield in Punjab, Pakistan. Data. For data collection, two secondary data sets are explored that are electronically available on the site of the Directorate of Agriculture (Economics and Marketing) Punjab, Lahore, Pakistan. Results. The maximum likelihood estimation technique is used for estimating the model parameters. For the selection of a better fit model, we follow some accredited goodness of fit tests. The efficiency and applicability of the ATOE-PF distribution are discussed over the province of Punjab (with RMSE = 4.9176) and Pakistan (with RMSE = 4.5849). Better estimates and closest fit to data among the well-established neighboring models offer robust evidence in support of ATOE-PF distribution as well.

1. Introduction

Being an inhabitant of the agricultural country of Pakistan, our masses’ primary source of income relies on agriculture. It has a dynamic role in developing the country’s foreign exchange, economic growth, and employment. Over the last 40 years, it has had an outstanding contribution to the development of Pakistan’s economy [1]. 65% fluctuating share of Pakistan’s population, 18.9% gross domestic production (GDP), and 42.3% of the labor force ultimately dependent on agriculture [2]. The total land area of Pakistan is 196.72 million acres, and 66.97 million acres are harvested, along with 20.51 million acres not harvested [3]. Reference [4] categorized Pakistan’s crops into food (wheat and rice) and cash food (cotton, maize, sugarcane), as both the crops have a 6.5% contribution to Pakistan’s GDP.

One of the oldest cultivated food a pearl millet, which the locals call Bajra. It is a fifth-ranked crop in Pakistan after sorghum, maize, rice, and wheat. This crop is significant for fodder and grain, along with high nutritional contents for poultry and livestock. From 2010 to 2011, this crop yielded 346 thousand tons with a grown area of 548 thousand hectares. However, it was quite an impressive increase (by 18%) as compared to 2009-2010 production [5]. Worldwide, pearl millet’s cultivation area is 31 million hectares [6], though, in Pakistan, 0.50 million hectares area along with 0.33 million tones production [7]. Pearl millet’s low yield in Pakistan incorporates many factors, including nonstandard crop, inappropriate time of seeding, fluctuating weather intimidations, competitor cereals, and watering issues [8]. Reference [9] explored it as the feeding of the pet birds. It is expected that if Pakistan imports 61,000 tons of pearl millet by 2030, it will be considered the second leading importer country after China [10].

1.1. Probability Models Used for Different Field Crops

Several statistical techniques to model crop yield have been developed and discussed in the past. For this, one can extend his knowledge by reading from [11–32] and many others.

2. Materials and Methods

2.1. Punjab and Pakistan Area and Production

Crop pearl millet has a very high potential of growing with dry heat and drought tolerance along with the low rainfall area (less than 350 mm) circumstances. Consequently, Sindh (Sanghar, Hyderabad, Nawabshah, Kairpur, and Dadu); Punjab (Gujranwala, Bahawalnagar, Rawalpindi, Gujrat, Chakwal, Mianwali, and Attock); Balochistan (Sibbi, Lorali, and Khuzdar); and NWFP (Bannu, D. I. Khan, and Karak) are considered the most suitable and favorable districts (cities) for appropriate cultivation.

Table 1 provides valuable information on the coordinates and region of Pakistan and Punjab. It is a useful resource for researchers and other stakeholders who are interested in understanding the geography and location of the region, and can be used for various analytical and research purposes.

Figure 1: A graphic representation of the area and pearl millet output in Punjab and Pakistan. The figure uses a map of the region along with data on pearl millet production to provide an easy-to-understand overview of the cultivation of pearl millet in this area.

Figure 1 

Area and production of pearl millet in Punjab and Pakistan.

Figure 2: An illustration of the ultimate shape of the pearl millet crop. This figure provides a clear visual reference for the physical appearance of the crop, which can be useful for those who are not familiar with it.

Figure 2 

Pearl millet picture.

Figure 3 includes two panels; the left panel displays a map of Pakistan, while the right panel displays a map of the Punjab province in Pakistan. The map of Punjab shows the major cities in the province, as well as the locations of pearl millet farms, providing valuable information on the geographic distribution of pearl millet cultivation in the region. The use of a map in this figure helps to provide a clear and visual representation of the information, making it easier for the audience to understand the distribution of pearl millet cultivation in the region.

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Figure 3 

Map of Pakistan (a) map of Punjab (b)

2.2. Pakistan Climate Conditions

Pakistan experiences a significant amount of climatic variability. Despite the fact the summer months of April to September are fairly nice, the winter is brutally chilly in the high mountains in the north and north west. The Indus Valley’s plains experience sweltering heat in the summer and freezing conditions in the winter. The southern coastline region experiences a mild climate. Rainfall is generally insufficient. The lower Indus plain’s northern regions receive an average annual rainfall of 16 centimeters, whereas the Himalayan area gets an average annual rainfall of 120 centimeters. Rainfall occurs late in the summer and has a monsoonal origin. Humidity is comparatively low because of the heavy rains and wide diurnal temperature fluctuation. High humidity only exists along the coastal strip.

2.3. Punjab Climate Conditions

In the majority of Punjab’s regions, the winters are gloomy and frequently rainy. The weather turns springlike by mid-February and stays that way until mid-April, whenever the summer heat arrives. Punjab is expected to experience the start of the monsoon season around May, although the weather has been unpredictable since the early 1970s. Either as the spring monsoon missed the region or it rained so heavily that flooding occurred. It is very hot in June and July. Media sources indicate that the temperature exceeds 51°C and frequently publish stories about persons who have passed away from the heat, despite the fact that official measurements of the temperature seldom go over 46°C. When the temperature reportedly reached 54°C in Multan in June 1993, temperature records were smashed. The “bars” (monsoon season), which give comfort once it passes, interrupt the intense heat in August. Even though the hottest portion of the summer is passed, colder temperatures will not arrive until late October. One of the most frigid winters in the province’s recent history dates back more than 70 years. Temperatures in the Punjab area average from −2° to 45°C; However they may get as high as 50°C (122°F) in the summer and as low as −10°C in the winter. Punjab experiences the following three distinct seasons:(1)Hot weather (April to June), with temperatures reaching 123 degrees Fahrenheit (51 degrees Celsius).(2)July to September is the rainy season. Average rainfall per year ranges between 96 cm in the submountain region and 46 cm in the plains.(3)From October to March, the weather can be cold, foggy, or mild. The temperature drops to 35.6 degrees Fahrenheit (2.0 degrees Celsius).

It should be noted that September through October is the ideal time to harvest the crop known as Bajra.

2.4. Climate Prerequisite

It may be sown at low soil temperatures before reaching 23°C. It germinates best in ideal conditions (25–30°C). The vapor pressure deficit (VPD) caused by the daily maximum temperature of 42°C during blooming directly reduces the pearl millet’s ability to set seeds [33]. At 40–45°C (base temperature of 10°C), tillering starts with the main tillers regions of the world depend on precipitation, which typically ranges from 150 to 750 mm (350 mm). Because of its resilience to very hot and dry weather conditions is becoming increasingly important in developing climate-resilient agricultural systems under changing climatic scenarios [34]. The pearl millet requires between 300 and 350 mm of rainfall to thrive. It is important to note that the water requirement of a crop can vary depending on various factors such as soil type, climate, and cultivation practices. The Figure 4 presented in the chart should therefore be considered as general guidelines rather than exact values.

Figure 4 

Water requirement of pearl millet in comparison with other crops.

2.5. Data Collection

For this study, we consider secondary data sets. For this, the first data presents the average yield of Bajra in Punjab (1947-48 to 2017-18) (Per Acre/000 Tonnes), and the second data relates to the average yield (Per Acre/000 Tonnes) of Bajra in Pakistan (1947-48 to 2017-18). The datasets are obtained from the agricultural statistics of Pakistan and are available at the electronic address provided in Appendix.

2.6. Model Description

In this paper, we develop a novel two-parameter probability model that performs so well not only in reliability engineering, hydrology, ecology, and medical sciences but has a vital role in agriculture sciences as well. We refer to it as the alpha transformed odd exponential power function (ATOE-PF) distribution. The associated cumulative distribution function (CDF) corresponding to the probability density function (PDF) along with the quantile function is, respectively, given by the following equation:where and , are two shape parameters.

Note that, the ATOE-PF distribution is one of the particular members of the ATOE-G class of distributions. Therefore, this paper uses ATOE-PF distribution as a modeling framework, and our ongoing project’s advanced complementary mathematical and reliability measures are under-processed.

2.7. Parameter Estimation

We use the maximum likelihood estimation technique for the parameter estimation of the ATOE-PF distribution. For this, we suppose be a random sample of size n taken from X, then the log-likelihood function () of X is given by the following equation:

The partial derivatives of for the parameters and are given by, respectively,

The ML estimates (, ) of the ATOE-PF distribution are derived by maximizing (2) or by solving the above nonlinear equations simultaneously. The following part has a detailed simulation with various parameter configurations to test the asymptotic capability of MLEs.

2.8. Simulation Study

The following algorithm discusses the performance of MLEs with the assistance of a simulation study: Step-1: a random sample x₁, x₂, x₃, …, x_n of sizes n = 100, 150, 200, 250, 300, 350, 400, 450, and 500 are generated from . Step-2: the required results are obtained based on the different combinations of the model parameters for  = 2, placed in S-I (), S-II (), S-III (), S-IV (), S-V (), S-VI (), S-VII (), S-VIII (), and S-IX () Step-3: average estimate (AE), bias, mean square error (MSE), and variance (Var) are presented in Tables 2–4. Step−4: each sample is replicated N = 1000 times. Step-5: gradual decrease in AE(s), bias(es), MSE(s), and Var(s) with increases in the sample size is observed. Step-6: finally, the estimates in Tables 2–4 help us specify that the method of maximum likelihood works consistently for the ATOE-PF distribution.

Note that, Figure 5 is a useful visual representation of the density function curves for various choices of model parameters for simulated data. The figure provides researchers with valuable insights into the impact of different parameter values on the shape of the distribution, enabling them to make more informed modeling decisions.

Figure 5 

Density function curves for various choices of model parameters for simulated data.

3. Results and Discussions

Now, we report the application of the ATOE-PF distribution. For this, we focus on the agricultural sector and engage two suitable datasets. The ATOE-PF distribution is compared with well-known competitive models. The CDFs of competitive models are listed in Table 5. The parameter estimates and standard errors are presented in Tables 6 and 7 for both datasets, respectively. Some typical results from descriptive statistics for both datasets are shown in Tables 8 and 9. These descriptive statistics are minimum value, 1st quartile, mean, median, mode, standard deviation (SD), 3rd quartile, maximum value, 90%, 95%, and 99% confidence intervals.

The goodness-of-fit statistics for the ATOE-PF distribution and competing models are presented in Tables 10 and 11. A better fit model is one with the criteria of a minimum value of Anderson–Darling (AD), Cramer-von Mises (CVM), root mean square error (RMSE), and Kolmogorov–Smirnov (KS) with a higher -value. Please note that a comprehensive list of standard measurement units and corresponding abbreviations can be found in Table 12 of this document.

The agriculture sector plays a crucial role in the economy of a country, and the ability to accurately predict crop yields is of utmost importance. In order to aid decision-makers in the farming industry, a new probability model was developed that is capable of accurately modeling agriculture data. This study utilized secondary data on pearl millet (Bajra) yields in Punjab Province, Pakistan and compared the alpha transformed odd exponential power function (ATOE-PF) distribution to its well-established rivals using various goodness of fit tests such as KS (-value), AD, and CVM. The ATOE-PF distribution showed a better fit for the average yield of pearl millet (Bajra) in Punjab and Pakistan than any of its competitors. The value (KS) was higher for the ATOE-PF distribution, indicating that it meets the minimal statistical value requirement for a better fit model. The empirical fitted PDF, CDF, Probability-Probability, and box plots of the ATOE-PF distribution are presented in Figures 6 and 7, which visually demonstrate the model’s adequacy. All numerical results and model estimates were obtained using the free statistical software R Studio version 1.2.5033 (cited therein) and its exclusive package AdequacyModel. This new probability model provides decision-makers in the farming industry with a reliable tool to aid in predicting crop yields. By utilizing the ATOE-PF distribution, farmers and related departments can begin implementing more effective predictive measures. The model’s superiority over its competitors in accurately modeling agriculture data provides valuable information for agriculture bodies. In addition, the use of various goodness of fit tests ensures that the model provides an adequate fit. Overall, the ATOE-PF distribution presents a promising solution for researchers and practitioners in the agriculture sector.

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Figure 6 

Empirically fitted plots for an average yield of Bajra in Punjab, Pakistan.

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Figure 7 

Empirically fitted plots for an average yield of Bajra in Pakistan.

4. Conclusions

In this work, a novel model called the alpha transformed odd exponential power function (ATOE-PF) distribution was established, and we introduced its PDF and CDF. A simulation study was carried out using the maximum likelihood estimation technique. To prove the superiority of the proposed model, we fitted two pearl millet datasets. The ATOE-PF distribution was considered the best fit model among the well-known rivals after passing the various goodness of fit tests. Referring to Tables 10 and 11, we found that the (ATOE-PF) distribution has the lowest K-S value and the highest value, proving the ATOE-PF distribution’s superiority. The efficiency and applicability of the ATOE-PF distribution are discussed over the provinces of Punjab (with RMSE = 4.9176) and Pakistan (with RMSE = 4.5849). Furthermore, outperforming estimates made it more relevant and encouraging for pearl millet farm decision-makers and other agriculture agencies.

5. Future Directions

The proposed technique would hopefully be adopted by agriculture experts and concerned agencies and implemented on maize, soybeans, rice, sugarcane, cotton, moong, mash, and jowar for a more appropriate prediction and a respectable predicted yield. Also, we have another critical future work: the study of COVID-19 infections and the mortality rate of the infected. Another expansion will be the competing risk resulting from death, whether it is from the disease or another cause.

Appendix

Data Availability

The data used to support the study are included in the paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study is supported via funding from Prince sattam bin Abdulaziz University project number (PSAU/2023/R/1444).