Narrow Results

Wireless signals often propagate in spaces containing large and small obstacles that affect the quality of such radiation. Obstacles cause a nonnegligible loss of the propagated power. The degradation of the signal quality is described by introducing fading and shadowing effects. A case of a wireless communication system that consists of a macro-level component represented by the selection combiner (SC) and two micro-level components consisting of the maximal ratio combiners (MRC) with $L$ branches is studied in this paper. The MRC component assumes the presence of a single-base station. The received signal is disturbed by simultaneous impact of the multipath Nakagami-$m$ fading and gamma shadowing. Consequently, the envelope is described by generalized-$K$ density function. On the other hand, SC component assumes the presence of two base stations, where the average power of the received signal is described using gamma distribution. The exact closed-form expression is obtained for probability density function of the signal at the output of the system, followed by the corresponding outage probability (P_out). Calculations are depicted graphically expressing influences of different fading values as well as the signal parameters.

In this paper, we attempt generalization of the Markovian software reliability model (MSRM). Defining the stochastic process whose state space is the cumulative number of corrected faults, we show the theoretical framework of the MSRM by assuming that the time interval between software failures is distributed generally. We also consider the imperfect debugging environment where the debugging activities are uncertain. Several software reliability assessment measures are expressed with the distribution of the transition time between arbitrary two states. Furthermore, we propose the approximation method for practical computation of the quantitative measures. Finally, we investigate the validity of our proposed approximation method.

We provide the exact likelihood ratio testing procedure of the scale parameter of the Erlang and gamma distribution when there is a missing time-to-failure information. This is an important result because the asymptotical χ²-test is oversized an thus inappropriate especially for small merged samples. The small merged samples can arise also for a large sample sizes, e.g. when individual times-to-failure are not available. Data sets with missing time-to-failure data can arise from field data collection systems. Real data and simulated examples are provided to illustrate the methods discussed.

Availability is an important measure of performance of repairable system. The steady state system availability has special importance since it demonstrates the performance of a system after it has been operated for long. The statistical inference about the steady state availability are particularly useful for practitioners. Much work has been done in this regard. Most of these researches proposed certain pivotal quantities for constructing confidence intervals of the steady state availability. Assuming both the lifetime and repair time follow gamma distribution with known shape parameters and unknown scale parameters, we propose a pivotal quantity for making inferences, and further derive the likelihood ratio tests. Tables of critical values are given for the convenience of applying the two-sided likelihood ratio test. Confidence intervals are also obtained by converting the acceptance regions.

The three-parameter Burr type XII distribution (3pBXIID) is quite flexible and contains a wide range of distribution shapes for fitting lifetime data. However, it is difficult to obtain reliable estimates of the 3pBXIID quantiles from censored samples for evaluating the reliability of lifetime data. In this work, a Metropolis–Hastings Markov chain Monte Carlo (M-H MCMC) procedure is proposed to obtain reliable maximum likelihood estimates (MLEs) of the 3pBXIID quantiles from a type II censored sample. Moreover, the parametric bootstrap percentile procedure is used to obtain the confidence interval of the quantile of the 3pBXIID. The performance of the proposed M-H MCMC method is evaluated in view of Monte Carlo simulations. Two examples, regarding the survival lifetimes of breast cancer patients and the reliability inference on the lifetimes of oil-well pumps for sucker-rod oil pumping systems, are applied to illustrate the applications of the proposed M-H MCMC method and bootstrap procedure.

The design of a system, for reliability purposes, often requires the exact expression of its reliability function, which, however, is seldom available in closed form. Approximate methods are non-optimal and could even lead to errors. For two-component systems, with independent Gamma-distributed components, it is shown here that we can accurately determine the second component, within various configurations, when the first component is supposed known. Both classical and Bayesian approaches are considered.

When assessing the reliability of a system or the viability of a material, it is necessary to consider the stress conditions of the operating system. Let X be the strength of the system and Y be the stress of the system caused by the environment. In a stress–strength model, the system’s reliability is defined as $P(X>Y)$, and due to ignoring environmental shocks or human failure, it is usually overestimated. To underestimate reliability, we assume that when X goes to the environmental conditions, the system’s strength reduces to $ZX$, where Z is a random variable on $(0,1)$. We propose a modified reliability model with $R^u=P(ZX>Y)$ for the gamma distribution. In this model, stress Y and the scale mixture of the strength of the gamma distribution, $ZX$, are modeled as independent random variables. The proposed model is compared with the traditional model through a simulation study. Additionally, the asymptotic distribution is used to construct an asymptotic confidence interval for $R^u$. Finally, we use two real datasets to demonstrate the practical applicability of our proposed method.

In a recent paper, using a splitting formula for the multi-dimensional Heisenberg group, we derived a formula for the vacuum characteristic function (Fourier transform) of quantum random variables defined as self-adjoint sums of Fock space operators satisfying the multidimensional Heisenberg Lie algebra commutation relations. In this paper we use that formula to compute the characteristic function of quantum random variables defined as suitably truncated sums of the Virasoro algebra generators. By relating the structure of the Virasoro fields to the quadratic quantization program and using techniques developed in that context we prove that the vacuum distributions of the truncated Virasoro fields are products of independent, but not identically distributed, shifted Gamma-random variables.

Assume that $n$ mobile sensors are thrown uniformly and independently at random with the uniform distribution on the unit interval. We study the expected sum over all sensors $i$ from $1$ to $n,$ where the contribution of the $i^{\mathrm{\text{th}}}$ sensor is its displacement from the current location to the anchor equidistant point $t_i=\frac{i}{n}-\frac{1}{2n},$ raised to the $a^{\mathrm{\text{th}}}$ power, when $a$ is an odd natural number.

As a consequence, we derive the following asymptotic identity. Fix $a$ positive integer. Let $X_{i:n}$ denote the $i^{\mathrm{\text{th}}}$ order statistic from a random sample of size $n$ from the Uniform$(0,1)$ population. Then

The gamma distribution, having location (threshold), scale and shape parameters, is used as a model for distributions of life spans, reaction time, and for other types of non-symmetrical data. It has been said that the inference for the three-parameter gamma distribution is difficult because of nonregularity in maximum likelihood estimation although numerous papers have appeared over the years. On the other hand, the methodology for inference for the two-parameter gamma distribution have been established over the years. It is usual to avoid fitting the three-parameter gamma distribution and to fit the two-parameter gamma distribution to data in practice. In this article, we propose a new method of estimation of the shape parameter of the gamma distribution based on the data transformation free from location and scale parameters. The method is easily implemented with the aid of table or graph. A simulation study shows that the proposed estimator performs better than the maximum likelihood estimator of the shape parameter of the two-parameter gamma distribution when the threshold is existent even though that is close to zero.

To predict the the ordering probabilities such as the probability that horse i wins and j finishes second), the Harville (1973) model has been the most popular. The model assumes the running time distribution is independent exponential. However, a recent empirical study shows that the Henery (1981) model has a better fit. In this paper, we consider the Stern (1990) model in addition to the two models above. We fit the Stern model in a Japanese data set and conclude that the Stern model with a particular value of the shape parameter is superior to the others. Under the assumption of the Stern model, we show that the Harville model has a systematic bias in predicting the ordering probabilities.

In this chapter, we introduce a variety of discrete probability distributions and continuous probability distributions that are commonly used in economics and finance. Examples of discrete probability distributions include Bernoulli, Binomial, Negative Binomial, Geometric and Poisson distributions. Examples of continuous probability distributions include Beta, Cauchy, Chi-square, Exponential, Gamma, generalized Gamma, normal, lognormal, Weibull, and uniform distributions. The properties of these distributions as well as their applications in economics and finance are discussed. We also show some important techniques of obtaining moments and MGF's for various probability distributions.

Quantum counterparts of certain classical systems exhibit chaotic spectral statistics of their energy levels; eigenvalues of infinite random matrices model irregular spectra. Eigenvalue spacings for the Gaussian orthogonal ensemble (GOE) of infinite random real symmetric matrices admit a gamma distribution approximation, as do the hermitian unitary (GUE) and quaternionic symplectic (GSE) cases. Then chaotic and non-chaotic cases fit in the information geometric framework of the manifold of gamma distributions, which has been the subject of recent work on neighbourhoods of randomness for general stochastic systems.

Gamma distributions, which contain the exponential as a special case, have a distinguished place in the representation of near-Poisson randomness for statistical processes; typically, they represent distributions of spacings between events or voids among objects. Here we look at the properties of the Shannon entropy function and calculate its corresponding flow curves, relating them to examples of constrained degeneration from ordered processes. We consider univariate and bivariate gamma, and Weibull distributions since these also include exponential distributions.

Analysis of rainfall data strongly depends on its distribution pattern. It has long been a topic of interest in the fields of meteorology in establishing a probability distribution that provides a good fit to rainfall. In this study three major statistical distributions are fitted to the daily precipitation data for nine meteorological subdivisions of NWI regions such as, Punjab, Haryana, Chandigarh &Delhi, Himachal Pradesh, Uttarakhand, East and West Uttar Pradesh, Gujarat, East and West Rajasthan, Jammu and Kashmir for the period 1982 to 2013. These distributions are Log-normal, Gamma and Weibull distributions. To study the goodness of fit, Chi-square test is taken over the DJF data among with individual December, January and February data for 91 days. From the Chi-square test it shows that lognormal distribution is more suitable for Daily Precipitation on NWI region among these three distributions.